An Introduction to .Projective Geometry - Forgotten Books · PDF filePREFACE MY object in writing the following pages has bee n to supply the growing n eed of t mathematical students

AN I NTRODU CTI ON

.PROJE CTI V E G E OM ETRY .

L . N . G . FILON ,M .A D .so

a"FELLOW AND L ECTURER OF UNI VERSITY COLLEGE , LONDONEXAM I NER IN MATHEMATI CS TO THE UNIVER SITY OF L ONDON

LO ND O N

EDWARD ARNOLD

4 1 43 MADDOX STREET,BOND STREET, W.

[All r ights reserved]

PREFACE

MY object i n w r iting the follow ing pages has been to supply the

gr ow ing need oftmathematical students i n th is country for a

compact text- book giv ing the theory of Conic Sect ion s on modernl ines. During recent years i ncreas ing space has been allowed, inUnive rs i ty syl labuse s and courses of i nstruct ion , to the more powerfuland general proj ect ive me thods , as opposed to the more spec ial me thodsof what is st i l l known as G eometrical Conics .The l ine of cleavage be tween the tw o has, however, been sharply

maintained , w i th the resul t that the already much overworked mathematical studen t has to learn h is theory of Con ic Sect ion s th ree t ime sover : ( 1 ) analytically ; (2) accor d i ng to Eucl idean methods ; (3) accord ingto Project ive me thods.The d ifficulty has been to reconc ile the Eucl idean and Project ive

defini tions of the curve ; i n fact to bring i n the focal propert ies i n toProject ive G eome try at a su ffi c iently early stage . Th e pract ice hasusually been

,i n orde r to pass from the project ive to the focal de fin i t ions,

to introduce the theory of i nvol u t ion . But the latte r requ i res for i tsfu llest and clearest treatment th e employmen t of imagi nary e lemen ts.I t seems undes i rable that the more fundamental focal propertie s of theconics

,e .g. the sum or d iffe rence of th e focal d istance s and the ang le s

made by these w i th the tangen t and normal , should appear to dependupon propert ie s of imagina r y poi n ts and l ines, even th o ugh th is migh tintroduce greate r rapid i ty o f treatment. The Un ivers i ty o f London hasrecognized th i s, for , w h ile adm i tt ing Project i ve Geome try into i tssyllabuses for the F i nal Examinat ion for a Pass Degree, i t has excludedi nvolut ion . Many teachers have fe lt that th i s exclus ion amoun ted to arigorous enforcemen t of the l ine of cl eavage men t ioned above . I n the

pre sen t book the ditficulty has been me t, i t is hoped ,successful ly .

Chapters I— V I I cover pract ical ly the w hole field of G eome try o f

“ C ‘ s-h . “

i v PREFACE

Con ics wh ich i s requ ired from the ave rage mathemat ical student w ho isnot read ing for Honours. I n these chapters no use has been made ofi nvolut ion . I n Chapter V I a proof is g iven that any con ic , proj ect ivelydefined , can be cut from a real r igh t c i rcular cone . The foc i ar e thenobtained from the focal spheres and the re st of the focal propertiesfollow .

I n Chapte rs I — V I I the only knowledge pre supposed i s that of

Eucl id , Books I — V I and Book XI ; al so e nough of A nalyt ical G eometryto understand the u se of s igns and coord inates and the meaning of the

equat ions of the straight l ine and c i rcle, so that the data of the draw ingexamples should be inte ll igible to the reade r. Of plane Trigonome trythe mean ing of the s ine , cos ine and tangent and the form u la for thea r ea

'

of a triangle s in 0) a r e all that is requ ired .

I nstead of bas i ng the treatment of the subje ct upon harmonicgroups , I have i ntroduced cross - rat io from the very beg inning.

A lthough ful ly apprec iat ing the supe rior e legance of the formerme thod , i nsomuch as i t enab le s Proj ect ive Geome try to be deve lopedw ithout any appeal to me trical propertie s, I th ink i t is hardly theone best su i ted to beginners. For th is reason I have used me tricalme thods whenever the i r use was obviously ind icated , although I hopei t w i ll be found that the spi r i t of the project ive me thods has beenadhered to.

The second part of the book is i ntended for students read ing forHonou rs or de s irous of mak ing themselves famil iar w i th the moreadvanced parts of the subject. I t has been imposs ible , i n such a shortspace

,to do more than bring the reade r to the threshold of the r ich

treasure house of Modern G eome try and to gi ve him a gl impse of

some of i ts more character ist ic methods. No attempt has been madeto g ive a comple te accoun t of the theorems obtained i n th is domain :those g iven have been chosen to i llustrate the methods. ‘Neverthe less i t i s be l ieved that an Honours student w il l find there most

,i f

n ot all , of th e fundamen tal resul ts w h ich he ough t to know.

The range of know ledge presupposed on the part of the reade r is,

o f course , much w ide r in the later than in the earlie r chapters . Thusthe whole theory of imagi naries and of homography has been allowed tore s t on an analyt ical bas is. Th is should present no d i ffi cul ty , for , bythe t ime the reader reache s these chapters , h e w i l l almost ce rtainlyhave acqu i red suffic ien t know ledge of Analyt ical Plane and SolidG eome try to make h is progre ss easy . On the other hand a purelygeome trical deve lopment of imagi naries w ould have be en too long and

laborious for inclus ion . But i t w i l l be found that those re sults wh ich "

depend on analyt ical cons iderat ion s a r e in every case broad general iza

PR EFACE

t ions , such as those relat ing to the propert ies o f conjugate imag inaries,to the operat ions w h ich lead to homograph ic re lat ions and to the numbe rof poi nts in w h ich curves and surface s of g iven degree and order in te rsect. I have carefully abstained from us ing analysis to prove part iculartheorems .W ith regard to homogr aphy the me thod of one - one corre spondence

has been made fundamental . I t i s true that d iscriminat ion has to beused in apply ing the princ iple , but th is may be said of almost anyprinc iple : and a studen t soon gets to know w hen a one - one correspondence geome trical ly given is re al ly algebraic. The not ion of

homograph ie i nvolut ions , w h ich appears to be a powerful instrument,has been introduced i n Chapter XI .

F inal ly Chapters X I I I and XIV deal w i th geome try of space . Manypropert ies of cones of the second order

,of sphero - con ies, and of quadrics

come most eas i ly from purely geome trical cons iderat ions and i t seemsa pity that the methods of Pure G eometry a r e not more frequentlyemployed at th is stage .

In the preparat ion of the book the class ical treat ise s of C remonaand Reye and a more recen t but very conc ise and instruct ive ex

pos it ion of the subj ect bv M . Ernest Dupor cq have been ch ieflyconsul ted .

I have ven tured to make ce rtain changes in the recognized nomenclatur e . Thus w hat is called by M r Russel l i n h i s Tr ea ti se of P u r e

G eometr y the axis and pole of homography I cal l the cr oss - ax is and

cr oss - cen tr e,as the name seems to bring more vivid ly be fore the mind

the fundamental property of the th ing defined . For a s imi larreason I have used the term i nciden t to denote two forms such thatthe e lements of one lie i n the correspond i ng e lements of the other.The term perspect ive , wh ich is employed by G erman wri ters i n th isconnexion , appears mislead ing, s ince i t would not then apply to whata r e universally known as perspec t ive ranges and penc i ls .The example s a r e taken mos tly from exerc ises set to classes at

Un iversi ty College , London , and from College and Univers i ty Examinat ion papers . Fo r permiss ion to u se these my thanks a r e due to the

Princ ipal of the Un ivers i ty o f London and to the Provost of Univers ityCollege , London . The example s contain also many theorems w h ich i thas not been found poss ible to i nclude in the text. A spec ial featureof those on the fi rst seven chapters is that they ar e d ivided into tw ose ts. Those marked (A) ar e theoret ical those marked (B) ar e draw ingexerc ises. My ow n experience as a teacher leads me to bel ieve thatsuch actual draw ing is of immen se value in ass ist ing begin ners tounderstand the subject , as we l l as i ntrins ically useful in pract ice .

PR EFACE

Cons iderable stre ss has there fore been laid upon draw i ng- board con

str uction s .

I w ish to express my deep sense of obl igat ion to my fr iend and

colleague , M r J . H . D ibb,B .Sc.

,of Un ivers i ty College , London , for the

help he has given me both be fore and during the pas sage of the bookthrough the Pre ss. To M r II . J . Harris

,B.A.

,I also ow e most hearty

thanks fo r some va luable cri t ic isms.

L. N . G . FI LON.

UN IVERS ITY OF L ONDON ,UNI VE RS lTY COLLEG E .

S ep tember , 1 908 .

I I .

I I I .

I V .

V I .

V I I .

V I I I .

IX.

XI .

XI I.

X I I I .

XIV .

CONTENTS

Projection

Cross - Rat io ; Project ive Ranges and Penc i ls

Project ive Propert ies of the Con ic

Pole and Polar

Non - Focal Properties of the Conic

Focal Propert ies of th e Conic

Self- corresponding Elemen ts

Imaginaries and Homography

Transformat ion of Plane F igures

Involut ion

The Homograph ic P lane Forms of the Second Order

Systems of Comics

The Cone and Sphere

Quadr ics

I ndex

PROJ E CTI V E G BOMETRY

CHAPTER I .

PROJECTION.

1 . C or r e sp on d i n g fi gu r e s . Geometri cal pr opert ies maybe obta in ed by corr espondence or transformation . Certa in r e

ations a r e assumed,whi ch intr oduce a corr espondence between

3118 elements of one figure (namely i ts points, l i n es and planes)ind the elements of another figure . The two figures ar e thena id to corr espond or to be tr ansf

’

ormable i nto one another . Tomy pr operty of a set of elements of one figu re corresponds ai r oper ty of the cor r espond ing set of elements of the correspondi ng,)1

‘ transformed , figu r e .

When the corr espondence i s such that to an element of ei therigure corresponds one element and one on ly of the other

,the

or r espondence i s sa id to be ( me- one.

The study of the r elat ions between such figu res,when the

or r espondence i s of a spec i al type to be expla i n ed i n Ar t. 3 ,onstitutes what i s call ed Proj ect i ve Geometry.

2 . Nota t i on . Points wi l l be denoted by Roman capi ta lsI , B ,

Straight l ines w i l l be denoted by smal l Roman letters a ,b,0,

lanes by smal l G r eek letters a , ,8, y ,

Two elemen ts are sa i d to be incident i f on e l i es i n or passesbrough the other .

Thus i f A l i es on a , then a, A are i nc ident .

If a conta i n s a ,then a

,a a r e i nc iden t.

When tw o symbol s are combi ned i n the form of a product,

he joi n t symbol denotes that elemen t whi ch i s i nc iden t wi th ther igi nal tw o.

F .

2 PROJECTIVE G EOMETRY [on

Thus AB denotes the stra ight l i n e pass i ng through thepoi n ts A , B ; Aa denotes the plan e determ i ned by the poi n tA and the l i n e a ; denotes the l i n e of i ntersect ion of thepla

léi es a

, ,8 afly i s the poi n t of i nter sect ion of the thr ee planes

a) 7'

Such a joi n t symbol i s not always i nterpr etable . Thusab has no mean ing i f a

,b a r e l in es in space whi ch do not

in tersect. I t has a mean ing only i f a , b are l i n es i n one

plane.In deal i ng w ith corr espondi ng figures

,cor r espond ing elemen ts

wi l l in var iably be letter ed al ike , the figur e to which they belongbeing i nd icated by suffixes or by accents . Thus A I cor respondsto A 2 , a ,

to a .) and so on . The student should be ver y carefu lto adhere r igidly to th i s practice, as random letter i ng obscu r esthe cor r espondence of elements

,wh ich i s the i r s ign ifican t

property and shou ld be brought i n to prominen ce by ever yposs ible means .The studen t i s supposed famil iar wi th the notion of a

segment on a stra ight l i n e as hav ing sense,as well as magni

tude. I n th i s con n ection i t shou ld be noted that the sense

pf a segment w ill be indicated by the or der of naming theetter s .Thu s AB BA

and, whatever he the order of the poi nts A , B ,C on the l ine

AB + B C =AG

When i t i s des i red to con s ider mer ely the length of a segmentAB , th i s w i l l be wr i tten length AB or mor e shor tly

[AB

When the symbol AB i s u sed i t wi l l i n gener a be ev identfr om the context whether the i nfin ite straight l i ne AB i s meant,or on ly the segmen t AB .

3 . P r oj e cti on . Fi gu r e s i n sp a c e p e r sp e cti v e . A

fundamenta l method of obta i n i ng correspond i ng plan e figures isthe followingLet a l (Fig . 1 ) be any plan e , V any fixed oi n t ou ts ide i t

a 2 any other plan e . Let P I be any point 0 a ] ; joi n VP ,

meeti ng a 2 at P 2 . Then th i s construction establ i shes a correspondence between the poi nts of the two planes, two correspond i ngpoi nts being always i n a l i n e th r ough V. This correspondence Itone- one for , P I bei ng known , P 2 i s u n iquely determin ed , ancconver sely (wi th certa i n cases of appar en t exception to whicl

w e shall r etu rn presently) .

I ] PROJECTION 3

Such a pr ocess of esta bl i sh i ng a correspondence between thepoi n ts of two planes i s termed a proj ect ion . V i s the v er tex ofp r oj ection ,

and we ar e sa i d to proj ect the poi n ts of a , from V upon

or the poi nts of from V upon al accord i ng as the figure i n the

first or the second pla ne , respec t i vely , i s r egarded as gi ven theplane up on wh ich w e projec t i s spoken of as the p la ne of p r oj ection .

The tw o figures thu s conn ected ar e said to be i n sp ace p er sp ective :they would appear coi nc iden t to an eye placed at V.

Let p ,be a stra ight l i n e of a P l a poi nt on P1 . P 2 i ts

projection on e } . VP 1~ lies . in the plane .Vp , , P 2 l i es in Vp l ,

P 2 l i es on _ ( Vp 1 , ag) , which i s a straight l i n e p g . Hence thelocus of poi n ts corre spond i ng to the poi nts ofp l i s p g . A stra ightl i n e therefore corresponds to a straight l in e i n proj ection .

We have immed iately the resu lts :The joi n of tw o poi n ts has for i ts cor respondi ng l i ne the jo in

of the two cor r espond i ng poi nts .The meet or i n tersec t ion of two l i nes has for i ts correspond i ng

poi nt the meet of the two correspond i ng l i nes .We may express thi s by say i n g that projection preserves

unal tered propert ies of i nc idence,that i s , i f two elemen ts are

4 PROJECTIVE GEOMETRY [err

inc i den t in one figur e, thei r corr espondi ng elemen ts a r e inc iden ti n the proj ected figu r e .Let 3 1 be any curve i n the plane a , , P Q, tw o poi n ts on s

l

the correspond i ng poi nts P 2 , Q2 l i e on the curve 32 which i s theproj ection of s. on 0-2 And when Q l approaches P I , VQlapproaches VP , and therefore Q2 approaches P 2 . Therefor ewhen P l Q1 appr oaches the tangent to sl at P 1 , i ts correspondingl in e l ’n app roaches the tangent to 32 at P 2 .

Pr oj ection therefor epreser ves unal tered proper t i es of tangency ,that i s, to a tangent to a cu rve at a gi ven poin t of one figu r ecor r esponds a tangen t to the cor r espondi ng cur ve at the cor r espondi ng poin t of the other figure .

4 . E l em e n ts a t i n fi n i ty . I t has been stated that whenP I i s known , i ts proj ect ion P 2 i s un i quely determ in ed , and

conver sely. But th i s w i l l be the case only i f the l in e VP ,P 2

meets a 1 and a,a t a fin i te d i stan ce. In th i s case both P I and P 2

ar e well determin ed poin ts . I f VP I P Q be paral lel to the plan e(1 2 then i n the language of Eucl i dean Geometry i t does not meetat2 at all and therefore no poi nt P 2 ex i sts .In li ke man ner i f VP 1 P 2 be paral lel to a

, no poi n t P I ex ists .In order to avoi d the compl i cat ion s wh ich wou ld continual ly

r esu lt fr om the n ecess i ty of con s ideri ng such cases of exception ,we intr oduce, by a conven t ion , a set of new i dea l el ements

,

poi nts,l i nes and plane, wh ich ar e cal led the elements a t infin ity .

By mean s of these e lemen ts the cases of exception are r emoved ,and theorems can be stated in a more general man ner.We shal l say that a gi ven d i rect ion i n space dete rmi nes on e

p oint a t i nfin i ty , through wh ich al l straight l i nes para llel to th i sd i rection ar e supposed to pass .Thi s gi ves a construct ion for the l i n e joi n i ng P to a given

poi n t at i nfin i ty, v iz. draw the paral lel thr ough P to the dir ect ion defin i ng that poi n t at i nfin i ty.

The studen t shou ld note careful ly that on any l i n e there i sone poi n t at i nfin i ty only, not tw o. For i f there were two po i ntsat i nfin i ty

,a parallel to the l i n e would pass through both of them ,

and two stra ight l i nes wou ld i n tersect i n more than one poi n t,whi ch would v iolate a fu ndamenta l postulate .

He may conv i nce himself of the iden t i ty of the two oppos iteinfin ities on a l i ne by imagi n i ng a ray through a poi n t 0 outs idethe l i ne and meet i ng the l i ne at P to r otate conti nuously abou t0. P travels conti n uously along the l i ne u nti l the r ota tin r aypasses through the pos i tion of paral lel i sm , when P sud en ly

I] PROJECTION 5

passes fr om one extrem ity of the l i ne to the other , showi ng thatthese oppos i te infin ities are not separate d .

To cal l attention to the fact that a poi n t A l i es at i nfin i ty ,the symbol 00 wil l be u sed as an i ndex, thu s A

“

.

Now let a be parallel to a plane a . Any plan e through a cuts0. i n a l i ne I) paralle l to a . I f A ” be the poin t at i nfin i ty on a ,

A” al so l i es on b and therefore on a . Therefore A “ i s the i ntersection of a and a .

With thi s convention we may say that there i s always onepoi n t correspond i ng to a gi ven po i nt of a l (Fig.

For draw through V a plan e 3, paral l el to a 1 and ,8 2 paral l el

to “2 . Let 2.

r

All poi n ts P I of a l not on 32 proj ect i nto poi n ts P 2 of a2 at a

fin i te d istance.A poi n t I , on i l proj ects i n to a poi n t If of or2 i n the d ir ection

paral l e l to VL .

S im i larly a poi n t J 2 on j 2 proj ects into a point L”

of a 1 i n thedi rection paral le l to VJ Q .

Conversely to find the correspond i ng poin t or cor r espondentof a oi nt J l

°°

of al in a d irect ion paral lel to p l (Fig . we jo in

VJ l i . e . draw through V a paral lel to p 1 to meet a 2 at J g .

VJ 2 be i ng para ll el to p l i s paralle l to al and therefore l i es i n B, .

Hence J ? l ies on a QB“ i . e . on j g .

Hence the poi n ts at i nfi n i ty i n the plane a1 proj ect i nto the

stra i ght l i n e of a 2 and s im i larly the poi nts at i nfin i ty of a? pr o

j cet in to the straight l i n e i 1 of a ] .

S i nce the poi n ts at i nfin i ty i n one plan e cor respond by pr oject ion to a s traight l i n e i n the other plane , i t i s n ecessary to regardthem as lyi ng on a straight l i n e.The locu s of the poi nts at i nfin i ty in any plan e i s cal led the

line a t infin ity in that plane .

Extend i ng the nota t ion already used we shal l say that toi l corresponds i?

"0 and to j ,” corresponds j g .

Two paral lel planes are looked upon as i ntersecti ng i n the i rcommon l i n e at i nfin i ty.

I t fol lows that al l planes paral lel to a gi ven plan e (I passthrough the l i ne at i nfin i ty of a and a r e thu s a parti cu larcase of a set of planes through a l i ne. A l i ne at i nfin i ty therefore corresponds to a defin i te or ienta tionfi as a poi n t a t i nfin i tycorresponds to a defi n i te di r ection .

The or ien tat ion of a plane i s the l ie of the plane relat i ve to fixedd irect ions. All paralle l p lane s have the same orien tat ion .

6 PROJECTIVE G EOMETRY [CH .

Two di sti n ct poi n ts at infin i ty determi ne a l i n e at infin i ty :for l et the poi nts be gi ven by two non - paral lel d irection s a , b all

planes paral lel to both a and b a r e paral lel among themselves , andthe i r common l i ne at i nfin ity i s the join of the two given poi ntsat i nfin i ty . Through any given poi n t C at a fin i te d istance oneplan e can be drawn conta i n i ng su ch a l i ne at i nfin i ty, namely theplane thr ough 0 paral lel to both a and b.

The aggregate of al l poi nts and l i nes at infin i ty i s met by anyother l i n e i n on ly one poi nt and by any plane in on ly one l i ne .

The aggr egate of poi n ts and l i nes at infin i ty therefor e possessesthe fundamental properties of a plan e : and hence w e speak of i tas the p la ne a t infin i ty .

The plane determi ned by three d isti nct non - col l i n ear poi n tsA

”

, B”

,C“m i s the plane at i nfin i ty. For i f i t were a plan e at

a fin i te distance , A“

, B”

, wou ld l i e on the l i ne at i nfin ity ofthi s plan e, which con trad icts the hypothesi s .

5 . V a n i sh i n g p o i n ts a n d l i n e s . The l ine i 1 whichcorr esponds to if i s cal led the van ishing line of a l . The l i ne j gwhich corr esponds to j ,

00 i s ca l led the van i shing l i n e of a 2 .

The poi n t I Jl of p , , whi ch cor responds to the poi n t at i nfin i tyof p 2 i s cal led the va n ishing p oin t of p l ; J 2 of p 2 wh ich corr es

ponds to the poi n t at i nfin i ty of p l i s cal led the van ish i ng poi nt

0 2 .pThe van i shing poi nt of a l in e i s that poin t in wh ich the l in e

meets the van i sh i ng l i ne of i ts ow n figure.S ince VI l (Fig . 1 ) pa sses through 1 2

” i t i s paral l el to p g.

Hence :The join of the vertex of proj ection to the van i sh i ng poi n t of

any l i ne i s parallel to the proj ected l i ne .

Cons ider i ng two pa i rs of correspond i ng l i nes (p l , p g) (ghg

g) , i fK1 be the van i sh i ng poi n t of q, , VK, i s paral lel to 72 an VI 1i s paral le l to p .. Hence the angle between p g , q2 angleL Vlf l , or :

The angl e between tw o l i n es i s equal to the angle subtendedat the ver tex of proj ect ion by the van ish i ng poi nts of the i r corresponding l i n es .

6 . C o l l i n e a t i on . Any tw o correspondi ng l i nes p l , p 2 l i ei n a plane 1 r through V. I f 7; meet the l i ne of i ntersection a: ofor l and a

2 at a poi n t A'

, X l i es on W and a l and therefore on p ,al soX l i es on W and a 2 and therefore on 2 , p l p 2 meet atX on

w . x i s cal led the axis of p r oj ection or tiie axis of collinea tion .

PROJECT ION 7

Thus cor r espondi ng l i nes of figures i n space perspecti ve meetm the ax i s of coll i neat ion .

The poi nts A’ of the ax i s are clearly self- cor r espond i ng, for :onsider X

'

as a poi nt of a , , VA’ meets a? at X,

wh ich thereforeor r esponds to i ts el f. For thi s r eason the poi nts X are not d i s;ingu ished by su ffixes .Conversely i f tw o correspond ing plane figures

,such that the

cins of correspond i ng poi nts are correspond i ng l i nes, l i e inl i ifer en t planes a «1 2 , and possess the coll i nea t ion proper tyInamely that every pa i r of correspond i ng l i nes meet on a fixedi ne a ) , they are i n space per specti ve.To pr ove th i s we first of al l observe that th i s l i n e a; can be no

ather than the intersection of the tw o planes ; for ther e i s no otherlocu s wher e a l in e in a

1 can meet a l ine inLet now P ] , Q be any tw o poi nts of one figu r e, P 2 , Q2 the

or r esponding poi nts of the other figure.S i nce P 1 62l meets P 2Q2 Q1 , P 2 , Q are coplanar . Hen ce

the l i nes P I P Q , Q1 Q2 a r e coplanar and must meet. Thus thejoi ns of correspond i ng po i nts meet i n pairs . This i s on ly poss ibleif they al l pass through one poi n t. For i f a th i rd joi n R IB ? donot lie in the plane of P 1 1

92 , Q1 Q2 , i t can meet both these l i n es

only by pass i ng thr ough thei r i ntersection V. I f i t do l i e in theplane of P 1 1

32 , Q lQJ , take a fou rth jo in 8n wh ich does not l i e

i n th i s plane. S 1 6 1, must pass through V. Hence i f 18l i s tomeet Sn , i t al so must pass through V. Hen ce al l the join sof correspond i ng poi n ts pass through V and the figures are inspace perspec t ive.

7 . R ab a tt i n g . For pr act i cal pu rposes , espec ial ly that ofdraw i ng pr oj ect ion s , i t i s conven i ent to rotate one of the

'

tw o

lanes about a u nt i l i t coi nc ides wi th the other plane . A figur ei n the former plan e rotates w i th i t, but i s fixed in i t . Such aprocess i s termed r aba tting.

8 . Fi gu r e s i n p la n e p e r sp e ct i v e . I f we have afigure 1 i n the plan e or1 which i s in space perspect i ve with afigure 2 i n the plan e a 2 and w e rabat the figure 2 upon theplane a l we obta i n a new figure 3 i n the pl ane a

] . The figure 3i s congr uent w i th or superposable to the figure 2 , but i s ina di li er en t pos i t ion and w i l l be cons idered d i sti nct from i t.We have now two correspond i ng figures

,1 and 3 , i n the same

p la ne. I t i s importa n t here to note carefully that the samepoi n t of the plane wi l l i n general have a qu i te d ifferen t s ign ifi

8 PROJ ECTI VE GEOMETRY [OH .

cause ( and be denoted by a di li er ent l etter) accor d ing as wetreat i t as belongi ng to the figure 1 or to the figu r e 3 .

The two figures 1 and 3 correspond to one another poi n t bypoi n t and l i n e by l i ne. For the poi n ts and l i n es of 1 havea correspondence wi th the poi n ts and l i n es of 2 and these againwith the poi nts and l i n es of 3.

Al so i f p , be corr espond i ng l i n es in the thr ee figures,

p 1 and p g meet at X on an But the poi n t X i s not moved bya rota t ion about .v . Hence p 3 passes through X The two

figures 1 and 3 ar e therefore coplanar cor responding figu r espossess ing the col l i neation proper ty.

Such figures are sa i d to be in p la ne p er sp ectiv e, or in

Izomology , or nomologica l. a i s then var iously cal led the axis ofcollinea tion

,or ofp er sp ective, or of lwmology .

9 . T w o fi gu r e s i n p l a n e p e r sp e ct iv e a r e p r oj e cti on s fr om tw o d ifl

‘

e r e n t v e r t i c e s of a th i r d fi gu r e i n

a n oth e r p la n e .

Let there be tw o figures 1 and 2 in plan e perspect ive ina plan e a

, so that thei r correspond ing l i nes meet on an axi s ofcol l i neat ion x.

Thr ough a: draw any other plan e B. From a vertex V,project the figure 1 upon B . We obta in a figure 3 . Al so i fp , p g , p 3 ar e three correspond ing l i nes of the three figures

, p l , p 2m eet at X on .v and (because 1 , 3 ar e in space perspect ive)p l , p 3 also meet on x , and therefor e m u st meet at X Therefore p 2 , p 3 meet at X on an The figu res 2 , 3 are in d i fferen tplanes and possess the col l i neation proper ty. Therefore byAr t. 6 they are i n space perspecti ve fr om some vertex V2 .

Figures 1 and 2 are ther efore proj ection s of figure 3 from VIand V2 .

I t fol lows fr om the above that al l the poi nts of a' a r e selfcorrespond i ng poi n ts .Figures i n plane perspecti ve , l ike those i n space perspecti ve,

possess va n i sh i ng l i nes , wh ich correspond to the l i ne at i nfin i tyof the i r plane treated as belongi ng to each of the tw o figur esi n tu rn .

S i n ce the poi n t at i nfin i ty on a: i s sel f- correspond i ng i t l ies onboth van i sh i ng l i n es . The lat ter are therefore par al l el to theaxi s of col l i neation .

https://www.forgottenbooks.com/join


i t take any tw o poi nts V V, . Let A A , be a pa ir of corr esponding poi nts . Then A A, meets V, . A A V, , V,are 00planar and A , V, meets A , V, at some poi n t A 3 .

Now let b, , c, be tw o l i nes through A , ; b, , c, the tw o corresponding l ines thr ough A , ; and let a , , a , be any other pa i r ofcor respond ing l i nes . W r i te a , c, = B a , h O, ; a , c,

=Ba , h, = 0, then as before (B , V, , B , B 3B and (0, V , , 0, 03 .

Let the plan e A 3B 303 cu t a i n .v . The triangles A ,B ,

A 3B 303 ar e i n space perspect i ve from V, A ,B , C, and A3B303a r e in s

Zofier specti ve from V , . Therefore the tr iangles

possess t e ineation property ; hence b, b, , c, c, , a ,a , al l l i eon .v . But b, , b, , a, , 0, may be kept fixed, and the tw o poin ts(6, a r e suffic i ent to determ ine .v . i f th en a , , a , be anypa i r of cor r espond i ng l in es whatever, l i es on the fixedaxi s .v .

1 1 . C y l i n d r i c a l p r oj e ct ion . An importan t case of pr ojection ar i ses when the vertex V i s i tsel f at infin i ty. Theproj ection i s then sa id to be c lindr ica l ; the l i n es jo i n i ng cor responding poi n ts are al l par al el and a curve and i ts pr oj ectionare sect ion s of a cyl i nder . The gen eral proj ection , when V 15 ata fin ite d istance, i s sometimes cal led centr a l or con ica l pr oj ect ion ,a cur v e and i ts pr oj ect ion bei ng her e sect ion s of a con e of

vertex VIn cyl i ndr ical pr oj ect ion , i f i ,

” i s the l i n e at i nfin i ty of theplan e a , , i , i s the i ntersect ion of a , by the plan e V

”

But theplan e V °°

i ,“ i s the plan e at i nfin i ty. It wi l l ther efore meet a , i n

the l ine at i nfin i ty of a, . Accordi ngly the van ish i ng l i n es are

the l i n es at i nfin i ty i n each plane, or i n cyl i ndrical proj ection

l i nes at i nfin i ty correspond . From the p1ojectiv e standpoi nt thi si s the fu ndamen ta l property of cyl i ndr ical projection .

If the j oi ns of cor r espond i ng points i n a cyl i ndrical p1 0j ect10na r e perpendi cu lar to the plane of proj ect ion , the pr oject ion i s sa idto be or thogo na l. Thus, i f the plane of proj ection be hor izon ta l ,the poi n ts of th is plane are vertical ly below their correspond ingpoi n ts . Thi s type of pr oj ection i s frequent i n pract ica l applicat ions .

12 . L ocu s of v e r tex of p r oj e c t i on du r i n g r a b a t

m e n t . If be two figu r es i n plan e perspec t i ve i n a

plane a and hav i ng .v for the i r ax1s of col l i neation ,and i f gt, be

rotated abou t .v through any angle 6 i nto a pos i t ion <i>3 , thefigur es (15, and ¢3 , hav i ng thei r cor r espond i ng l i nes meet ing on .v .

PROJECT ION 1 1

i n space perspecti ve fr om some vertex V, and for a l ikeson qt, and gt, are i n space perspec t i ve from some vertex Vthe angle 6 i s al tered

,l'

, and l'

, wil l al ter.I n order to determi ne the vertex of proj ection , i t i s suffi c i en tk now two pai rs of correspond i ng poi nts ; the ver tex i s theni nte r section of the tw o jo i n s of correspond i ng poi nts .Let Fig . 3 represent the section of the plan e a and of the

.ne of ¢3 , which w e may cal l ,8 , by a plan e perpend icu lar to .v

1 pass i ng through the pole of perspecti ve 0 of gt, and ct, Let

s plane meet .v at .Y. Let the poi nt at i nfin i ty O11 OA’ be1oted by J ,

“

or accor di ng as we treat i t as belongi ng toor to qS, .

The poi n ts J , and I , l i e O11 0X,s i nce and each

i ta in 0.

The rotation 9 br i ngs J , , I ,” i nto pos i tions J 3 , I ,

“

on a l i ne'

Ougl1 X mak i ng an angle 0with A’

O and x J A’J , .

V, i s now the inte rsection of and that i s,i t i s

fourth angular poi n t of the par al le logram of whi ch XL , .YJ 3s ides . Hence I , V, XJ 3 X

'

J_

= a con s tant length indepenut of the angle 0.

The locus of V, as 0 varies i s therefo re a c i rcle cen tre I , , theI I , V, turn i ng r ou nd at the same rate as the rotati ng figure .

V, i s the i n tersect ion Of J , J , andbe i ng the joi n of two poi n ts a t i nfin i ty i s the l i ne at

i n i ty in the plane of the paper . Therefore i t mee ts J u l , a ti n ity.

1 2 PROJECTIVE G EOMETRY [1

Thus V, is at infin i ty on J ,J , .

Hence by Ar t. 10, qt, bei ng pr oj ection s of qt, fr omV, respectively , thei r pol e of perspective 0 l i es on V, ti s, on the para l lel through V, to J ,J , .

The tr iangles I , V,O, XJ sJ , hav ing the ir s ides paral lel e .

to each , and I , V, =XJ 3 , are congr uen t. Ther efor e

Hence the locus of V, passes thr ough 0,or V, may be

tained by r otat i ng 0 about the van i sh i ng l i n e of the fixed figlConver sely i f the vertex of pr oj ect ion be k nown and

figur e be r abatted upon the other, the pol e of perspect i ve nbe con structed by rabatti ng the vertex— not abou t the ax iscol l i n eat ion , l i ke the rest of the figur e— bu t about the van ishline of the figur e tha t does not move.

That 0 shou ld be on the locus of V, i s a lmost i ntu i tivev i den t i f we con s ider the l im i t ing pos it ion of V, whenrota t ion 9 i s made to approach zer o. As a proof, however, 1would be u nsati sfactor y, for a proper ty whi ch i s tru e howesmall 9 may be

,i s not n ecessari ly sti l l true when 9 i s actu:

zer o.

1 3 . C on str u cti on of fi gu r e s i n p l a n e p e r sp e cti'

When the pole of per spective 0 and the ax i s of collin eatioare g i ven ; and al so a pai r of cor r espondi ng points A , and(Fig. 4) the poin t P , corr espond i ng to P , may be constructedfollows.Join P ,A , meeti ng .v at X Where XA , meets OP , i s

For P , l ies 011 OP , and A ,P , , A ,P , , bei ng corr espond i ng l i 1meet on .v .

I n th i s w ay the second figure may be constr ucted fromfir st by poi n ts , or , by revers i ng the construction , the first fimay be der i ved from the second .

When 0,x and a pa i r of cor r espondi ng l i nes a , ,

are gi ven,w e constru ct the l i ne p , correspond i ng to p ,

Join p , a , to 0. Where the joi n mee ts a , i s the 0i ng poi n t p , a , . Joi n p , a , . to the poi n t X

'

, where p ,then X i s a poi nt of p , i s p , .

These con struction s are s1mplified i n practi ce i f, i nany tw o cor respond i ng poi n ts or l in es, one of the y an i sh isay i , , i s g i ven .

Take any poin t I , 011 i , . By the proper ty of thel in e and that of the pole of per spective, I , l i es

PROJECT ION 1 3

If P , I , meet .v at X X l i es on or P , l i es on theallel to through A’. Where th i s paral le l meets OP , 13 P .

If any l i ne p , be gi ven , meeti ng i , at I , , .v at X i ts cor

pondent p , i s X'

I , and 18 therefore paral lel to the j oi n of thee of perspec t i ve to the van i shi ng poi nt of the gi ven l i n e. I fg, be two g iven l i nes , the angle be tween p , and g, i s equal to thegle subtended at O by the van i sh i ng poi nts of p , g, (of. Art .

14 . P a r ti cu la r c a s e s of fi gu r e s i n p la n e p e r sp e c

e . If the axi s of col l i n eat ion a: be at i nfin i ty , correspond i ngas are paral le l . The tw o figures In perspect i ve ar e then s im i larl s imi la rly s i tuated and the cen tre of perspect i ve becomes thetr e of s im il i tude .

Symme try about an ax i s i s l ikewi se a pa1 ticular case of plan especti ve. The axi s of symmetry i s here the ax1s of coll i n ea t ion ,symmetri ca l l i nes meet 011 i t. Al so the joi n of two sym

;r ica l poi nts i s perpend icu lar to the ax i s of symmetry . Thee of perspecti ve 13 therefore at i nfin i ty 1 11 the d irect ion perdicula r to the ax i s of symmetry.

A un i form str etch Of a figu1 e m any d i rec t ion 18 a part i cu lar3 of plane per Spectiv e . A stretch 1s defined as fol lows . i f P ,

g. 5) be any poi n t and the l i ne through P , paral le l to a fixedac t ion cal led the direc t ion of s tre tch meet a fi xed l i n e termed

1 4 PROJECTIVE GEOMETRY l

the axi s of stretch at X the correspond ing poi n t P , i s onand i s such that XP , :XP , a constant r at io, which i s cal les tretch - rat io. Cl early oi n ts on the ax is of stretch areLet p , (F ig . 5) meet t i s axi s .v at Z. I f Q , be anyon p , , Q, Y the d is tance of Q, from .v measu red i n thestretch , then YQ, YQ, =XP , XP , or , by a wel l ki n s im i lar tri angles, P , , Q, , Z are col l i nea r, and thei s a stra ight l i n e p , . Thus in a stretch a str a ir esponds to a stra ight l i n e, and s i nce p , , p , meetaxi s of col l i neat ion . The pol e of perspect ive i s the pointi nfin i ty i n the d ir ect ion of str etch .

A translation w ithout r otat ion of a figure i n i ts own plan eplane per spective transformation . For here also the joi n scorr espond i ng poi nts ar e ar a llel to a fixed d irection ,that of the translation , an correspond i ng l in es are para]i s , meet on the l i n e at i nfin i ty . The lat ter i s thereforeof col l i n eation , and the pol e of perspective i s the poin t atin the di rection of the tr anslat ion .

1 5 . D r aw i n g of p r oj e ct i on s . I f i t be r equ i red011 paper the proj ec t ion upon a plane B of any gi venplan e a from a given vertex V

, or , what i s the samesection by ,

8 of a con e whose vertex i s V and basefigure , the method adopted i n prac t i ce i s to rabat thbe drawn upon the plan e a about aB.

1 ] PROJECT ION 1 5

pr oblem aB i s k nown . Al so draw i ng through V a plane parall elto B, th i s plane cuts a i n the van i sh i ng l i ne i , of the given figure.The pol e of perspect i ve i s then obtai ned by r abatti ng V abou t i ,i n the same sense that the proj ection i s rabatted abou t aB. We

have now the pole of perspect i ve, the ax i s of col l i neat ion and onevan i shi ng l i ne . The rabatted proj ection , which i s of cou rse i nplane perspect i ve w i th the or igi nal figure , may now be drawn bythe ru les gi ven i n the latter par t of Ar t. 1 3 . I f at any stage theconstruction becomes awkward , so that l i nes or poin ts employedi n the construction come off the paper, two su i ta ble correspondi ngpoi n ts (or l i nes) may be found and the construct ion s given in theearl ier part of Ar t. 13 can then be u sed .

If the pr oj ection be cyl i ndri ca l,the constr uction by the

~van ish i ng l in e fai l s

,for by Ar t. 1 1 , both van ish i ng l i n es are

then at i nfin ity . Thu s to a poi nt I ,” at i nfin ity corr esponds a

poi nt I ,” al so at i nfin i ty ; and I f

”

, I ,” are in general d i st i nct,

s ince the axi s of col l i n eation i s not here at i nfin i ty . Thei r joi ni s therefore the l i n e at i nfin i ty ; and O, wh ich i s on th i s

l i ne, i s a poin t at i nfin ity. I ts pos i t ion i s then to be found byconstr ucti ng, i n any mann er, some on e pa i r Of correspond i ngpoi n ts A , , A , . O

°° i s then the poi n t at i nfin i ty on A,A , . The

construct ion for correspond i ng poi nts wh ich i s g iven first inAr t. 1 3 may then be used, remember ing that, where a l i n e i ssta ted to be drawn “ through 0 i n that constr uct ion ,

it shouldi n the present case be drawn paral l el to A ,A , .

Notice that such a cyli ndrical proj ect ion , when r abatted i ntothe plan e of the or ig i nal figu r e, i s equ ival en t to a str etch.

1 6 . P r a c t i c a l E xam p l e . A c i r cl e of r ad iu s 4 u n i ts andcentre 0 l ies in a hor i zontal plan e 01. V i s a poi n t 3 un i tsver t ically above a poi n t A , of the c i r cle . B , i s a poi nt of thec i rcle 90° di stant from A , . The c i rcle i s proj ected fr om V on to

a plan e B passi ng through a l i n e .v i n a wh ich bisects OB , atr ight angles . The plan e B i s i ncl i ned at 60

°

to the hor izontalplan e. There ar e two such plan es B . To completely define Bw e suppose tha t i t i s the one whose upper hal f i s fur ther fr om A , .

Cons i der the lan e 7 wh ich passes through V and i s perendicu lar to .r . e shal l n eed , for the pract ical constr uction , tw ogures (Fig. one i n 7 which we shal l cal l the elevation figure,and one i n a which w e shal l cal l the plan figure . I n the el evat ionfigu re the planes a

, B appear as s tra ight l i nes , v i z . the l i nes i nw 1 i ch they cu t y ; these are cal led the traces of the planes on 7 .

Sim i larly i n the plan figure y appea rs as i ts trace 011 a . I t i s

1 6 PROJECTIVE G EOMETRY [CH

conven i ent to place the figu r es on e above the other, the two lineswhich repr esen t (1 7 in the two figures be i ng paral lel

,the poi nts

which repr esent the same poin ts bei ng on the same perpendicu lar sto a y .

Mark in the elevat ion figu r e the poin t A , and the poin t Awhere .v meets a y . Thr ough X draw a l in e mak i ng 60° wi th a y

Thi s i s the tr ace of B. V i s 3 u n i ts above A, in the elevatiorfigure. Through V draw VI , paral lel to the trace of B to meela

'

y at I , . I , i s thu s a poi n t on the van i sh i ng l i n e Of the or ig i nafigure. Rotate V about I cou nterclockw i se i n to a pos i tion 0 O1

a y . 0 i s the '

pole of perspective when the figure i n plane B i


PROJECTIVE GEOMETRY

18 . P a r ti cu l a r c a se s of p r oj e ct i v e fi gu r e s .

A r otat ion of a plane figure about any axi s perpendicular ti ts plan e 18 a pr o] ectiv e tran sformation .

Let OZ be the axi s of rotat ion ,9 the angle of r otat ion .

OX, be any l i n e through 0 ly i ng l n the plane a of the or ifigure , P , a poi nt of that figure. Let B be the planewhich i s perpend icular to a . Choose any vertex V, inand proj ect the figure 1 on to B ,

so that P , comes to P 3 .

r otate the figures 1 and 3, the vertex V, and the plan e B a

a r igi d whol e through an angl e 9 abou t OZ . V, takes uppos i t ion V , , the plane B takes up a pos i t ion y , the figur ebecomes a congr uent figu re 4 i n the plane y and the figu rebecomes the figure 2 . The figure 2 i s therefore the pr 0j ectiofrom V, on to y of the figure 4 . Al so the figures 3 and 4 , beinobta inable on e from the other by rabatmen t abospace perspect i ve from a vertex V3

°°

(Ar t. Hence2 from 1 we project 1 from V, upon B as 3 , then 3 from V,

“

y as 4 , then 4 from V, upon a as 2 . The figures 1 and 2therefore proj ect i ve .

They are not, however , i n space perspecti ve , for correspondl i n es . make a constant angle 9 wi th each other . Thusin tersections of correspondi ng l ines through tw o corrpoi nts P , , P , i s a c ircl e at the c ircumferencesubtend an angle 9 . There 1s accord i ngly no ax i sS i nce any r igi d d isplacement of a figure i n i ts

be broken up i nto a translat ion and a r otatio

tran slation i s a proj ecti veplacement of a figure 1 n

format ion . And s ince the tu r n ing over of a planeequ i valent to constructi ng another figu r e symmetr icalfirst

,the ax i s of r otation bei ng the ax is of symmetry

,

last transformation i s a plane perspecti ve one (Ar t.

that tw o coplanar figures,wh ich can be superposed w i th or

out turn i ng over, are proj ecti ve .

n like man n er two s im ilar figu r es , however pl acedproj ecti ve . For by r otation , or by rotat ion combi ned w i th tuover, they can be s im ilarly placed and they are then m pperspect i ve (Ar t.

1 9 . P r ob l em s i n p r oj e ct i on . I t i s often usefu lbe able to constru ct a pi ojection so that the pr oj ected figshal l sat i sfy certa in cond it ions . We wi l l cons ider threethese.

1] PROJECT ION 1 9

I . To proj ect a figure so that a gi ven l i n e i,i s projected

to i nfin i ty. Thus i , i s to be the van ish i ng l i ne . Hence , thever tex l

' be i ng arbi trar i ly selected , the plan e of proj ec t ion i sany lane parallel to V i , .

ll)

. To proj ec t a figu re so that a gi ven l i n e i , i s proj ectedto i nfin i ty and the angle between tw o gi ven l i nes a , , I) , i s proj ectedinto a gi ven

'

angle a .

First sol ve the r oblem : to construct a plan e perspecti verelation sati sfy i ng the requ i red condi t ion . Let A B

1be the

poi n ts where a , , b, respect i vely meet i , . On A , B , describe asegment of a c i rcl e conta i n i ng an angle a . The pole of per

spectiv e O l i es on th i s segment. Take for 0 any such poi n t andfor .v any l i ne paral lel to i , . Th i s defines a plane perspec t i verela t ion sati sfy i ng the gi ven cond i tions . Now rotate 0 abou tiIthrough any angl e 9 i n to a pos i t ion l ', and at the same t ime

rota te the plane of the or igi nal figure abou t .v through the sameangle 9 i n to a pos i tion B . A pr oj ection fr om V on to B e fTects

wha t i s r equ i r ed .

20 PROJECTIVE G EOMETRY

I I I . To proj ect a figure so that a s imple quadri l(Fig. 7) becomes a square of gi ven s ize . As in

wi ll sol ve the problem firs t for plane perspect i ve .

Let B , , F, be the‘

in ter sections of oppos i te s idesB , O,) respect i vely ; let G , (not marked i n the

be the i ntersection of the di agonal sTake E ,F, as van i sh i ng l i ne i , ; then B , , F, are at infin

and i s a paral lelogram .

If the angle at G , (the angle between the new diagoni s a r ight angle

,the paral lelogr am A ,B , i s a rhombus .

fur ther any one of the angles at A B , , O, , D , i s a r ight anA ,B , i s a square .

Descr ibe on E ,F, a sem ic i rcl e ; i f 0 li e O11 th ithe angles at A B , , B , , wh ich stand on E ,F, ,

r ight angles .S im i lar ly i f A, O, meet E ,F, at H, and B ,D , meet E ,F,

J , O l i es on a sem ic i rcle on J ,H , . I t i s therefore the inter sectiof these two semic ircles .Now the side A , B , must be paral lel to OE , , for E ,

van ish i ng po i n t of A ,B , .

Place be tween OA , , OB , , paral lel to OE , , a leng th A ,B,

the s ide of the given square th i s w i l l be the l i n e corr esto A ,B , . Where i t meets A ,B , i s a poi n t X on thecol l ineat ion . Through X draw a parall el to the v an ishi

E ,F, th i s i s the ax i s a of col l i n eat ion .

To Obta i n the requ i red result by direct pr oj ection ,0 about E ,F, through any angle into a pos i tion V, and pr ojfr om V 011 to a plane through a: parallel to VE ,F, .

EXAMPLES 1 A.

1 . Show that i f tw o figure s ar e i n plane perspect ive the l 1111601111111 .

joi n ing any pai r of correspond i ng points is a se l f- correspond ing l ine .11 111111131‘l’illlllglurjl,2 . Show that w hen tw o figure s ar e in plane perspect ive there

tw o poin ts i n each figure , one O f them be ing the pothat every angle at e i ther poin t corresponds to an equal angle , ana construction for the second point.

3. Show that when tw o figures ar e in space pe rspect ive theretwo poin ts in each figure such that eve ry angle at any one of such poprojects i n to an equal angle .

w w/EXAMPLES 21

4 . Prove that i f tw o figures be in plane or in space perspect ive theages of the axis of col l ineat ion i n the van ish ing l ine s a r e correspond inges such that corre spond ing segments on them ar e e qual .5 . Prove (w i thout us ing the property of the po le of perspect ive )

0 fig ures i n plane perspe ct ive a r e en t i rely g iven by the axis ofation and tw o pairs of correspond ing po ints . Deduce a constr uc

n for the point correspond ing to a given poin t w ith the above data .

6. G iven a pair of correspond ing l ines and the tw o vanish i ng l inestwo figure s i n plane pe rspect ive , cons truct (a ) the pole of per spect ive ,the poin t corre spond ing to any given poin t.7. G iven the pole of perspect ive

,the axi s of col l ineat ion and a pair

correspond ing poi n ts of tw o figure s in plane pe rspect ive , construct6 tw o vanish i ng l ine s.8 . Construct the l ine correspond i ng to a given l ine , g iven the polepe rspect ive , the axi s of coll ineat ion and a pair of correspond i ng

9 . Prove that the d istance of the pole of perspective from e i therl ine is equal to the di stance of the axis of coll ineat ion fromvanish ing l ine .

10. Know ing one van ish i ng li ne , the axis of coll ineat ion and a pairf correspond ing po i nts, construct the pole of perspective .

1 1 . Two figure s, d) , ar e i n plane or in space perspect ive . L i nesq, of a r e paral le l to fixed d i rect ions and a r e such that the angle

etw een them correspond s to a constan t angle in Show that theIte r sec tion ofp 2g, describes a c i rcle .

1 2 . Show that, g iven any tw o triangles i n a plane , a th i rd triangle

h ich is in plane perspect ive w i th each of them may be constructed i nu infin i te numbe r of ways .13. I f a figure is i n plane pe rspect ive w i th and (I) , in plane

er specti ve wi th O, , 0 be ing the poles of pe rspect ive and .v , , .T , the

xes of coll ineat ion in the tw o cases, show that is self- corre)0 11d i 11g in and find a po in t not 011 wh ich i s also sel f) r r espond ing i n d) ,

14. G iven any two triangle s in space , a th i rd triangle can alwayse found wh ich is in space pe rspect ive w i th each of the original tw o.

15 . Prove that tw o non - COpIana r congruent figure s a r e always p r owfi va

I 6 . l’r ove that tw o non - coplanar s imi lar figures a r e always p r o

act ive:


17. I f i n a plane perspect ive relation i t i s given that thperspect ive and the axis of coll ineation a r e at infini ty, showperspect ive relat ion must be equ ivalent to a translation w ithouin the plane .

1 8 . Show how to project a g iven l ine to i nfini ty and any tw oangles i nto angles of given magnitude. I s th is problem capablesolut ion in all cases 2

1 9 . Show how to project a given l ine to infini ty and a given trianin to a tr iangle congruen t w i th a given triangle .

20. A triangle AB C has i ts s ides AB ,AC cut at D and E

parallel to the base . Show how to construct an equ ilate ral trigiven s ide wh ich shall be in plane perspect ive w i th AB C,taken as the vanish ing l ine .

2 1 . I n Problem I“ of Ar t. 1 9 show that there ar e tw o

pos i tions of O and two poss ible pos i t ions of .v and that thesecomb ined in pairs i n four ways

,so that there ar e four pe r

re lat ions g iving a solution of the problem.

22 . Show that i f tw o figure s ar e s imilar (but not ne cessarily slar ly s i tuated ) the van ish ing l ine s ar e at i nfinity.

23. Three coplanar triangle s ar e tw o by tw o in perspect ivehave a common axis of coll ineat ion . Show that the poles of pe r specar e coll inear.

24. Three coplanar triangles ar e tw o by tw o i n perspect ivehave a common pole of pe rspect ive . Show that the axes of coll inear e concurren t.

EXAMPLES I B .

1 . Two figures in plane perspect ive have .v = 0 for axis of coll int ion. O) ; O) ; B, 1 ) ar e pai rscorrespond i ng poi nts. Construct po in ts corre spond ing to P , (2,Q2 = (5 , 1 1

” at i nfini ty on y =0; J ,” at i nfin i ty on x + y =0.

that A , A , , B , B , ,P , P , , Q, I , I , , J , J , all pass through a poi

2 . The pole of perspect ive be ing the origin , the axis of col l inthe l ine .v + 2 =0 and the vanish ing l ine of the figure be ingconstruct the poin ts of correspond ing to t , l

, ( l ,(2, construct also the poi n ts of correspond ing to the same p

3 . G iven the pole of perspe ct ive (3, O) , the axis of coll ineationand the pair of correspond ing l ines a , @ .v ) andby tangents the curve correspond ing to the c ircle .v

" + y?

figure

EXAMPLES 23

4 . Two planes a , , a , cut one another at an angle of On the

sac b isect ing the angle of 1 20° be tween them a vertex V is taken

t 4 inche s from the i r l ine of i ntersect ion .

a figure i n a , i s projected from V O I1 to a , construct the van ish ingrabatted project ion . I f the axi s of

be taken for axis of y and the foo t of the perpend icularV upon i t as origi n an d i f the pos it ive half of the axis of .v be the

one nearer to V,find th e poin ts i n a , correspond ing to (2, O) (5 ,

(3, 4) in a , .

5 . A pyramid 80 fee t h igh stands on a square bas e of s ide 100 fee t,the s ide s of the base runn ing N . and S.

,E . and W . Draw the section of

th is pyramid by a plane at 30° to the horizontal pass ing through a linerunning fromW .N W. to ES .E. th rough the S W . corner of the pyramid ,the plane ris ing as one move s N .

6. A righ t c i rcular cone of semi - vert ical angle 60° is cut by a plane

mak ing an angle of 30° w i th i ts axis and cutt ing that axis at a d i stance

of 3 inches from the ve rtex. Draw the curve of sect ion.7. A hori zontal square ABOD of 2 s ide is projected from a vertex

17” above the corne r A . D raw its project ions upon the tw o planes

through the d iag onal BD i ncl i ned at 45°

to the plane of the square .

8. A convex quadrilateral ABCD is such that AB : 4 AD = 5

CD =2 CB = 3”

,A 0 : F i nd the pole and axis of col l ineat ion wh ich

w i l l transform ABCD i nto a square of s ide 1” and draw th is square .

9. The axis of a: be ing taken as vanish ing l ine , construct an equ ilateral triangle of s ide 2 un i ts wh ich is i n plane perspect ive w i th thetriangle whose vert ices ar e ( l , (223, 2 a ) , (3, l ) ; and construct the poleof perspect ive and axi s of coll ineat ion for th is case.

10. A c i rcular cyl inder of rad ius 2 i s cut by a plane mak ing anangle of 37

° w ith i ts axis. Draw the sect ion .

1 1 . A hori zontal c ircle i s projected on to a vert ical plane throughi ts centre from a point at infini ty 0 11 a ray incl ined at 45 ° to the verticaland such that the vert ical plane through i t is incl ined at 60

°

to the

plane of project ion.12. The entrance of a skew tunne l is in the shape of a c i rcular arch ;

the hori zon ta l proj ect ion of the axis of the tunnel make s an angle of 15°

w ith the normal to the plane of the arch and the axis i tsel f slopesupwards at D raw the sect ion of th is tunne l by a hor i zon tal plane.

CHAPTER I I .

CROSS - RAT IO ; PROJ ECT IVE RANG ES AND PENC ILS.

20. C r os s - r a t i o . Let A , , B O, , B , , etc . (Fig . 8 beset of points on a str a ight l i n e i i , . Let them be proj ecte fr o

any vertex V i n to poi n ts A B , , O, , B , ,upon anothe

stra ight l i n e a , .

We r equ i re to find a relation between the mutual d i stan ceof the poi nts A , , B , , O, , I) wh ich wi ll not be al tered bproj ection .


26 PRO.) ECTI V E GEOME'

I‘

RY 1]

stances the first poi n t A , must n ecessar i ly be i n terchanged wi thsome other, the three cases to be cons idered ar e therefore thosewhere A , i s in terchanged w i th B , , 0, and 0 , r espect ively. Wehave to pr ove that

{8 1 1 4 10 101} AI B I}

or,wr i ti ng out the cross - ratios ,A ,B , . B ,A ,

A 1 0 1 Ol B l B 10} D lA l Cl B l o A ID ] B IA ] . B IC],

equal i t i es whi ch are obv iously tr ue .I t fol lows that d isti nct cross - r atios can be der ived only fr om

those permutations i n w h ich A , s tands first. For,i f we

permutat ion in which A , does not stand firs t, i t may bei nto a permutation in which A , does stand fi rst by permuA , wi th the leadi ng elemen t and i nterchangi ng the remai n i ngelements, and thi s w ithou t alteri ng the cross - r at io.

We have then on ly s ix d i st i n ct cross - r at ios , namely thosewh ich A , stands first, the remai n i ng three B 0 , bei

permu ted i n al l poss ible ways .To find the r elation among these r at ios, proj ect A , to infin i

that i s,cu t the fou r rays through V by a stra ight l i n e a, (Fig.

par all el to VA , . We have by Ar t. 20

{A 1 3 1010 1} {As sél

s }

Af B , A 0 3B3

f i 3s A s

ia

-D 3 3

s i nce the r at io of a fin i te to an infin i te segmen t i s zero.

0 0

CZB:I nter change even letters 0 and B . Then

0303 1

030 3 A

In te rchange m iddle l e tters B and 0,B 30 3 0 3B3 O,B 3

M9 1 3 1“0,B , 0, B ,

1] CROSS - RATIO 27

Inte r change i n (2) the m iddle l etter s ,1 A l

{A IOID IB 1}= l —X_

A,

I nterchange second and fourth letters in1

{A ,D 1 8 101}l — A,

I nterchange second and fourth letter s inA

These gi ve the s ix d isti nct cross - rat ios of four poi n ts .

2 2 . C r os s - r a t i o of fou r r a y s . From Ar t. 20 i t fol lowsthat all transversal s

,that i s, al l straight l i nes wh ich cut a set of

four rays or l i nes thr ough a poi n t,have a constant cross - rat io.

This cross - rat io i s therefore a property of the set of four rays andis cal led the cross - rat io of the fou r r ays .The analytical expression for the cross - r atio of four such rays

i s eas i ly wri tten down . For

A ,B , . A A , VB , A 0, V0 ,

A ,0 , . 0,B , A A , V0 , A O, VB ,

VA , . VB , s i n A , VB , . V0, V0 s i n 0, V0VA , . V0 , s i n A , V0 , . V0, . VB , s i n 0, VBsinA , VB , s i n 0, V0 ,

sinA , V0 , . sin 0, VB ,

’

the s igns of the angles bei ng attended to; A, VB , be i ng measuredby$1

5rotation (pos i t ive cou nterclockw ise) which br i ngs VA ,

to

S i nce fou r concu rr en t rays project i nto fou r concu rr ent raysand transversals i nto transversal s

,i t fol lows fr om the permanence

of cross - rat io of four poi n ts i n proj ection that the cross - r at io of

four rays i s l ikew ise u nal tered by project ion .

The cross - ra t io of four rays abccl wil l be denoted by {abcd}.If the rays be 0A , 0B , 00,

00 ,i t wi l l al so be denoted by

O {AB00 }.

2 3 . R a n g e s a n d p e n c i l s . A r a nge i s a set of poi n ts ona strai ht l i ne. Afl a t p enci l, or shortly a p en ci l, i s a set of raysthr ougha po i n t wh i ch i s the ver tex or cen tr e of the enoi l .Ranges and pen c il s are cal led One - d imens iona elemen tary

geometr i c forms . The str a ight l i n e conta i n i ng the r ange or the

28 PROJECT IV E G EOMETRY

vertex of the penc i l i s spoken of as the base of the formcomponen t poi nts or r ays ar e spoken of as i ts elements.

A form may be denoted by bracketi ng a n umber of

elements, thu s (A ,B ,0, denotes a range of which A , , B , , 0,are poi nts . More frequ ently i t w i l l be denoted by tak i ng aelement and enclos i ng i t i n square brackets . Thus [P ] i sof which the po i n t P ,

which i s then cons idered var iabletypical el emen t [p ] i s a penc i l of which p i s the typi cal0 [P ] i s a penc i l w i th vertex 0, of which OP i s the typiOfte n when no confus ion i s l i kely to r esul t, a form w i l ldenoted s imply by i ts base

,thu s a wi l l denote a r ange on

l i n e u , U a penc i l whose vertex i s U.

24 . P r oj e cti v e r a n ge s a n d p e n c i l s . The elementtwo forms may be made to cor respond , each to each . Whenforms are of the same type

,that i s

,when both are r ang

penc i ls, they ar e sa i d to be pr oject i ve when the correcan be establ i shed by means of a fin i te number of projooperation s . I t w i l l then fol low from Ar ts . 20, 22 that pr ojeranges and penc i ls are also equ i - anharmon ic, that i s , anyelements of one form have the same cross - ratio as thecorrespond ing elements of any other fb rm proj ect i ve w i th theAn importan t parti cu lar case of proj ective ranges and pencr

i s when the tw o r anges are sec t ion s of the same pen c i l by twdiffer ent tr ansversal s

,or when the two penc i l s are obta i ned b

join i ng up the po i n ts of the same r ange to two differen t ver ticesIn the first case the joi n s of cor respond ing poin ts of the twranges pass through a fixed poin t in the second case the meetof corr espond ing r ays of the tw o penc i ls l i e 011 a fixed l i n e . Tw

su ch ranges and penc i ls are sa i d to be p er sp ective : they a i

clearly par t i cu lar cases of figu r es i n plane or in space per spectiwand a r e therefore proj ective .

I f tw o ranges be perspective the poi n t where the ir basei ntersect i s sel f- correspond i ng , and i f two coplanar penc i ls 1perspec t i ve the ray joi n i ng the two vertices i s selfcorrespond i ngS imi la r ranges are correspond i ng ranges in which cor r espondin

segments ar e proportional .E qua l ranges or penc i l s ar e ranges and penc i ls wh ich can t

superposed so tha t correspond i ng elemen ts coi nc ide .

Equal penci l s i n one plane are sa id to be dir ectly , or 0 p ositelgequal accord i ng as they can , or can not, be super pose w ithou

bei ng tu rned over.S i nce i t has been shown (Ar t. 1 8) that congr uen t and simils

figu r es are par ti cu lar cases of proj ective figures,i t follows the

CROSS - RATIO 29

ila r r anges are proj ecti ve and also that eq ual r anges and equa'

oils are proj ective .

I n tw o s im i lar ranges the poi n ts at i nfi n i ty correspond . FOI= a fin i te ratio A,

i f A ,B , i s i nfin i te , so musHence i f A , , A , he po i n ts at a fin i te dista nci nfin i ty

,B , i s at i nfin i ty.

ranges i n w hi ch the poi n ts at infin itjs im i lar . Let be corre

groups of fou r poin ts of tw o such ranges, then , since cross - ratio i s u naltered ,

A ,B , . A ,B , .

0,B , 0,B ,

r emember mg (Ar t. 2 1 ) that I I I—w

A I ,” 1 , w e hav

A 1B 1 : 01B 1= A 2B 2 1 02B 2 ,

the ranges are s im i lar .

Proj ecti ve ranges and penc i l s may be coba sa l, that i s, a pr oe can be establ i shed between points of thr ays pass i ng through the same vertex . I ]

par ti cu lar poin t of the base has a differ ent sign ifirding as we cons i der i t to belong to one range , or ts im i la rly a parti cu lar ray through the vertex ha

t s ign ificance,accord i ng as i t be longs to one or to th

e tw o penc i ls . Such r anges ar e termed collinea r am

ls concentr ic.

s of two project i ve penc i l s by transversalw ar e proj ect ive .

For i n the set of proj ect ions wh ich tr ansform [p ,] tot a l i ne i t, not .belonging to [p ,] transform i nto a

The r ange i t, [p ,] i s proj ecti ve wi th u ,

But v [p ,] i s perspect i ve and pr oject1v e w i th am

i s perspecti ve and proj ect i ve wi thence v [p ,] i s r ojective w i th wS im i larly i f [P , [P ,] be tw o proj ec t ive r anges , O,

S any twvertices

,the penci ls 0 [P ,] S [P ,] are projecti ve .

115whichon“ For 1 11 the set of proj ect ions which transform [P ,] to [P , ] lel‘ a pomt U, n ot belongi ng to [P ,] transform i n to U, .

The penci l U, [P ,] i s proj ec t i ve w i th the penc i l U,

But O [P ,] i s perspective and proj ecti ve w i th U, anS [P ,] i s pers ective and proj ec t i ve wi th U,

Hence O P ,] i s pr oj ect i ve wi th S [P ,]

30 PROJECT IV E GEOMETR Y [c

2 5 . Tw o c ob a s a l p r oj e ct iv e fo rm s a r e i d e n t i c

i f th ey h a v e th r e e e l em e n ts s e l f - c o r r e sp on d i n g .

Cons ider tw o ranges . Let A ,B , 0 be the selfcorrespond i

poi n ts , P , , P , any tw o corr espond i ng poi nts .Then {ABOP ,}

AB . 0P , AB . 0P ,

AP , . 0B AP , . 0B’

OP , OP ,

AP,AP ,

’

0A + A P , 0A + AP ,

AP,

AP ,

0A AP , 0A AP , .

0A i s not zer o, s ince by hypothes i s the points A ,B

,0 a

d i stin ct, AP ,=AP , or P , , P , are coi nc ident. Hence eve

poin t i s self- cor r espondi ng and the ranges a r e i den tical .Con s ider now two concen tr ic penc i ls. They determin e 1

any l i n e tw o col l in ea r proj ect ive r anges . If three r ays of t.pen c i l s ar e self- cor r espond i ng, three poin ts of the r anges ar e secorr espondi ng. Therefore every poin t of the r anges i s 8

correspond i ng and in consequ ence ever y r ay of the penc i lsself- corresponding .

It fol lows that two d istin ct cobasal pr oj ecti ve forms canhave more than two self- cor r espond i ng elements .

C on st r u cti on of p r oj e ctiv e r a n ge s a n d p en c

fr om c or r e sp on d i n g t r i a d s .

R a n ge s . I f the gi ven ranges ar e i n d iffer ent planes ,i f they are i n the same str a ight l i n e, l et them first of al lproj ected i n to two Coplanar non - col l i n ear r anges .therefor e con s ider on ly the p r oblem of establ i sh i ng acorrespondence between two r anges of the latter typeLet a , , a , (Fig. 9 ) be the two r anges . Let A , , B , ,

three giv en'

points of a , , A, , B _0, the thr ee corr espond i ng p

of a , . Join A ,A , and 011 i t take any tw o poi n ts S , , S , .

S ,B , , S ,B , meeti ng at B 3 , and S , S , 0, meet i ng at 03 .

B , 03= i t3 meet A ,A , at A Then A, B , , 0, are perspect iveA , , B 3 , 03 fr om S , and A B 3 , 03 i n tu rn are perspect ive wA B , , 0, from S , .

Take now P , any poi n t of the range u , . Proj ect P , fromas vertex i nto P , on u , and then P , from vertex S , i nto P , on

The r anges [P ,] ar e pr oj ective. The r ange

CROSS - RATIO 3 1

ectiv e with the or ig i nal range it, and l ies i n the same stra ightw i th i t. But these two col l i near r anges have cl early A B

_0,

three sel f- correspond i ng poi nts . Hence the range [P ,] i swi th the or igi nal range i t, and the construc t ion given

onnects cor r esponding points of the tw o r anges a, ,u

I I . P e n c i ls . F irst of al l , i f the tw o penci ls are not alreadyoplana r and non - concentr ic , proj ect them i nto penci l s wh ich are“oplanar and non - concen tr i c . We shal l then cons ider the twopenc i ls to be of th i s type .

Let U, , U, (Fig. 10) be tw o such penc i l s . Let a , , b, , c, be anythr ee gi ven rays of U, cO1TespOI1ding to a , , b, , e, of U Throughx, c, (=A ) draw any two tr ansver sal s s, , 3 , meeti ng 9 1 , at

32 PROJECTIVE G EOMETRY7

B , , 0, and b, , c, at B, , 0, re spectively. Let B ,B , , 0, 0,U3 . Let U3A ,

U, B, , U, 0, be a , , The sets of r aya 3b3c3 are perspecti ve and so are a 3b, c3 , If p , be anyof u, (not shown in the figure) meeti ng 3 , at P , , jo in U3P ,

meeting 3, at P , and join The penc i ls [p , ]proj ective. Hence and the or ig i nal penc i l U, ar e pr ojecand have thr ee self- correspondi ng rays a , , b, , c, . Theyther efore iden ti cal , and the g iven construct ion connects al lsponding r ays of the two or ig i na l penci ls .

Fig. 10.

I t fol lows from the above constr uctions :(a ) That the r e lat ion between two proj ective forms

enti rely determi ned as soon as three corr espondi ng pa i rselements a r e gi ven .

(9) That a proj ective relation between tw o l i ke formsalways be establ i shed in whi ch three arb i trary elements of

shal l corr espond to three arb i trary elements of the other , whiosometimes expressed by say i ng that groups of three elementsalways proj ective .

(c) That a proj ect i ve re l at ion between tw o l i ke formsalways be establ i shed in which any four elements of the


34 PROJECTIVE G EOMETRY [c

lza rmon ic, or to form a ha rmon ic r a nge, and the two wh ichinterchanged ar e sa id to be Iza rmon ica lly conj uga te with r eg:

to the other tw o.

By Ar t. 2 1 , in ter changing both A and 0,B and D

{OBAD} {AB OB}.

Hence {AB OB}I t fol lows from (c) of Ar t. 26 that i f A , B , 0, D can

pr oj ected in to 0,B , A D

,they can be pr oj ected in toA ,

D, 0,

Fig. 1 1 .

So that i f A , 0 ar e conj ugate wi th r egar d to B , D , so arew i th regar d to A , C.

I f we join the poin ts of a harmon i c r ange to a pthe r ange , we obta I n a penc i l of fou r r ays possess i ng theproperty, namely that i t i s proj ect i ve w i th i tsel f, two raysi nterchanged. The i n ter changeable rays are termed conjand the pen c i l i s termed a ha rmon ic p enci l.

2 8 . C r os s - r a ti o of fou r h a rmon i c e l em en ts .

ll ] CROSS - RATIO 35

k be the cross - ratio of fou r harmon i c el ements,say four poi nts

A, B , C, D of a r ange .1

I f {ABCD} A,then by Ar t. 2 1 {ADOB} X {OBAD }.

Hence or A= + I .

Now i f Awere 1 , we should haveAB CD AD 0B

(AB + B C’+ CB ) CB ,

AB (BC + CD ) (BC CB ) CB ,

(AB + B0) (B C + CD ) = 0,A0 . BD 0.

That i s,ei ther 0 and A , or B and D coinc ide. But th i s i s

Iot the case , by hypothes is . Hence the cr oss - rat io of fou rIarmonic elemen ts, i n which conj ugate el ements are not coinci

len t, i s 1 .

The cross - ratio of a harmon i c penci l i s al so 1 , s i nce suchLpenci l stands on a harmon i c r ange .

It follows at once that every transversal cuts a harmon ic penc i lalso that four harmon ic elemen ts a r e

ly proj ecti ve w i th four other harmon ic elements , s i nce thehave the same cross - ratio.

The relation {ABOD } 1 can be pu t in to tw o otherifieI'ent forms , which are of great importance .

We have AB . CD + AD . CB z O,

AB (0A AD ) AD (0A AB ) : 0 ;

AB . AO + AD . AO : 2 . AB . AD ,

(1 div id i ng by AB . AC . AD

1 1 2

AB+AD

_

A C'

To get the other form ,l et 0 be the poi n t m idway between tw o

njugates, say A and 0. Substi tu t i ng i nto the relat ionOB = O,

have(AG 4» 0B ) (00 OD ) (AO 00 ) (00 0B ) 0

,

CO + (0B + 2 .

But AG = OG = — CO,

0B .- AO . CO 01 1

2.

36 PROJECTIVE GEOMETRY [c

When {ABCD } 1 , AB : AD CB : CD , or the poi ntsand 0 div i de BB i nternal ly and exter nal ly in the same ratiHence by Eucl i d v 1 . 3 the tw o bi sectors of the angles formed bypa i r of straight l i n es are harmon i cal ly conj uga te wi th r egardthe tw o gi ven l i n es.Conversely i f, in a harmon i c penc i l , one pa i r of conj uga

l i n es are at r ight angles, they b isect the angles formed by t‘

other pa ir . For l et a ,c be at r ight angles. Then i f b

,d be n

equal ly incl in ed to a , c l et 6, d' be equal ly incl i ned to a , 0 : th.

{a , b, c, d’

} 1 {a , b, c, d}, d = d’

, that i s b, d are equali ncl i n ed to a

,c.

If on e of the poin ts of a harmon i c range be at i nfin ity i ts coj ugate i s m idway between the other two. For

,l et A °° be tli

poi n t,then

A°°

B . CD

A°°

D . 0B

tha t i s BO : CD or 0 bi sects BB .

If B : D,or 0 A

, the defin ition of Ar t. 27 apparently leato an indeterm i nate resu lt. Let u s agree that the equat ion

AB . CD + AD . 0B = O

sha l l hold in al l cases . If we now put D =B,we have

QAB 0B 0.

Hen ce e ither AB 0 or OB 0, that i s , e i ther A or C coinci

wi th B and D . That the same resu lt holds for penc i l s i sseen on cu tt i ng by a tr ansversal .

1 , or

2 9 . H a r m on i c p r op e r t i e s of th e c om p le te qu

r i la te r a l a n d qu a d r a n gl e . A complete quadr i la teral i sfigure formed by four stra ight l i nes a , b, c , d, cal led i ts s ides .has s i x verti ces ab, a c, ad, bc, bd, cd formeds ides in pai rs . The three pai rs of ver ti ces a b,such that the two i n each pa i r do not lie on

termed oppos i te vertices the thr ee l ines joi n i ng them a r e

d iagonal s of the quadr i lateral . The tr iangle formed bthe d iagonal tr iangle of the quadri lateral .A complete quadrangle i s the figure formed by fou r

B , C'

,I ) cal led i ts ver t i ces . I t has s ix s ides AB

,AO,

B0 , OD formed by tak i ng jo i n s of vertices i n pa i rs .pa i rs of s ides AB ,

CD AO, B0 ; A l ) , B C such thaIn each pa i r do not pass through a commonoppos i te s ides . The i r three meets ar e cal led

I I] CROSS - R ATIO 37

of the quadrangle . The tr iangl e formed by them i s the d iagonaltr iangle .

The harmon ic propert ies of the complete quadri la teral andquadrangle ar e a s fol lows :I . The tw o verti ces of a complete quadrila teral on any

diagonal are harmon i cally conjugate w i th regard to the tw o

vert i ces of the diagonal tr i angle on that d iagon al .I I . The tw o s ides of a complete quadrangle through a

diagonal poi nt are harmon ical ly conjugate w i th regard to the tw os ides of the diagonal tr iangle through that d iagonal poi nt.To prove these resul ts, refer to F ig . 1 1 . Here AA '

, A'

O, CC'

,

C'

A are the fou r s ides of a complete quadri lateral , of wh ich A’O

'

,

A0, ST are the three d iagonals . The d iagonal AO i s d i v idedharmon ical ly at B and D (Ar t. But B and D a r e thepoi nts where AO i s met by the other two diagonal s . The resu l tfor the other diagonals fol lows by symmetry.

Agai n A , O,A

’are the four verti ces of a complete quadrangle

, of wh ich S , B , T are the thr ee d iagonal poi nts . The two

s ides through S , SA and S0, ar e harmon ical ly conj ugate w i thr egard to SB and SD (s i nce A , C are harmon ical ly conj ugatew i th r egard to B , D ). But SB , SI ) are the two s ides of thed iag onal tr iangl e through S .

From the above properti es w e obta i n the fol low i ng constr uctions for the e lemen t harmon ical ly conjugate to a gi ven elementwi th regard to tw o gi ven e lements .I . Through the poi n t B

,to w hich a conj ugate i s r equ i red

wi th r egard to A and 0,draw any l i n e and on i t take any tw o

poi n ts A'

, 0'

(Fig. Join AA '

, 00' meet i ng at S

,A C

'

,A

'

C

meeti ng at T. TS meets the or igi nal l i n e i n the poi n t I )requ ired.

I I . On the r ay SB b,to wh ich a conj ugate i s requ i red w i th

r egard to SA a,SO

'

= 0,take any po i n t B , a nd through i t draw

any two l i nes a '

,c'

. Let s : joi n of na’

,cc

’

,t joi n of a c

'

,a’

c.

The joi n of i s T) to the vertex S gi ves the r ay (I requ i red .

In the above cases i t i s often sa id that I) i s a fourthha rmon ic to A , B , C and d a fou rth harmon ic to a

,b, 0, r espec

tively .

30. C r os s - axi s a n d c r oss - c e n t r e o f c op l a n a r p r o

j e cti v e r a n ge s a n d p e n c i l s . I f i n cons truc t ion 1 of Ar t.

26

(Fig. 9 ) S I be taken at A2 and S , a t A 1 we obta i n an i n ter

me i iate range 11, perspect i ve w i th n , from A , and w i th u, from A , .

To cons truct by means of u , the poi n ts wh ich correspond tothe poi nt of i ntersec t ion of a , , i t Let th is poi n t cons idered

38 PROJECTIVE GEOMETRY

as a oint of a , be cal l ed U, , and cons idered as a poin t of u, becal l edV , .

A , U, meets a , at U, ; A ,U, meets a , at U, . But A , U, isi tsel f Therefore U,

= I n l ike manner V, a m,Now the projective relat ion between the r anges bei ng given

,

U, , V, ar e fixed poi n ts and therefor e U, V, i s a fixed l ine.A , B,C

’2 may be any cor respondi ng tr iads whatever of the

given r anges.I t follows that i f A, I 2 , B ,B , be any two pa i rs of cor r espond

ing poi nts of two pr oj ec t i ve r anges the meet of cr oss - joinsA ,B ,) l i es on a fixed straight l i n e . Thi s stra ight l i n e may

be termed the cr oss - axis of the tw o pr oj ective r anges .S im ilarly i f in construct ion I I of Ar t. 26 (Fig . 10) s, be

taken coi nc ident wi th a 2 and s, wi th a , , the ver tex of the i ntermediate penc i l i s a poi nt U4 . I f we now cons ider U, U, and treati t as a r ay n , of the penc i l U, , i t meets a 2 at U, ,

But the penc i ls U4 , U2 bei ng perspect ive, U, U, i s self- corr esponding, hence U4U2

= u, . S im i larly i f U,Hence U4 i s the i n tersect ion of the tw o rays corr espond ing toU, U2 U4 i s therefor e a fixed poi n t. a, b, c2 ar e any cor r esponding tr iads . Hence i f a , a 2 , b,b, be any two pa i rs of correSponding poi n ts of tw o proj ecti v e penc i ls the joi n of cross - meets(a , a ,o,) passes thr ough a fixed poi nt. Th is fixed poin t maybe termed the cr oss- centr e of the tw o proj ect ive penc i ls .

I f the r anges (or penc i ls) i n the above theorems be per spect ive the r eason i ng employed fai ls , for then mu, (Fig. 9 ) andU, U2 (Fig. 10) ar e selfcorrespond i ng. Ther efore U, , V, (Fig. 9)and a , , i ) , (Fig. 10) ar e coi nci dent , and al l we have proved i s thati t, passes through one fixed poi n t, v i z. the intersect ion of ther anges

,and that U, l i es on one fixed l i n e, v iz. the join

verti ces of the penc i ls .I n the case of perspective r anges and penc i ls

,howe

di rect proof of the ex i stence of cross - ax is and cross—centr e i sgiven as fol lows :I . For ranges . Let 0 be the pol e of perspect ive, S the

sect ion of the ranges, A ,A , , B ,B 2 tw o correspondi ng pa i rs .A ,A ,B 2B , are verti ces of a complete qu adrangle of wh ich

A ,B ,) are d iagon al poi n ts . Hence,by the bar

pr o er ty of the complete quadrangle,to A ,B , , A ,B ,) are harmon ical ly co

bases of the two ranges. But S0 and

Hence the l i n e jo i n i ngfore (A ,B A ,B ,) l i es

n ] CROSS - RATIO 39

I I . For penc i ls . Let x be the axi s of coll i neation , s the joinof the vertices, a , o , , b,b, two correspond i ng pa i rs . Thenar e s ides of a complete quadr i late ral of which

i .e . s, a: and (a , b, , ar e the d i agonals .U,U, i s harmon ically d iv i ded by x and But .r meetsU,U, at a fixed poi n t : U, , U, are themsel ves fixed . Hence thefour th harmon ic i s also fixed and a ,o,) passes through afixed poi n t on s. Thi s i s the cross - centre.We wi ll close the presen t chapter w i th the followi ng two

theorems on the tr iangle , which a r e of importan ce .3 1 . C e v a ’s th e or em . I f the stra ight l i n es joi n i ng the

ver t ices A , B , C of a tr i angle to a poin t 0 of i ts plan e meet theoppos i te s ides at P , Q, B , then

BP CQ AR

P 0 QA'

BB

Fig. 1 2.

Through 0 draw a l i n e L il/[ N (Fig. 1 2) par al l el to B0,meet

i ng BC at L”

, CA at A] , AB at N . Then the r anges (BPLm

C) ,(QAMO) are per spect ive from 0. Hence

BP . L°°

O BP AQA AI O—

QC . AIA

S im i larly (OPLQ

E ) , (RANB) are perspect i ve from 0. Hence0P . L

°°

B OP RA N B

GB RB . NA

D iv id i ng ( I ) byB_ _

P QA 0111 RB AN

P 0 CQ'

MA°

A 1 i" NB

40 PROJECTIVE GEOMETRY [0

BP CQ AR CM AN

0" QA'

RB_

MA’

NB

s in ce AIN be i ng par all el to B 0, AN . N B = AM : M0.

Note careful ly that I II the above the segments have totaken wi th proper sign be pos i t i ve sen se on each s ide oftr iangle may be arb i trar i ly selected . I t i s u sual to take i tthat i f we go r ound the tr iangle keepi ng the area on our 1

we ar e mov ing i n the pos i t ive sense throughou t.3 2 . M e n e la u s’ T h e o r em . If any str a ight l i n e

the s ides B C,CA , AB of a tr iangle at P , Q,

R r espectiv

BP CQ AR

P 0 QA RB

F ig.

Thr ough Q (Fig. 1 3) draw a par al lel to B0 meet ing ABand B 0 at S

(BP CS’0°

(BRAT) are perspective from Q.

BP . CS°°

BR . AT BP AR AT AQ—

BT . AR CP'

BR BT OQ’

s ince TQ i s paral l el to B0.

BP CQ ARHence

OP'

AQ'

BR_

or , r evers ing the signs of the three denominators ,BP CQ AR

P U'

QA'

nD

The theorems converse to those.

of Ceva and Menelauseas i ly proved and are left as an exerc i se for the student.


42 PROJECTIVE GEOMETRY u]

1 2 . A, B a r e tw o fixed poi nts : P , , P 2 ar e harmonicallyw i th regard toA , B. Show that the ranges [P 2] ar e project ivefind a geome trical construct ion by projections to pass from one to

other. What a r e the correspondents of th e poi nts A,B ?

13 . Show that i f {A PB Q } {A then {AP BP’

} {AQBDeduce tha t i f A , B ar e se l f- correspond ing elemen ts of tw o col

proj ect ive ranges, any tw o corre spond ing points de termine w i th A ,

a constan t cross - rat io.

1 4. Prove that i f{A, B, C, P ,} {A2 B 2 02 P 2}{A 1 B 1 01 Q 1} {A2 B a02 Q2}{A I B 1 01 R 1l= iA2 132 02 132}{A I B 1 01 S I}2 {A2 B 2 02 S 2}{P l Q 1 R I Q2 R 2 S Z}

1 5 . Prove that i f tw o correspond ing range s be such that anye lemen ts of one have the same cross - rat io as the correspond ingelemen ts of the othe r they a r e project ive .

1 6. I f EFG be the d iagonal triangle of a comple teABCD and the s ides of EFG mee t the s ides of the quI,J , K,

L,M

,N

, show that I , J ,K

,L

, J”, N a r e the

of a comple te quadrilateral having for i ts d ia17. I f efg be the d iagonal triangle of a.

a bcd and the vert ices of efg be joined to the vert ice s of thelateral by l ine s i , k

,l,m

,77,show that i

, j , k, l , m ,n ar e the s

of a complete quadrangle having efg for i ts d iagonal triangle .

1 8 . G iven the cross - axis of tw o project ive ranges and acorrespond ing poi nts , show how to construct the point of 0corre spond ing to a given poi n t of the other. I n part icularth e vanish ing points.

19 . G iven the cross - centre of tw o project ive penc i ls and acorrespond ing rays , find a construction for the ray of one penc i l cospond ing to a given ray of the othe r.

20. A ray through a fixed point 0 cuts a l ine a at P , and the

at i nfin ity at P 2”

. P , , P 2°° then describe projective ranges on u

on the l ine at infin i ty respectively. Show that the cross - ax is oftwo range s is a paralle l to a at a d istance from u equal to the d iof 0 from u .

2 1 . The arms OP , OQ of an angle of fixed magni tude wh ichin one plane about i ts fixed ve rtex 0 i ntersect two g iven str aigh

at I ’and Q respectively. Show that the ranges [P ] , [Q] ar e p r oj

1] EXAMPLES 43

22 . I f in Ex. 2 1 one of the given straight l ines is the l ine at infin i ty,

onstr uet the cross - axi s of the range s [P ] ,23. Through a poi n t 0 a ray OP Q i s drawn mee t ing tw o fixed l inesQ . I f R be harmon ical ly conjugate to 0 w i th regard to P , Qthat the locus of R i s a straigh t l i ne .

24. A,B ar e tw o fixed points

,i t a fixed l ine . I f P be any point of

and p be harmonically conjugate to u w i th regard to P A , P B ,show

at p passes through a fixed point.

25 . I f the vertices of a polygon lie on fixed concurren t lines, wh i les ides but one pas s through fixed poin ts

,the last s ide also

through a fixed poin t.

I f the s ides of a polygon pass th rough fixed coll inear points,

all th e vertices but one move on fixed straigh t l ine s, the locus ofe last remaining vertex is a straigh t l ine .

27. Prove the conve rse of Menelaus’and Ceva’s Theorems.[Exs. 28— 36 follow from the theorems of Menelaus and Ceva ]28 . Prove that the three med ians of a triangle mee t at a point.

29 . Prove that the three perpend iculars from the vert ices of atri angle on the Oppos i te s ide s meet at a point.

30. I f ABC be a triangle , D the mid - poi n t of B C,and if AD

’be a

l ine mak ing w i th AB , AC the same angles that the med ian AD make sw ith AC, AB ,

then AD is called the symmed ian th rough A . Show thatthe three symmedian s mee t at a poi nt.

3 1 . Apply Menelaus’Theorem to prove Desargues’Theorem that i fABC

,A

’B

’C’be tw o coplanar triangles such that AA’

,BB’

,00

’ar e

concurren t,then d a '

, bb’

,cc

’ar e coll inear and conversely.

32. AB C i s a triangle , 0 any point i n i ts plane . I f 0A mee t B Cat P

, 0B mee t CA at Q , 00mee t AB a t R , and i f P’be the harmonic

conjugate of P w i th regard to B C, Q’the harmonic conjugate of Q w i th

regard to CA ,R

’the harmonic conjugate of R wi th regard to AB , show

that P ’

, Q’

, R’ar e coll inear.

33. I n Ex. 32 prove that the middle po i nts of P P '

, QQ’

,R R

’a r e

coll inear. Hence prove that the midd le poin ts of the d iagonals of aquadrilatera l a r e coll inear.

34. The vert ices of a triangle a r e joined to the poi nts of con tact ofthe oppos i te s ide s w i th on e of the e scribed c i rcles . S how that the l ine sthus formed ar e concurren t .

44 PROJECTIVE G EOMETRY [e

35 . Pairs of poi nts P and P’

, Q and Q’, R and R’ar e taken 0

s ides B C, CA ,AB of a triangle

,such that AP and AP

’

, BQ and BCR and CR

’ar e equally i ncl ined to the b isectors of the angle s A ,

B,

Prove that i f A P,B Q , CR ar e concurren t, then so also ar e AP

'

, BCR

’

.

36. Pairs of poin ts P and P’

, Q and Q’, R and R’ar e taken on

s ide s B C , CA ,AB of a triangle and equ id i s tant from the i r mid-

poi

Show that i f AP,B Q , CR ar e concurren t, then so also ar e AP

’

,B

CR'

.

EXAMPLES I I B.

l . A ,B,C , a r e g iven by d istance s from a fixed originequal to 2

,1, 3, 4 ; 1

, 5 , 2 respect ively.Construct geome trically a poi nt D 2 such that

and ver i fy your resul t by calculat ion.

2 . A = (2. C =t 1

D = (O, Construct a ray C’D

’such that 0 {ABCD } 0’

3. Construct the locus of cross - joins of the ranges defined bytriads (O, (O, (0, l ) ; ( l , O), (O, (3, 0) respect i vely, the axe

ord inates be ing i ncl ined at Hence construct any pair of cspend ing po i nts of the ranges and the envelope of the joins ofpoi n ts .

4 . A ,B

,C ar e three poin ts of a straigh t l ine , AB = 2 , B O= 1 . C

struct poi nts P, Q, R wh ich shall be harmon ically conjugate to A w

respect to B C, B w i th re spect to CA , C w ith respect to AB .

5 . Construct a ray 0D harmon ically conjugate to CD w i th r e

to 0A,00 where the angle s AOB ,

BOC ar e 30°

and 15° respect ively.

6. Us ing the ruler only, draw a l i ne through a given poin t P and

the i nacce ss ible mee t Q of tw o straigh t l ines a , b.

CHAPTER I I I .

PROJ ECT I VE PROPERT I ES OF THE CON IC .

3 3 . D efi n i t i on of th e C on i c . A con ic section or con ic

the proj ect ion of a c i rcl e, or the plane section of a con e (r ightobl ique) on a c ircu lar base .

S i nce i n general a stra ight l i n e meets a c i rcl e i n two poi n ts ,same i s t ru e of a con ic, because properti es of i nc idence ar elter ed by proj ect ion : and s i nce from any poi n t tw o tangentsi n general be drawn to a c i rcle , the same holds for the con i c

tangency a r e u Ii alter ed by proj ection .

fol from the defin i t ion that any property of the ciI cleI i ch i s proj ect ive

,i .e. u nal tered by proj ect ion , can be tran s

rr ed at once to the con ic .

34 . T y p e s of C on i c . There are three types of con ic ,cord i ng as in the or ig i nal figure the van i sh i ng l i n e cu ts therole i n tw o real d isti nc t poi n ts, or i n tw o real coi nc iden t poi n tse . touches i t) or does not cu t i t i n real po i n ts .I n the first case there are tw o dist i nct poi nts at i nfin i ty on

.e con ic , name ly the poi nts I f , J ,” correspond i ng to the i nter

ctions I , , J , of the ci rcl e w ith the va n i sh i ng l i ne . Such a con i ccal led a fig/p ei 'bola .

The tangents to the c i rcle at L , J , proj ect i nto the t angen tsI f , J ,

”

to the con ic . These two tangents are cal led the.y/mp totes of the con ic . The curve has tw o bra nches . cor

sponding to the tw o parts i n to which the van i sh i ng l i ne d i v idesIe c i rcle

,cf. Fig . 6

I f the van i sh i ng l i ne touch the c i rcl e L , J , coi nc ide .

The con i c has two coi nc iden t poi n ts at i nfin i ty , i .e . i t hasre l ine at i nfin i ty for one of i ts tangents . Su ch a con ic i srlled a p a r abola . I t cons i sts of one branch ex tend i ng to i nfin i ty .

I f the van ish i ng l i n e do not cu t the c i rcle i n real poi n tsIer e a r e no real poi n ts at i nfin i ty on the con ic . The con ic

46 PROJ ECTIV E GEOMETRY

con s i sts of an oval ly ing ent i rely at a fin i te di stance and i san ellip se.

S i nce i n cyl indr i cal proj ect ion the van i sh i ng l in e i s atthe ell ipse i s the on ly one of the con ics wh ich can be obfrom the c ir cl e by cyl i ndr i cal proj ect ion .

There ar e al so two other types of con ic, v iz. the line-

p a ir

and point-

p a i r . These are to some exten t anomalous , and wi llbe d iscussed In Ar t. 44 .

3 5 . Nota ti on for p r oj e c t iv e r a n ge s a n d p e n c i ls .

C u r v e a s e n v e lop e a n d locu s . To abr idge proofs, the words“ i s proj ective wi th ” wi l l In future (ex_

ce t i n enu nc iation s) bedenoted by t e symbol T Thus [P , AA P[

)

Is to be read : ther ange descr I ed by P , IS pI‘OJectiv e w i t the range descr ibed by P .

The terms locus and envelop e wi l l fr equently occu r i n whatfol lows . A cu rve may be generated I n tw o ways : (a ) by a mov i ngpoi n t P ; we then speak of the cu r ve as the locu s of P and w e

con str uct it gr aph ica l ly from a la rge number of pos i t ions of P ,

form i ng a close ly i nscr ibed polygon (b) by a mov i ng tangent pw e then speak of the cu r ve as the envelop e of p and we con structi t gr aphi ca l ly fr om a large n umber of pos i t ion s of p ,

forminga closely c ircumscr ibed polygon .

3 6 . C h a s l e s’ T h e or em . I f P be a var iable poin t ona c i rcle

, p the tangen t at P , 0 any fixed poi n t on the c i r cle,15 any fixed tangen t to the c i rcle , then the penc i l 0 [P ] I s equ ia n lza r mon ic wi th the range t [p ] that I s, i f P , Q, R , S be anyfour pos i t ion s of P , p , q, 7 , s the cor r espofi ding tangents, then

0 {P QRS }Let C (Fig . 1 4) be the cen tre of the c i rcle

, T the poi n t ofcon tact of t, P

’=p t. Then P '

P , P’

T be i ng tangents to a c i rcle,the angle P ’

CT gP CT angle at the c i rcumference P OT.

Therefore by plac i ng 0 on C and CT on CT the pen ci ls0 [P ] , C [P

'

] are superposable. Hence they are d i rectly equal .Hence by Ar t. 24

0 [P ] T C0 {P QRS } t { pqr s}.

3 7 . P e n c i l s ob ta i n e d b y j o i n i n g a v a r i ab l e p o i n t

to tw o fi xed p o i n ts . I f 0’ be any other fixed poi n t on thec i rcle

,w e have by Chasles’theorem

0'

[P ] A C [P’

] A 0 [P ]Hence the joi n s of a var iable poi nt P on a c ircle to two fixepoints 0,

0’

on the c i rcl e sweep out proj ective pen c i ls.

P80

PROJ ECTIVE PROPE RTIES o r THE come 47

I n the case of the c i rcle these penci ls ar e clearly dir ectly equa l,the angle TOP z angle TO

'

P (Fig . by the wel l - knownroper ty of angl es in the same segmen t .

Fig. 14 .

R a n ge s ob ta i n e d b y i n te r s e c t i on s of a v a r i ab l e

t a n d tw o fi xe d ta n ge n ts .

Let t'

(Fig. 1 4) be any other fixed tangen t. Let p t’= P

Then by Chasles’TheoremC [P

”

] T 0 [P ] T CCutti ng the proj ect ive penci ls C C [P

”

] by t, t'

[P T [P l.or the i nter sect ions of a var iable tangen t p to a c i rcl e and tw ofixed ta ngents t

,t' descr i be tw o proj ective ranges .

3 9 . C o r r e sp on d i n g p r op e r t i e s for th e c on i c . S i ncecross - r atios are not al te red by proj ect ion and proj ect i ve rangesand penc i l s proj ect i nto proj ect i ve ranges and penc i ls respect i vely ,the proper ties stated i n Arts . 36— 38 hold for the con i c , exceptthat now the penc i ls w i l l no longer be equa l, for equal angles ar enot, in general , proj ected i n to equal angles . But the propert ies

0 {P QBS } t {p qi'

s} ( l ) ,0 [P ] T 0

’

[P ]t [P] T t

'

[P]hold equal ly i f for the word “ c i rc l e i n the la s t three ar t i cleswe read “ con i c .From the proper ty (2 ) i t follows that every con i c may be


obta i ned as the locu s of meets of correspond ing rays of

proj ect ive penc i ls . From the proper ty (3) i t fol lows that econ i c can be obta i n ed as the en velope of joi n s of corr espoints of tw o proj ecti ve ranges .If i n Fig. 14 P approaches 0 , OP approache s OO

'

,

pr oaches the tangent at 0. Hence to 00' cons idered asof the penci l 0 corresponds the tangen t at O’

. S im i lar ly to Ocons idered as a r ay of the penci l 0

’ cor r esponds the tangent atThe cross - centre (Ar t. 30) of the tw o pen c i ls through 0,

ther efore the poi nt of i n ter section of the tangents at O, 0 .

Agai n,i fp approaches t, P

' approaches T and P " approachthe i ntersect ion U of the tw o tangents t, t

’

. Hence to tt’ cosider ed as a poi n t of range t’ corresponds the poi n t of contactof t. S im i larly to tt’con s idered as a poi n t of range t cor r esponthe poi n t of contact T’

of t’

. The cross - axi s of the two rangestherefore the chord of con tact TT’

(Ar t.

I n the above rea son i ng i t i s immaterial whether the cu rveFig . 1 4 be a c ir cle or a con i c .

40. P r op e r ty of ta n ge n ts to a p a r a b ola . Th

per ty (3) of the last arti cle takes a partienthe con i c i s a par abola. For then the l i neto the curve by Ar t. 34 . Hence the poin ts at infin i ty of the

r anges t,t’ correspond, and by Ar t. 24 the ranges ar e s im i lar .

But T, P'

,U correspond to U, P

"

, T'

: henceTP

’

P’U z UP P

"

T'

,

or the i nter cepts made by a var iable tangent to a parabola ontwo fixed tangents ar e i n versely proport ional . Thi s furn i shes aneasy gr aphical method of draw i ng a parabola as an envelope, twotangents UT, UT and the i r poi n ts of contact T,

T' be i ng gi ven .

i t bei ng a large i nteger , take leIIgths aUT and UT and laythem off in success ion any n umber of t imes upon UT'

,TU

r espect i vely, starti ng from U along UT’ and from T along TU.

Joi n correspondi ng poi n ts of d iv i s ion each of these i s a tangentto the parabola .

4 1 . T h e p r odu c t o f a n y tw o p r oj e c t i v e p e n c i ls i s a

c on i c . We shal l cal l the locu s of meets of correspond ing r aysof tw o penci l s the p r oduct of the penc i ls and the en ve lope of j oi nsof correspond i ng poi n ts of tw o ranges the p r oduct of the r angesWe have seen that every con i c ca n be ob tai ned as the product

of tw o proj ect i ve penci ls . But these penci l s m ight be r ojectiv epenc i ls of a Sp ecia l type (as in the case of the c i rcle

,wher e they


50 PROJECTIVE GEOMETRY [e rr

the r abatted proj ect ion of th i s c i rcle upon another plane . It i stherefore a con i c by defin i t ion . Note that the con i c and c ircletouch at 0 ; i f they i ntersect aga i n at Y, Z, then Y, Z must beself- correspond i ng poi n ts and the axi s of col l i neat ion .v passesthrough them .

4 2 . T h e p r odu ct of a n y tw o p r oj e c ti v e r a n ge s i s a

c on i c . Let [P ] , [P ,, (Fig . 1 6) be the r anges , t, .v thei r bases,

p P P O . Let T be t e poi nt of r ange .v cor r espond ing to thei ntersect ion U of ta'. Draw any c ircle touch i ng .v at T and from

Fig. 16.

U, P 0

draw tangents t’, p’to th i s c i rcl e. Let Then

Al so U of r ange t’ corresponds to T of

r ange .v and T of range .v corresponds to U of r ange t. Hencethe r anges t

,t’— or [P ] , [P

’

]— have a self- corresponding poin t U.

They are therefore perspec t ive ranges by Ar t. 26. Hence PP ’

passes through a fixed point 0. The l i nes p’

, p are obta i n edfrom one another by the construct ion for figures in plan e perspectiv e, 0 bei ng the pol e, .v the ax i s, t, t

’ a pai r of gi vencorrespond ing l i n es . For p , p

’ meet 011 a ; (p t, p’t’

) passesthrough 0. Hence the enve lope of p i s i n plan e perspectivewith the envelope of p

’ and,as i n the last article, must be a

con i c.

I I I] PROJECTIVE PROPERTIES o r TH E CON IC 51

The con i c and c i rcle both touch a' at T. I f they have otherreal common ta ngents y , these must be selfcorrespond i ng l i nesand so pass through 0.

4 3 . D e du ct i on s fr om th e a b ov e . I n the proofs of

Arts . 4 1 , 42 the c i rcle may clearly be replaced by any con ic.For the on ly proper t i es of the c i rc le made u se of in the proofsare also

,by Ar t. 39 , propert i es of the con i c .

I t fol lows that two comics i n conta ct can be brought i n toplane perspecti ve i n two w ays , v i z . ( 1 ) by tak ing the poi n t ofcon ta ct to be the pol e of perspecti ve the ax i s of col l i nea t ion i sthen a l i n e pass ing through the r ema i n i ng two i n tersect ions ofthe tw o comes ; (2) by tak ing the common tangent to be theax i s of col l i neation : the pol e of perspect i ve i s then a poi n tthr ough wh ich pass the r ema i n i ng common tangen ts of the tw ocomes .These

,however, ar e not the only ways in which such con ics

can be brough t i nto plane perspect ive (of. Exs . I I I A.

Note also that the product of tw o proj ect i ve penc i ls passesthr ough the verti ces of the penc i ls .Thu s the product of two proj ect i ve penc i ls of paral lel r ays i s

a hyperbola whose poi nts at i nfin i ty and therefore the d i rection sof whose asymptotes ar e given by the d i rections of the tw o pen c i ls .S im i larly the product of two proj ecti ve ranges touches the

bases of the ranges . Thus the product of a range at a fi n i tedi stance and a proj ect ive range on the l i n e at i nfin i ty i s apar abola .

I f ther efore thr ough a fixed poi n t 0 a ray OP be dr awn meetina fixed l i ne i t at P , and P Q

°° be drawn through P mak i ng a fixedangle a with OP , P Q

°° touches a fixed parabola . For draw OQparallel to P Q

“

,the penc i ls 0 [P ] , are superposable by

means of a rotat ion a abou t 0. They are therefore equal andproj ecti ve . Hence the ranges [P ] , [Q

”

] are proj ect i ve . Thelatter bei ng on the l i ne at i nfin i ty, the resu l t sta ted fol lows .I t wi l l al so fol low from Ar t. 4 1 , s i nce proj ect i ve penc i l s

proj ect i n to pr oject i ve penc i ls , that the proj ect ion of a con i c i s acome

4 4 . L i n e - p a i r a n d p o i n t - p a i r . I f the two proj ect i vepenc i ls of Ar t. 4 1 are p er sp ective the i r produc t breaks up i ntotwo straight l i nes , name ly the axi s of col l i nea t ion a nd the sel fcorrespondi ng ray , s i nce the latter may be regarded as i n tersec t i ngi tse l f at any one of i ts po i n ts , and therefore m ust figu re i n thelocus of i n te rsec t ion of correspond i ng rays.

52 PROJECTIVE G EOMETR Y [CH .

A line-

p a i r i s therefor e a spec ial case of a coni c locus .I f the tw o proj ecti ve r anges of Ar t. 42 are perspect i ve, thei r

product breaks up i n to tw o sets of l i nes , one passi ng through thevertex of perspecti ve, the other pass i ng through the self—cor r eSponding poi nt , s i nce any r ay through the latter may be lookedupon as the join of the poi nt to i tself. The envelope thenreduces to these two po i nts

,so that :

A point-

pa i r i s a spec i al case of a con ic en velope.The l i ne - pai r and poi n t - pa i r pr esen t certa in anomal i es wh i ch

shou ld be noticed .

Let the compon ents of a l i ne - pa i r be a,b and thei r meet C.

Then a l i n e through a poi nt P of the plane meets the l i ne - pa irin tw o d i sti nct poi n ts , u n less the l i n e passes thr ough C whenthe in tersections coi nc ide . Thus fr om any poi nt one tangen t,and one only, can be dr awn to a l i n e - pa i r. We may

,to keep

the proper ti es of con ics perfectly gener al,look upon these as two

coi nc ident tangents,but i t i s then no longer tru e that a po in t,

the two tangents fr om which to a con i c ar e coi nc iden t, i s i tself onthe con i c .

On the other hand, l et the components of a point pai r beA , B and the i r join 0. From a poi n t P two d ist inct tangentsP A ,

P B can be drawn to the poi n t- pai r, except i f P be on 0

when they coi n cide . The poi nts Of c may therefore be lookedupon as be longi ng to the poi n t - pa i r con ic . Any str a ight l i nethen meets the poi n t - pa i r in one poi n t, and one on ly. I f we lookupon th i s poi n t as two coi nc ident poi nts , to preserve the proper tythat a straight l in e meets a con ic in two poi n ts , i t wi l l appearthat a s tra ight l i n e can meet a con ic i n tw o coi n c iden t po i nts,wi thou t be i ng a tangent to i t, for l ines not through A and B arenot tangen ts to the point pai r .

The tru e s ign ificance of the l i ne - pai r and poi n t - pa i r wi ll bemore apparen t later on when we come to study the centr al andfocal proper t i es of the con i cs .I t i s i n teresti ng to note the manner i n w hich the l in e - pa i r

an d poi nt - pai r appear as projection s of a c i rcl e . To obta in thel i n e - pair

,proj ec t the c i rcl e from a vertex V outs ide i ts plane

and cut the cone so formed by a plan e through V,i .e . take

the vertex i n the plan e of proj ect ion . To Obta i n the point - pai rtake V i n the plan e of the c i rcle and proj ect on to anyplan e .

4 5 . A c on i c i s d e te rm i n e d b y fi v e p o i n ts o r b

fi v e ta n ge n ts . For l et 0, A , B , C be five poi nts oncon i c, P any s ix th poi n t, then the penc i ls

I I I] PROJECTIVE PROPERTIES OF THE com e 53

O (ABCP ) , O’

(AB CP )are proj ect ive . But O, O

’

,A , B , C determi ne complete ly the

corr espond i ng tr iads G (ABO) , 0’

(ABC) , and these i n tu rndeterm i ne completely the rel ation between the penc i l s . HenceOP bei ng given , O

’

P i s known , that i s , P i s determ i ned . Everyoi n t on the con ic i s therefore fixed when five poi n ts a r e fixed .

t fol lows that tw o d isti nct con ics can not have more than fourpoi nts of i nter section .

The poi n ts A , O (and al so the poi nts B , O’

) may coinc idew i thout mak ing the con str uctions i ndete rmi nate

,prov ided w e

i nter pret 0A , O’

B as the tangents at O, Accor di ngly be i nggi ven a poi n t OII the con i c and the tangent at th i s poin t i sequ ivalen t to be i ng given tw o poi nts .S im i lar ly i f t, t

’

, a , b, c be fiv e tangen ts to a con i c, p anys ixth tangent

, p t, p t’ are correspondi ng poi nts of the proj ecti ve

ranges defined by the tr i ads t (abc) , t’

(abc) . Therefore when p ti s known

, p t’ i s k nown , and p i s determ l ned. Thus every tangent

i s determ i nate when five are g i ven . I t fol lows that tw o d i sti nctcon ics cannot have more than fou r common tangents .As before a

,t (and al so I) , t

’

) may coi nc ide withou t mak i ngthe con struction s i ndetermi nate, prov ided we i nterpret a t, bt

’asthe poin ts of contact of t, t

’

. Thus being given a tangent and i tspoi n t of con tact i s equ ivalen t to bei ng gi ven tw o tangents .

4 6 . C on st r u ct ion fo r a c on i c th r ou gh fi v e p o i n ts

Take two of the g i ven poi n ts for vert i ces of tw o proj ect i vepenc i l s and obta i n correspond ing tr iads by joi n i ng to the thr eer emai n i ng poi nts . Construct pai rs of cor r espondi ng rays by themethod of Ar t. 26, or any other. The i n tersect ions of corr esponding r ays ar e po i nts on the con i c .A pr ec i sely s im i lar method can be appl ied to obta in a con ic

as an envelope when five tangen ts are given .

EXAMPLES I I I A.

1 . Show that wi th four poi n ts A ,B

,C ,D on a conic may be assoc iated

a defini te cross - rat io : and also that w i th four tangen ts a , b, c, ( i maybe as soc iated a defini te crossrat io ; and Show that the cross - r at io of

four such tangen ts is the cross - rat io of the i r fou r poin ts of contact.2 . O,

O’

,A ,B

, C, D ar e s ix poi nts on a con ic . I f (0A,

R, (OD , 0 1 1 ) = S , prove that ifP , Q ,

R,S,O,O

54 PROJECTIVE G EOMETRY [on

h e on a conic the rays OB , OD ar e harmonically conjugate w i th regardto OA

,00.

3. A variable tangent LL’mee ts two fixed paralle l tangents to a

con ic whose points of contact ar e A and A’at L and L

’

; prove that therectangle AL .A

’L’ is constan t.

4 . A variable tangen t mee ts the asymptote s of a hyperbola at P ,P

’.

I f C be the i ntersect ion Of the asymptotes , prove that CP . CP’z con st.

5 . On the tangen t at O to a con ic any poi nt P i s taken and P T isdrawn to touch the conic at T. I f S be any other fixed poi nt on thecon ic

,show that the locus of the intersect ion of OT

,SP is another

con ic,wh ich touches the origi nal con ic at O and S .

6 . O,O’a r e tw o fixed poi nts on a conic s l i s a fixed straigh t l ine .

P is any point on s ;OP , O’P ar e jo ined, mee t i ng I at R ,

R’respect ively;

OR’

, O’R mee t at Q . Show that the locus of Q is another conic.

7. The arms OA,OB of an angle of fixed magn i tude a mee t a fixed

straigh t l ine at A and B , and through A and B respect ively l ines AP ,

BP ar e drawn parallel to fixed d i rections. Show that the locus of P isa hyperbola and find i ts asymptotes.

8 . Tw o oppos i tely equal penc i ls have tw o d ifferent vert ices. Showthat the i r product is a rectangular hyperbola (i .e. one whose asymptotesa r e at righ t angle s). [F i nd when corre spond ing rays of the tw o penci lsar e parallel ]

9 . O, S ar e fixed poi n ts, a , b fixed straigh t l ines. A l ine through 0meets a at P ,

b at Q . Through Q i s drawn a parallel to S P . Showthat th is paralle l touches a fixed parabola.

10. OT,0V ar e tangen ts to a parabola whose points of con tact ar e

T and V. Show that the tangent to the parabola parallel to TV b isect sOT, 0V.

1 1 . Tw o circles touch one another. From a po int on the i r commontangen t l ines ar e drawn touch ing the c ircles at P , P

’. Show that P P ’

passes through a fixed poi nt.1 2 . Two c i rcles touch one another. From the i r poi n t of con tact a

ray i s dr awn mee ting the c ircles at P ,P '

. Show that tangen ts at P ,P ’

ar e paralle l .1 3. G iven three poi n ts OII a hyperbola and the d i rect ion s of i ts

asymptotes, show how to construct the penc i ls of para llel rays wh ichgenerate the hype rbola and deduce a construct ion for the asymptotes.

1 4. Show that any tw o conics a, , a, i n a plane can be brough t i n to aplane perspect ive relation by tak ing one of the i r common chords as axis

I I I]"

EXAMPLES 55

of coll ineat ion . [Let XY be the common chord , A ,A , a commontangen t touch ing s, at A , , a, at A , ; Z a point O II XY : let A , Z mee t a,at B , , s, at B Take then the pe r spective relat ion defined by pole 0,

axis XY and pair of correspond ingpoin ts A A transforms a, i nto a conic through the th ree poi ntsX,

Y,

and touch ing 0A . at A r e . into15 . From any poin t O II one of the i r common chords tangents ar e

draw n to tw o con ics , touch ing the conics at P, Q. Show that P Q

passes th rough one of tw o fixed points.1 6. Show that any tw o con ics a, , A2 i n a plane can be brough t i nto

a plane perspect ive relat ion by tak ing the mee t of tw o of the i r commontangen ts as pole of pe rspect ive .

1 7. Through the mee t 0 of the common tangen ts to tw o con ics al ine is drawn mee t ing one coni c at P and the other at Q . Show thatthe tangents at P , Q meet on one of tw o fixed l ine s.

1 8 . A tangen t to a conic at P mee ts a fixed tangen t at Q and

QR i s draw n through Q para lle l to OP where O is a fixed poin t on theconic. Show that QR touche s a parabola.

1 9 . The side s of a polygon pass through fixed points and all thevert ices but one lie on fixed l ines . What i s the locus of the lastremaining vertex ?

20. The vert ices of a polygon lie on fixed l ines and all the s idesbut one pas s through fixed poin ts. What is the envelope of the

last remaining s ide ?

EXAMPLES I I I B .

1 . There a r e tw o project i ve penc ils of rays whose centres ar e at thepoi n ts ( 1 , 2 ) and (4 , The rays

y 2.c —y : 0

,.v — g + 1 =0

Of the fi r s t pencil correspond to the raysx 2.v .v

respect ively of the second penc i l .Construc t the ray of the second penc il correspond ing to the ray

4x of the fi rst penc i l and the con ic w h ich i s the locus of thei n tersections of corre spond ing rays of the penc i ls . Construct the tangents to th is con ic at the points ( 1 , 2 ) and (4 ,

2 . AB ,AC a r e tw o l ines i ncl ined at AB = 2 AC : 4

Construc t by tangen ts the parabola w h ich touches AB ,A C at B and C.

56 PROJECTIVE GEOMETRY [OH . I I I

3. Two project ive penc ils of parallel rays ar e given by the tr iadsx z o

,1 , 3, y =0,

— 2,l, the axes be ing incl ined at D raw the locus

of i nte rsect ions of correspond ing rays of the pencils and con structasymptotes.

4 . D r aw any two c i rcle s i n con tact. V eri fy that i f tangents bedrawn to the circles from any poi nt on the i r common tangent, the joinOf the poi nts of contact passes through a fixed point.



metr y the angles P UO, S UO ar e equal . Hence i f th i s perpendicu lar meet OP Q at R ,

the penci l U (OPR Q) , and ther ange (OPR Q) i s harmon ic . The locu s of R i s therefor e a fixedl i n e, namely the perpend icu lar through U to OU.

Thi s locu s i s cal led the p ola r of O wi th regard to the c i rcle,

conversely O i s cal l ed the p ole of UR with regard to the c ircle.G i ven UR ,

i ts pole w i th regard to the c ircle i s a well determi nedpoi n t , be i ng that poi n t on the d iameter perpend icu lar to UR

which together w ith UR div i des that d iameter harmon i ca l ly.

4 8 . P ola r a s c h or d of c on ta ct . If r eal tangents canbe drawn from O to the c i r cle, thei r poi n ts of contact lie 011 thepolar of O.

For i f the conjugate poi nts P , Q coi n ci de R must coinc idew ith them by Ar t. 28 .

Hence the polar of O i s the chor d of contact of tangen ts fr omO to the c i rcle .

4 9 . R e la ti on b e tw e e n d i sta n c e s of a p o i n t a n d of

i ts p ola r fr om th e c e n t r e of th e c i r c le . Let C (Fig. 1 7 a )be the centre of the c i rcl e.Then by Ar t. 28

,C bei ng halfway between the two poi nts A ,

B , wh ich are harmon i cal ly conj ugate w ith r egar d to O,U

CU . CO CA2 square of radiu s of c i rcle .

5 0. P o la r of a p o i n t on th e c i r c l e . If 0 be at A,

CU . CA CA 2 gives CU= CA, or U is at A . Hence UR i s

perpend icular to the radi us at A ,that i s, i t i s the ta ngen t at A .

Thus the polar of a poi n t on the c ircle i s the ta ngen t at theoi n t.pConver sely the pol e of a tangen t to the c i rcl e i s i ts point

of con tact.5 1 . C onj u ga te p o i n ts a n d l i n e s . If a poin t P l i es

on the polar of Q - wi th regard to a c ircle, then Q l ies OII thepolar of P .

If the l i n e P Q cut the c i r cl e i n real points S , T the proofi s obv iou s . For P i s on the polar of Q P , Q are harmon ical lyconj ugate w ith regard to S , T ; but thi s i s the cond i tion that Q ,

shou ld l i e on the pol ar of P .

But i f P Q do not cu t the c i rcl e i n real po i n ts,let 0 (Fig. 1 8)

be the cen tre of the c i rcl e . Let the polar of Q ,P Q

’meet OQ atQ’

. P Q’ i s perpend icu lar to OQ,

c i rcle OII‘

PQ as d i ameter'

passes through Q’

. Let thi s c ircl e meet OP again at I”. ThenQP

’ i s per pendicu lar to OP and OP . OP’= OQ . OQ

’= square

IV] POLE AND POLAR 59

of r adi us (s i nce Q i s pol e of Hence P ’

Q i s the polar of P ,

or the polar of P passes through QTw o poi nts wh ich have the property that the pola r of ei ther

passes through the other ar e sa id to be conj uga te p oin ts w it/ir ega r d to tire cir cle.

Cal l i ng p the polar of P, g the polar of Q , the theorem

sta ted at the begi n n i ng of th i s art icle may be wr i tten : i f the

Fig. 1 8 .

pol e of p wi th r egard to a c ircl e l i es on g, then the pol e of ql i es on p .

Two su ch l i nes p , q ar e sa i d to be conj uga te lines w ith r ega r d

to tire ci r cle .

5 2 . C i r c l e on l i n e j o i n i n g tw o c onj u ga te p o i n ts a s

d i am e te r . Referr i ng to Fig . 1 8 , s i nce OP . OP’= squ are of

ta ngen t from O to c i rcle O II P Q as d i ameter : square of rad iu s Ofor igi nal c i rc le , the rad iu s of the or igi nal c i rcle i s equal to the

tangen t from O to the c ircle OI I P Q as d iamete r the two c i rclestherefore cu t at righ t angles . Hence the c i rcle on the line jo i n i ngtwo poi n ts conj ugate w i th regard to a gi ven c i r c l e i s or thogonalto the given c i rcl e .

60 PROJECTIVE GEOMETRY

Conver sely i f any c ircle be or thogonal to the given c ir cle,extrem i ties P Q of any d iameter ar e conj ugate poin ts w ith r egto the g i ven c i rcle. For let 0 be the centre of gi ven c ircle joiOQ, OP meeti ng the orthogonal c i rcle aga i n at Q

’

,P

'

.

Then OQ OQ’ square of tangent to orthogonal c i r cl esquare of rad i us of given c ircl e .

P Q’

(bei ng perpend icu lar to OQ) Is the polar of Q. HencP

, Q are conj ugate poin ts .I f P Q meet the given c i rcle at S , T,

then S,T are har

cally conj ugate with regard to P Q.

Hen ce i f two c ircles are orthogonal , every d iameter of eit

wh ich meets the other in r eal points i s harmon i cal ly d iv idedthat other.

5 3 . Tw o c onj u ga te l in e s th r ou gh a p o i n t

h a rm on i c a l ly c onj u ga te w ith r e ga r d to th e

ta n ge n ts fr om th e p o i n t . For i f from Q (Figtangents QU, Q V be drawn to the c i rcle , UV i s the pI f any l in e through Q meet U V at P , , the pol e Pon UV. QP , QP 2 ar e conj ugate l in es through Q. But

being the polar of P wi th r egard to the c ir cle P , V, U areha rmon i c poi n ts . Therefor e Q (P VP 2U) i s a harmon i c pewhich proves what was requ i red .

5 4 . T h e p ola r s of a r a n ge fo r m a p e n c i l e qu i

a n h a rm on i c w i th th e r a n ge a n d c on v e r s e ly . Con s iderpoi n ts P on a l i n e 9 (Fig. Let p be the polar of P w i thr egard to the c i rcle whose centre i s 0, Q the pol e of g wi th r egar dto th i s c i rcle. Then p passes through Q . Hence the l i nes p forma pen c i l . Al so OP i s perpend icula r to p . Hence by a translationof O to Q and rotation through a r ight angle the pen c i l s 0 [P ]and [p ] are super posable .

[P ] T 0 [P ]But [P ] i s a sect ion of 0 [P ] and equ i - anharmon ic w i th

O [P ] , [P ] i s equ i - anharmon i c wi th [p ] .Conversely i t may be shown that when p descr i bes a penc i l

wi th ver tex Q, i ts pol e P descr i bes an equ i - anharmon ic r angeon g.

5 5 . C onj u ga te r a n ge s a n d p e n c i l s a r e p r oj e c t i v e .

To any poi n t P there i s poI nt P , on any l i n e 3 andon e only , namely the p lar p of P cu ts 3 (Fig .

But by the last ar ticl T O [P ] . Hence taki nthe r anges descr ibed

I v ] POLE AND POLAR 61

on two given l i nes by poi nts conj ugate wi th r egard to a gi venci I cle are proj ective . Such ranges may be called conj ugateranges w i th regard to the c i rcle .

In l i ke man ner there exi s ts one l i ne conj ugate to any l i ne pthrough any gi ven poi n t S and on e on ly

,namely the l i ne p , joi n i ng

S to the pole P of p . Let p t u r n abO IO

I t Q , then P moves on q :we have [p ] A

T

O [P ] . But A these penc i ls bei ngperspect i ve. Hence p ] A S [P ] A l1

0

h

Therefore the penc i ls described by conj ugate l i nes throughtwo gi ven poi n ts ar e proj ect i ve . They may be cal led conjugatepenci ls w i th regard to the c i rcle .

The conjugate ranges may be on the same l i n e . Thu s i f P beany poi n t of g and the polar p of P meet g at P , (Fig . thenP

,P , are conj ugate poi n ts on g and [P ] T [P ,]S im i larly conj ugate penc i ls may be concen tri c. Thu s to p

through Q, QP i s conj ugate and Q [P ] T [p ] .Note that s i nce when a poi n t i s on the ciI cle, i t l i es on i ts ow n

polar,the poi n ts where a l i ne cu ts the ci I cle are se l f correspond i ng

poin ts of the conjugate ranges on that l i n e , and I n a prec i selysi In ilaI man ner the tw o ta ngents to the c i rcl e from a poi n t ar eself- corr espond i ng rays of the conj ugate penc i l s through thatpoI nt.

5 6 . C onj u ga te r a n ge s on a ta n ge n t a n d c onj u ga te

p en c i ls th r ou gh a p o i n t on th e c i r c l e . An impor tantexceptional case occu rs when a l i ne i s a tangent to the ci rcle . I n

th i s case the polar of any poi nt on i t other than the poi n t ofcon tac t passes through the po i n t of contact. The poi n t of con ta cti s therefore conj ugate to every poi n t on the tangent and conversely.

I II l i ke man ner i f we have a poi nt 011 the cu rve , the pole of

any ray through the poi n t l ies on the tangent at the poi n t . The

ta ngen t at the po i nt i s therefore conj uga te to every other l i nethrough the poi n t and converse ly.

5 7 . P o la r o f c e n t r e . S i n ce chords through the centreare bi sected at the cen tre , the poi n ts harmon ically conj uga te tothe cen tre on these chords a r e a t i nfin i ty by Ar t. 28 . The polarof the cen tre i s therefore the l i ne at i nfin i ty.

Conversely the polar of a po i n t a t i nfin i ty , say P”

, passesthr ough the centre , i . e . i s a d iame ter. Also this polar i s thechord of con tac t of the two tangents paral le l to the d i rec t ionof P

”. Bu t th i s chord of contact, be i ng a d iame ter , i s pe r

pendicular to the tangents at its extremi t ies . Hence the polarof P

” i s the d iameter perpend icu lar to the d i rec t ion of P ”

. The

62 PROJECTIVE GEOMETRY [GEL

l ine through the centr e conj ugate to th i s d iameter passes thr oughP

”

,i .e . I s perpend icu lar to th i s d i ameter . Ther efore conj ugate

diain eter s of a c irc le ar e at r ight angles .

5 8 . R e la ti on b e tw e e n d i sta n c e s of c onj u ga te

p o i n ts on a l i n e . Cons ider the po in t I ” at i nfin i ty on g

(Fig . I ts polar i s the d iameter OQ perpendicu lar to g.

Therefor e i ts conj ugate poi n t on g i s Q’. Let R , R , be any other

pa i r of conj ugate poin ts on g, and note that i f I”

i s conj ugateto Q

’

, so i s Q’ conj ugate to I ”

. Thus In the conj ugate rangeson g the set of fou r P Q

’ ”

R I corresponds to the set of fou rR , Q

’. Equat i ng the cross—ratios ,

P Q’. R I

”

R , Q’

P I”

. R Q’P , Q

’

. R , I”

P Q’R , Q

’

R Q’P zQ

Q’P . Q

’P , Q

’R . Q

’R , .

Hen ce the product of the d i stances of conj ugate poi n ts fr omthe foot of the per pend icular fr om the centr e on the g iven l in e i scon stan t.To find thi s con stan t, con s ider a poin t U wher e g meets the

c ir cle. U is a se lf- cor r esponding point of the conj ugate r anges ong by Ar t. 55 . Therefor e

Q’P . Q

’P ,

= Q’U . Q

’U = Q

’U f

.

I f however g does not cu t the c i rcl e in real points, Q IS ou ts i dethe c i rcle . With centre Q

’and r adiu s = tangent from Q

’to the

c i r cle descr ibe another c i rcle. Thi s la tter c i rcle wi l l be or thogonalto the g iven c i rcle . Hence by Art . 52 i t meets g at two poi n ts,say T

, T, wh ich ar e conj ugate w ith r egard to the g iven c i rcl e .But we have clearly

Q’T Q

’T,

and Q’P . Q

’P , Q

’T. Q

’T, Q

’T2

square of r ad iu s of orthogonal c i r cle,that i s , P , P , are on oppos i te s ides of Q

’and P Q’. Q

’P , (tangen t

fr om Q’to given

5 9 . S e l f - p ola r t r i a n g le s Let P (Fig. 1 9) be any poi n t,p i ts polar wi th regar d to a c i rcle . On p take two conj ugate poi ntsQ , R . Joi n PR : (1 , P Q = r . Then becau se Q Is on p , the polarof Q passes through P . Al so becau se Q , R are conj ugate, thepolar Of Q passes through R . Therefore the polar of Q Is P R

,

I v ] POLE AND POLAR 63

i .e . q. S im i larly the polar of R i s r . The tr iangle P QR i stherefore su ch that each vertex i s the pole of the oppos i te s ideand i ts vertices (and l i kew i se i ts Sides) are conj uga te i n pa i rs .Such a triangle i s sa id to be sel/l cong

'

uga te or self -

p ola r withregard to the c i r cl e.G i ven a selfpolar tr i angle P QR and a poi nt A on the

con ic , three other poi nts B , C, D can be fou nd . Tor joi nPA meet i ng QR at E . TaTe B harmon ical ly conj ugate to Awi th regard to P , E . Then s i nce P i s pol e of QR , B i s 011 thecu r ve . S im i larly joi n AQ meeti ng RP at F, AR meeti ng .PQat G . Joi n RB meeti ng AQ at C, QB meeti ng AR at DQ (RA CD ) i s harmon ic , QP i s a s ide of the d iagonal tr iangle

of the quadrangle AB CD ,P i s a d iagonal poi nt.

Hence {D GAR }= 1 , {CFAQ} 1 , i .e. D , C ar e on the

Fig. 1 9 .

con ic . And D C passes through P . N0 n ew.

poi n ts can thereforebe obta i n ed by repeat i ng th i s construct ion w 1th B , C or D .

60. T r a n s fe r e n c e of p o l e a n d p o la r p r op e r t i e s to

th e c on i c . S i nce properties of i nc iden ce and tangency andharmon i c propert i es a r e n ot al tered by proj ection

,i t fol low s that

i f w e proj ect the ci rcle i n to any con ic al l the resu l ts which can besta ted i n terms of propert ies that ar e not altered by proj ectionwi ll hold equal ly of the con ic — even though i n the proof we mayhave u sed proper t i es which do not pers i s t i n proj ect ion . I t i sthi s fact w hich gi ves value to the method .

Resu l ts wh ich i nvol ve the l i ne at i nfin i ty , or perpend icu larl i nes , or measurement of l engths (apart from cross - ratio) do not

64 PROJECTIVE GEOMETRY [CH .

gener al ly pers i st in proj ect ion and therefore are not n ecessa r ilytrue of the con ic .

The fol low i ng properties fol low for the con ic .I f a ray through 0 meet the con ic at Q, R the locus of poi nts

harmon ically conj uga te to O wi th regard to Q, R i s a straightl ine, the polar of O wi th regard to the coni c : O i s the pol e of thestra ight l i ne w i th regard to the con i c .If the polar of P pass thr ough Q the polar of Q passes

thr ough P P, Q are conj ugate poi nts wi th regard to the con ic.

If the pol e of p l i e on g the pol e of g l i es on p p , q ar e con

j ugate l i nes wi th regard to the con ic .Tw o conj ugate l i nes through a point are harmon ical ly conju

gate w ith r egard to the two tangen ts to the con i c fr om thatpomt.

The polars of a range wi th regard to a con ic form a penc i lequ i - anharmon i c wi th the r ange and conversely.

I f P , P’be conj uga te poi nts on tw o fixed l i nes (which may be

coi n c ident) the ranges [P ] , [P’

] are proj ect i ve .

Ifp , p’be conj ugate l i n es through two fixed poi n ts (which may

be coi nc ident) the penc i ls [p ] , [p’

] are proj ec t i ve .

The polar of a poi n t on the con i c i s the tangent at that poin tand the poi n t i s conj ugate to every other poi n t on the ta ngen t.Conversely the pol e of a ta ngent i s i ts poi nt of contact and

the tangent i s conj ugate to e very other l i n e thr ough th i s poin t ofcontact.A po i n t and two conj ugate poi nts on i ts polar form a self

polar tri angle wi th regard to the con ic .6 1 . C on str u ct i on s fo r p o l e a n d p ola r w i th r e ga r d

to a. c on i c . Let 0 (Fig . 20a ) be any poi n t : to construct i tspolar wi th regard to a con ic draw any tw o l i nes through 0meeti ng the con i c at P and Q,

R and S . Joi n P R, QS meeti ngat T;P S ,

RQ mee t i ng a t U. Join TU. Then TU i s the polarr equ i red .

For l et TU meet P Q at L ,RS at X. By the harmon i c

property of the quadrangle P QRS ,the penci l T (ORKS ) i s

harmon ic . Hence the ranges (ORKS ) (OPLQ) a r e harmon ic,that i s If , L are poi n ts OII the pol ar of O w i th regard to the con ic .Therefore II’L ,

i .e . TU, i s th i s pol ar .Note that the same cons tru ction , star ting ..fi:om U, gi ves OT

as the polar of U w i th regard to the con ic and OU as the polarof T. TOU i s therefore a se lf- polar tr iangle for the con ic , orThe d iagonal tr i angl e of any quadrangle i nscribed in a con i c

i s sel f- polar for the con i c .


66 PROJECTIVE GEOMETRY [OH .

the polar of Q i s a l in e g thr ough C. Hence the pol e of P Q i s C.

Thu s every l i ne not through C has C for i ts pol e and every poin tnot C has i ts polar pass ing through C.

I f the con i c be a poi n t - pai r whose components ar e A , B , of

wh ich 0 i s the join ,the pol e of a l i n e p i s the fixed poin t thr ough

wh ich pass the harmon i c conj ugates to p wi th r egard to the twotangents from poi n ts of p to the con ic . If Q be any poi n t ofQA , QB are the tw o tangents fr om Q to the poi nt- pa i r. I f t eharmon i c conj ugate to p w i th regard to QA , QB be drawn , i tmeets c at a fixed poi nt P ,

which i s the harmon i c conj ugate ofp c with r egard to A and B . Hence the poles of every l in eother than 0 l i e on 0.

Al so the polar of any poi nt not on c i s cl early c. For AB i sthe chord of con ta ct of ta ngents from ever y such poi n t.

6 3 . R e c ip r oc a l p ola r s . I t appears from the pr ev iou stheory that a con i c s establ i shes a r ec iproca l correspondencebetween the elemen ts of i ts plane

,thu s : to a poi nt P cor r e

sponds i ts polar p , to a l in e 9 corresponds i ts pol e Q. To anyfigu r e i n the plan e

,made up of poi nts and l in es , wi l l cor respond

another figu re , made up of l in es and poi nts, wh ich are the polar sand poles r espect i vely of the poin ts and l i nes of the first figurewi th r egar d to the con i c s, whi ch i s call ed the ba secon ic.

It wi l l al so be seen that properti es of incidence '

ar e pr eser vedin th i s r ec ipr ocal corr espondence , for i f P l i es on g, p passesthr ough Q and to the meet of two l in es pg corr esponds the joi nof thei r pol es P Q. Also from Ar t. 54 i t fol lows that cr oss - r atiopr operties ar e una l ter ed . Accor di ngly two proj ect ive r anges w i l lr eciprocate i n to tw o proj ect ive penc i ls , self- cor respond ing poin tsw i l l rec ipr ocate i n to self—corr espondi ng r ays

,in par ticu lar per

spectiv e r anges wi l l r ec iprocate in to perspective pen c i ls, fourharmon i c poin ts w il l r ec iprocate in to four harmon ic r ays ; andconver sely .

To a curve gi ven as a locu s of poi n ts wi l l cor r espond acurve gi ven as an envelope of ta ngents the degr ee of one cu r ve— that i s

,the n umber of poi nts in wh ich i t i s met by a stra ight

l i n e— becomes the cla ss of the correspond i ng curve, that i s, thenumber of tangents wh ich can be drawn to i t fr om any poin t.The joi n of tw o coi nc iden t poi nts at P on the first cu rve , i .e . thetangent at P ,

r eciprocates i nto the meet of tw o coinc iden ttangents p of the second cu r ve

,that i s

,the poi n t of con tact

of p .

64 . T h e r e c ip r oca l of a c on i c i s a con i c . Let the

Iv ] POLE AND POLAR 67

'

ven con ic be obta ined as the product of two proj ecti ve penc i l s.The two proj ect i ve penc i l s r ec iprocate into tw o proj ec t i ve rangesand i ntersec t ions of correspond i ng rays i nto joi n s of correspondi ng poi nts . The product of tw o project ive penc i ls thereforer ec iprocates i n to the product of two project ive ranges, that i s ,i nto a con i c .

6 5 . P r i n c ip le of D u a l i ty . I t fol lows fr om the tran sformat ion by reciprocal polars that to every theorem concern i nga figure made up of poi nts and l i nes there corresponds anothertheorem con cern i ng a correspondi ng figure made up of l i nes andoi n ts respect i vely, so that geometr ical theorems appear i n pa i rs .Severa l i n stances of th i s pr inc iple of dual i ty have al ready beenmet with and i t wi l l be an i n s truct i ve exerc i se for the studen tto trace such dual theorems as have already been gi ven . As

examples of such theorems we may quote the fol lowi ng,corre

sponding theorems appeari ng s i de by s ide :

I f in tw o correspond ing figure smee ts Of correspond ing l ine s lie ona fixed l ine , joins of correspond ingpo in ts pass through a fixed poin t.

The harmon ic property of the

comple te quadrangle .

The mee ts of cross -joins of correspond ing points of two project iveranges lie on a fixed l ine .

The harmon ic conjugates of afixed poi n t w i th regard to the tw opo i n ts at wh ich any l ine throughi t mee ts a fixed conic lie on afixed h ue .

I f in tw o correspond ing figuresjoi n s of correspond ing poi nts pas sthrough a fixed poin t

,mee ts of

pom-

espond ing l ines lie on a fixedme .

The harmonic property of thecomple te quadrilate ralThe joins of cross - mee ts of cor

re spond ing rays of two project ivepenc i l s pas s through a fixed point.The harmonic conjugate s of a

fixed l ine w i th regard to the tw o

l ines wh ich can be drawn from anypoi nt on i t to touch a fixed con icpass through a fixed poi nt.

Rec iprocal theorems are obta i ned at once one from the otherby s imply translati ng the language

,the fol low i ng bei ng the

terms i n terchangedstraigh t l inejoi n

tangen t to a curvepo in t of con tac t of a tangen t

lie on

range/

coll ineardegreelocus

I t shou ld be noticed,however

,

po intnl ee t M W 1

poin t‘ on a curvetangen t at a poih t on the curve

pas s through’penc i lconcurren tclas s

enve lopethat theor ems tru e of sp ecia l

5— 2

PROJECTIVE G EOMETRY [CH .

cur ves r ec iprocate i n to theorems tru e only of the curves whichar e the r eciprocal s of these Spec ial cur ves . Al so that pr operti esof length and angular magn i tude (whi ch are termed metr ica lproperti es) do not general ly rec iprocate i n to l ike properties.I t w i l l be fou nd that the properti es to which the pr i nc ipl eof dual i ty can be appl ied successful ly are the p r oj ective pr opor tIes.

6 6 . C e n tr e a n d d i am e te r s of a con i c . The pol e ofthe l in e at i nfin i ty w ith regard to a con ic i s cal led the centr e of

the con ic . The centre of the con i c cor responds , in the plane ofthe origi nal c i rcle , not to the centre of the c i rcle, bu t to the pol eof the van i sh i ng l i n e .

L in es through the centre of a con ic ar e cal led i ts diameter s.

S in ce‘ the centre and the poi nt at i nfin i ty d i v i de a diameterharmon i cal ly, a d iameter i s b i sected at the centre .

Conj ugate d iameters are conjugate l i nes thr ough the centr e.Hence the pol e of ei ther i s the poi n t at i nfi n i ty on the other.Ther efore the tangents at the extr em i ti es of a d iameter areparal lel to i ts conj ugate .

By the pol e and polar proper ty chords para l lel to a d iameterar e d iv i ded harmon ical ly by the poi n t at i nfin i ty on the chordsand by the conj uga te d iameter, that i s, they ar e b i sected by theconjugate d iameter .I f C be the centre of a con ic

,P any poi n t

,the polar p of P

wi th r egard to the con i c i s conj uga te to the d iameter CP and

ther efor e b isected by i t. For i f c be the l i n e at infin i ty, clearlyp c, tha t i s , the po in t at i nfin i ty on p ,

i s the pol e of P C.

All d iameters of an el l i pse meet the cu rve in rea l poi n ts .For the van i sh i ng l i ne being out-s ide the c ircl e, i ts pol e i s i ns ide.Every l in e through thi s pol e therefore cuts the c i rcl e in realpoi nts , and the same holds good after proj ection .

On the other hand , of tw o conjuga te d iameters of a hyperbola,One and one only meets the curve in r eal poin ts . For con s iderthe or igi nal c i rc le . The van i sh i ng l i n e cu ts the c ircle i n r ea lpoi n ts I , J (Fig. 2 1 a ) . The tangents at I , J meet at C, whichi s the ole of the van i sh i ng l i n e and i s ou ts ide the c ircl e . Of ther ays t rough C, those wh ich l i e i n s ide the angle I OJ meet thec i rcle, the others do not. Now by Ar t. 5 3 any two conjugatel i nes thr ough C are harmon ically conj ugate wi th r e ard to CI , CJ .

I f they mee t I J at P , P’

, (P I P’

J ) i s harmon ic . ence i f P ' beins i de I J ,

P i s ou ts ide and conversely,s i nce P , P

’d i v ide I Jin ternal ly and external ly i n the same r atio (Ar t. i fCP

’ cuts the c ircl e in r eal points,CP does not, and conver sely.

Iv ]‘

POLE AND POLAR 69

Proj ecti ng I J to i nfin i ty the proper ty sta ted fol low s for diameter sof a hyperbola .

Al so note that i n the hyperbol a the proper ty that CP , CP’

ar e harmon ica l ly conjugate w i th regard to the tangen ts from Cbecomes :Two conjugate d i ameters of a hyperbol a are harmon ical ly

conj ugate w i th regard to the asymptotes .I II the case of the parabola the van ish i ng l i n e 0 touches the

origi nal c i rc le at C (Fig . 2 1 b) , and every l i n e CP i s conj ugate to c.

The centre of the parabol a i s therefore at i nfin i ty, and i ts d i rcet ion gi ves the poin t of con tact of the l i n e a t i nfin i ty w ith thecurve . All d iameters of a parabola are paral lel to th i s fixed

Fig.

direction,and are to be looked upon as conj ugate to the l i n e at

i nfin i ty . The l i n e at i nfin i ty has no defin i te d i rect ion , but i tmay be shown that to each diameter there i s a defin i te conjugatedi r ection . For l et L (Fig. 2 1 b) be the pol e of CP ,

chordsthr ough L are conj ugate to CP . Proj ec t c to i nfi n i ty the c i rclebecomes a parabola , CP a d iameter

,the chords through L a

sys tem of paralle l chords b i sec ted by that d iameter , P L theta ngent at i ts extremi ty

,which tangen t i s paral le l to the chords .

Hence a d iame ter of a parabola bi sects chords paral lel to thetangent at its extremity .

Becau se the el l i pse and hyperbola have the i r cen tr e at afin i te d ista nce, they are termed cen tr a l con ics .

70 PROJECTIVE G EOMETRY [on ”I

6 7 . S u pp lem e n ta l ch o r d s of a. c e n tr a l c on i c a r e

p a r a l le l to c onj u ga te d i am e te r s . Let AOB (Fig. 22) bea diameter of a con ic . I f P be any poi n t on the con i c , P A , P B

ar e know n as supplemental chords . I f D be the m iddle poi n t ofP B

,then , C be i ng the middle poi n t of AB , CD i s parallel to

PA . Al so i t i s the d i ameter conj ugate to the d irection P Bs i nce i t b isects PB . Therefore PA , PB are par all el to conjugate d iameters . Conversely, i f through A , B par a ll els be drawnto tw o conj ugate d ir ect ion s, they meet at a poi n t of thecurve . For i f CD be the d iameter conj ugate to RP meeti ngBP at D ,

the poi n t P on BP whose d i sta nce from D : D B

Fig. 22 .

must l i e on the cu r ve (s ince CD bi sects the chord) , hence theparal l el through A to CD meets BP at P on the cu r ve.

6 8 . Axe s of a c on i c . An axis of a con i c i s a diameterperpendicular to i ts conj ugate .

To pr ove that a central con i c has one and only one pa i r ofaxes .Let AOB (Fig. 22) be a d iameter of the con i c . Assuming

for the moment the ex i stence of a pa i r of axes , draw through Aa paral lel to one of them ,

through B a paral le l to the other .The in tersection Q of these two l i nes l i es on the con ic , by the

pfeccding ar ti cle . I t also l ies on a c i rcle on AB as d iameter

.

ence the i ntersect ions of supplemental chords paral le l to apa i r of axes are the intersections of the con ic wi th the c i rcl e OIIAB as d iameter.

I V ] POLE AND POLAR 71

The c i rcl e and con i c have already two i n ter sections , v iz. atA and B . Hence, by Ar t. 45 , they can have at most two otheri ntersect ions . Now a c i rcl e has an i ns ide and an ou ts ide

,and to

pass fr om ins ide to outs ide i t i s necessary to cross the c i rcle .

Projecti ng 011 to any lane and bear i ng I II mi nd that pr oject iondoes not al te r the r ei

J

a tive or der of poi n ts, w e see that a con ichas also an i nsi de and an ou ts ide, which are the projections ofthe i ns ide and outs ide, respec t i vely, of the or igi nal c i rcle . And

to pass from in s ide to outs ide, or conver sely, i t i s n ecessary tocross the con ic .Now l et u s go r ound the c ircl e of Fig. 22 i n an ass igned

sense,i nd i cated by the arrows i n the figure . If the tangen ts to

the con i c at A and B (which are paral le l , be i ng tangents at theextr emities of a diameter) ar e not perpend icu lar to AB ,

the c ircl ecrosses the con ic at A and B . Ow i ng to the centr al ssymmetryof both con i c and c i rcle, a rota tion of two r ight angles abou t Cbri ngs both figures i nto coi nc idence with themselves

,the poi nts

A and B bei ng interchanged . Hen ce, i f at A the c ircle crossesthe con i c from ou ts ide to in s ide, i t cr osses at B al so from outs ideto i ns ide . Ther efore between A and B i t crosses fr om ins ide toouts ide, giv i ng an in ter sect ion R , and between B and A i t crossesfrom i ns ide to outs ide , giv i ng an i n tersection Q . A , B , Q, R ar e

then the four rea l i ntersect ions of the c ircle and con ic .

Also the poi nt symmetr ical to Q with respect to C l i es on

both c i rcl e and con i c .

.

It mus t accord i ngly be R . QR , AB

bi sect each other at C, . . AQBR Is a paral le logr am and,s i nce

the angle at Q i s a r ight angle, i t must be a rectangle.Hence the points Q, R give the same pai r of axes , namely

the ar allels to the s ides of AQBR .

uppose now the tangen ts at A and B to the con ic areperpend icular to AB . Then the c i rcl e and coni c touch at Aand B ,

that I s,they have fou r real common poi nts, coi nc i dent

i n pa i rs at A and B . They have then no other i n tersect ions .AB i s an axi s, be i ng perpend icu lar to the tangent a t A , and no

axes ex ist,other than AB and the perpend icu lar to AB .

Thu s for any central con ic there ex i sts one pai r of axes ,which are always real , and one pa i r only.

Both axes of an el l ipse mee t the curve in real poi n ts ; thelonger and shorter axes ar e cal led the maj or and minor axesof the e l l ipse respect i vely.

By Ar t. 66, one ax i s of a hyperbol a mee ts the cu rve in realpoi nts . Thi s Is cal led the tr a n szer se ax i s . The ax i s wh ich doesnot meet the cu rve i n r eal poi nts i s cal led the conj uga te ax is .

72 PROJ EeTI V E G EOMETRY [CH .

I II the parabola an axi s i s a diameter wh i ch bi sects chordsper pend icu lar to i tself. Sin ce all diameters are parallel, w e haveto take that one which bi sects chords perpendicu lar to al ld iameters . Hen ce a parabola has on ly one ax is .The poi nts where an ax i s meets a con i c are cal l ed ver tices

the curve.Note that a con i c i s symmetr ica l with r egard to each axi s.For i f P be a poi n t 011 the curve the chord through P

per pend icu lar to an ax i s i s b i sected by that ax i s and therefor emeets the curve aga i n at the symmetr ical poi nt P ’

.

Also the axes are harmon ical ly conj ugate w ith r egard toa symptotes, and therefore (be i ng per pend icular) they bisectangles between the asymptotes by Ar t. 28 .

Fig. 23.

6 9 . C on str u ct i on of a n e l l i p s e w h en tw o p a i r s of

conj u g a te d i am e te r s a r e giv e n i n p os i t i on a n d le n gth .

Let AOB ,COD be the tw o conjugate d iameters (Fig.

Complete the paral lelogram EFCH , of wh ich they are medianl i n es . Dr aw any l i ne QR Q

’ paral lel to the d iagonal E C of th i sparal lelogr am meeti ng BF, CD , CH at Q, R , Q

’ res ectiv ely .

Joi n AR , BQ meeti ng at P . QR i s para l lel to a ed l i ne,

the ranges s im i lar and therefor e proj ective . HenceA [R ]T B locu s of P i s therefore a con ic . Thi scon i c passe and B

,these be i ng ver t i ces of the two

penc i l s . Al so i f QR i s along E C , Q i s at E , R i s at O,hence

AB corr esponds to BE i f QR be at i nfin i ty, Q i s at i nfin i ty OIIFE ,

i .e . O II BA ,R ' is at i nfin i ty OII D C, i .e . on AF ; hence BA

corresponds to AF. The con i c locus of P tou ches EH at B ,



10. I f P and Q he poi nts OII the rad ical axis of tw o c ircles andP

, Q be conjugate poin ts for one c i rcle, they ar e also conjugate for theother .[The rad ical ax is of tw o c ircles i s the locus of points the tangen ts

from wh ich to the two c i rcle s a r e equal. I t i s the common chord i f thecircles cut i n real points ]

1 1 . I f th rough two poi nts A and B conjugate penc i ls be drawn w i thregard to a conic

,the product of these conjugate penc ils is a conic

pass ing th rough the poi n ts of contact of the tangents from A ,B to the

or iginal con ics .12 . I f on tw o l ines a , b conjugate ranges be taken w i th regard to a

conic, the product of these conjugate ranges is a conic touch i ng the

tangents to the origi nal conic at the points where the latter is met bya,b.

1 3. A ,B ar e two fixed poi nts i n the plane of a con ic s. P is a

point such that the tw o tangen ts from P to s ar e harmon ical ly conjugatew i th regard to P A ,

P B . Show that the locus of P is a conic.1 4. State and prove th e theorem Obtained from Ex. 13 by r e

cipr ocation .

1 5 . The locus of the i n tersect ions of the polars w i th regard to tw ofixed c ircle s of a poi nt P ly ing on a fixed s traigh t l ine is a con ic.

1 6. Th e product of two conjugate penci ls w i th regard to a circle,the vertex of one of wh ich i s the cen tre , is a c i rcle .

17. I f T be any poin t, C the cen tre of a conic,N the point where

the polar of T mee ts the d iame ter through T,A a point where th is

d iame ter mee ts the con ic, prove that CN . CT : CA 2

1 8 . I f a d iame ter of a parabola mee t the curve at P and a conj ugate chord at V ,

Show that i f T be the pole of that chord,T l ies on

the d iame ter and TP =P V .

1 9 . I f two pairs of conjugate d iame ters of a conic ar e at righ tangles the con ic is a c i rcle .

20. Show that a con ic is comple tely de termined i f two points andthe i r polars and a poin t on the curve be g iven .

2 1 . Show that a coni c is comple tely de termined i f tw o poi n ts andthe i r polars and a tangen t to the curve be g iven.

22 . Prove the follow ing construct ion for a conic, given a d iameterAB

,a poi n t P on the curve and the d i rect ion conjugate to AB .

Comple te the parallelogram ADP E OII AP as d iagonal and whose s idesAD ,

D P ar e a long and conjugate to AB re spect ive ly . Let a paralle lto D E mee t P D at Q , P E at R . The rays AR

,B Q mee t on the con ic.

I v ] EXAMPLES 75

23. G iven a sel f- polar triangle for a conic and a tangent to the

coni c,construct three other tangen ts .

2 4. A pair of conjugate d iameters of a given conic mee t a givenstra igh t l ine at A and B ; on AB i s described a triangle APB s imilarto a given triangle . Prove that the locus of P i s a hyperbola and findits asymptotes.

25 . Through a fixed poin t 0 a straigh t l ine is drawn to mee t a fixedstraigh t l ine I at P and i n tersects the polar of P w i th respect to afixed conic s at Q . Show that as P describes the straigh t l ine l ,Q describes a conic pass ing through three fixed po i nts independen t ofthe l ine I chosen .

Show that ( l ) i nvers ion is a part icular case of th i s con struction and

that (2) when 0 l ies on s the conies correspond ing to tw o l ines such as lof the plane have s imple con tact w i th s at 0, but contact of the secondorder w i th each other.[P ,

P’ar e said to be i n ver se points w i th regard to 0 i f (0,

P,P

’

be ing coll inear) OP

EXAMPLES IV B .

1 . Tw o conjugate d iameters of an ellipse ar e respect ively 8 and

in length and the angle between them is draw the el l ipse,and

measure the lengths of its princ ipal axe s.2 . Us ing the ruler only construct the polars of the points ( I , 1 ) and

(6, 2) w i th regard to the c i rcle x2+y 2= 9 . I n any manner construct thepolars of the point (3, 0) and of the points at infin i ty on and y z x

w i th regard to the same c ircle .

3. Con struct a l ine pass ing through the po in t (3, O) and conjugateto w i th regard to the c i rcle (ar l .

4 . Construct the envelope of the polar of a po int P on the c i rclew? 4x y

? 0

w i th regard to the c i rcle5 . D raw the coni c th rough the five points (0, 3) (O, 5 ) ( 1 , O) (4 , 0) (2 , 2 )

and construct i ts axe s .6. I f the pole of perspect ive be ( I , the axis of coll ineat ion

x : — 2,construct the axis of the parabola in plane pe rspect ive w i th the

c i rcle w2+ y 2 — 3a: =O, the vanish ing l ine for the c ircle be ing .z': 0.

7. Th e coord i nate s be ing rectangular, A , B a r e the poin ts 4,

(3, 0) respec t ively. I f AR be any ray th rough A ,P the pole of AR

w i th regard to the c i rcle zv2+ y

9= 4, and i f B P mee t A ] ? at Q ,

construc tthe locus of Q .

CHAPTER V .

NON- FOCAL PROPERT IES OF THE CON IC .

70. P a s c a l’s T h eor em . If AB ’

OA'

BO’

(Fig. 24) be ahexagon " i n scr ibed in a con i c, the meets of Oppos ite s ides

(AB’

,A

'

B ), (BO'

,B

’O) , (OA

’

, O'

A )a r e col l in ear.Let P be (AB

'

, A’B ) ; Q : (BC

’

, B C) ; C'

A ) ;L : (AO

'

,M : (BO

’

,

Fig. 24 .

Proj ect the four poi n ts A'

, B’

,C’

,B fr om A , 0 w e have by

the pr operty of the con i cA

Cu tti ng the fi rst penci l by A'

B,the second by BO'

,

(A'

P LB ) 7x

The hexag on cons idered here is not generally , and i n graph icalexamples not conven iently, a con vex figure . A s imilar remark appl iesto polygons i n general , except where the con trary i s d ist inctly stated.

CH. v ] NON - FOCAL PROPERT IES or THE come 77

These tw o proj ect ive r anges have a sel f correspond ing poi n t B,

they are perspect i ve andO

'

the j oi ns of corr espond i ng po i n ts ar econcu rrent

A’Jll , P Q, C

'

L are concurrent,(A

’

lll , C’

L ) l i es on P Q, I t l i es on P Q.

71 . B r i a n ch on ’s T h e o r em . I f ab'ca '

bc' be a hexagon

cu cumscn bed to a come, the j oms of oppomte v er t1ces

p (ab'

, a'

b) q (bc'

,9'

(cd'

,0 ( 6

ar e concu rren t .Thi s theorem i s obta i ned immed iately from Pascal’s theorem

by rec iprocat ion . The student w i l l find i t i n s truc t i ve to con

str uct a proof of Br ian chon’s theorem from the proof gi ven aboveof Pascal’s theorem

,rec iprocat i ng each step .

Pascal’s and Br iauchon’s theorems are conven iently expressedby the fol lowi ng numerical ru l e :

P a s c a l . I f 1 , 2 , 3, 4 , 5 , 6 be the s ides of a hexagon i 11scribed in a con ic taken in order then 1 4 , 25 , 36 are col l i near.The li ne on w hich they l i e 1s cal led the P a sca l line of the

i nscr i bed hexagon .

B r i a n ch on . I f 1 , 2 , 3 ,4 , 5 , 6 be the vertices of a hexagon

c ircumscribed to a con ic taken i n order, then 1 4,25 , 36 are

concurrent .The po i n t through which they pass i s cal l ed the B r ia na/zen

poi nt of the c i rcumscr ibed hexagon .

72 . C on st r u c ti on of c on i c th r ou gh fi v e p o i n ts .

By means of Pascal’s theorem we can construct the con i c through

five poi n ts .Take the poi nts i n any conven ien t order

,l e tter them i n thi s

or der AB '

CA'

B . Number the s ides AB'= 1 . B

'

C’= 2

, CA'

= 3 ,A

'

B 4 . Then P 1 4 i n Pascal’s theorem i s k now n . Draw anyPasca l l i n e P R Q mee t i ng 2 at Q and 3 at B . Joi n Q to the freeend of 4 , v iz. B , and R to the free end of I , v i z . A . The i ntersect ion of AB , B Q i s a poi n t 0

'

on the con i c .

By tak i ng var iou s Pascal l i n es through B w e can constructany number of poi n ts 011 the con i c .

73 . C on st r u ct i on of c on i c tou c h i n g fi v e l i n e s .

S im i lar ly let fiv e tangents to a con ic be gi ven . Le tte r them i norder ab

'

ca'

b. Number the cer taces a b = 1,b c 2 , cd

'

= 3,

a'

=b 4 . Then p m 1 4 i s a fixed l i n e . On 19 ta ke any Br ianchoripo i n t B . Let g be the joi n of B to 2

,7 the jmn of B to 3 .

78 PROJECTIVE GEOMETRY [CH .

q meets b the open s ide through 4 at the ver tex 5, r meets a theopen s ide through 1 at the vertex 6 . 56 i s the tangen t c’to thecome.By tak i ng d ifferent Br ianchon points on p , we can constr uct

the con ic by tangents as an envelope.

74 . C o i n c i d e n t e l em e n ts . Impor tant par ticu lar casesof Pascal’s and Br ianchon’s theorems occur when tw o elementscoin c ide. In th i s case i t i s importan t to bear in mi nd that i f thecoi n c ident elements ar e poi nts , these poi nts have to be taken asconsecu t ive vert ices of the Pascal hexagon and the s ide of thehexagon joi n i ng them i s to be i nter pr eted as the correspond i ngtangent. I f the coinc i den t elemen ts are tangents, these a r e consecu tiv e s ides of a Br ianchon hexagon , and the vertex of thehexagon common to them i s i n ter preted as the correspond i ngpoin t of contact.I n al l ca ses w e shal l wr ite repeated elements tw ice over when

con s ider i ng Pasca l and Br ianchon hexagons, thu sAABODD

w ill be con s ider ed a hexagon , and i ts s i des taken in order ar eAA (tangen t at A ) ,AB ,

BC,

CD ,

DD (tangen t at D ) ,DA .

75 . A sym p tote p r op e r t i e s of th e H y p e r b ola . Let0 (Fig. 25) be the centre of a hyper bola, A

“

, B°° the poi n ts at

infin i ty on the two asymptotes, P , Q tw o poi n ts 011 the curve.Cons ider the Pascal hexagon A °°

A i ts s ides taken inor der ar e as fol lows 1 the asymptote A

°°

P 2

the par allel P L to CA PB“

3 the paral lel PM to 0B

4 asymptote 0B B”

Q 5 the paral lel QL to 0B

QA°°

6 the paral le l QB] to CA .

Hence 1 4 0, 25 = .L, 36 = rlI and C/L ,IV] are col l i near, that

i s , i f on P Q as d iagonal a paral lelogram be descr ibed whose s idesare paral le l to the asymptotes the other d iagonal passes throughthe centre .

I t fol lows that the par al lelograms PN CN '

, QKOK’

, bei ng

v ] NON - FOCAL PROPERTIES o r THE CON IC 79

made up of LK'

CN and of the complements LPN '

K KQLNr espect i ve ly, ar e equal in area . Hence i f through a poi nt P on ahyperbola paral lels are drawn to the asymptotes

,th i s paral lelo

gr am i s of constant area .

Al so K'

N i s paral lel to P Q, for the parallelograms CNLK’

,

LQMP are clearly s imi lar and s im i larly placed . Hence i f P Qmeet the asymptotes at R , S , PR NK’

QS oppos i te s ides ofparal lelograms) . Hence the d ista nces i ntercepte on any straightl i ne between the cu r ve and the asymptotes are equal .Thi s last proper ty furn i shes an easy method for draw i ng a

Fig. 25 .

hyperbol a when the asymptotes and one poi nt P on the curve a r egi ven .

,

Draw a var iable r ay through P meeti ng the asymptotesat R and S . On th i s ray take a poi n t Q such that SQ = P R . Qdescribes the hyperbola .

Care must be ta ken that 1 11 al l cases PQ and BS shal l havethe same med -

p omt'l hus 1 11 Fig . 25 ,

when the r ay i s draw n asP B S

'

, Q mus t be taken at Q ou ts ide 15'

B and not a t Q" i n s ide .

I f the poi n ts P , Q coi ncide the property las t proved becomes :the i ntercept of a tangen t to a hyperbola between the asymptotesi s bi sected at the po i n t of contact .I f TU (Fig . 25) be draw n through P paral lel to NN

'

,

TP NN'= P U. Hence TU 13 the tangen t at P .

80 PROJECTIVE G EOME'

I‘

RY [GEL

Al so the tr iangle TCU tw i ce paral lelogram P N ON '

const. by property proved above .

Hence a var i able tangent to a hyperbol a cuts off fr om theasymptotes a tr i angle of constan t area .

76 . C on st r u c t i on o f a h y p e r b ol a , g i v e n th r e e

p o i n ts a n d th e d i r e c t i on s o f b oth a sym p tote s . Wefirst of al l proceed to construct the centre .

If A , B ,0 be the three gi ven poi nts, cons truct the paral lelo

gr ams 011 AB , BC as d i agonals whose s ides a r e paralle l to theasymptotes . The centre i s then the i ntersection of the other twod iagonal s (Ar t. The asymptotes ar e now k nown i n pos i t ionand the hyp erbola may be constructed by the me thod of Ar t. 75 .

77 . G i v e n fou r p o i n ts on a h y p e r b o l a a n d th e

d i r e ct i on of on e a sy m p tote , to c on s t r u c t th e d i r e c

t i on of th e oth e r a sy m p tote . Let A, B ,

0, D be the fou rpoints let B

“0

be the d irec t ion of the gi ven asymptote, F°° that

of the r equ ired asymptote . Then,cons ider i ng the hexagon

ABODE”

F”

, the poi n ts P i ntersec t ion ofAB,DE Q

°° i ntersect ion ofB0and l in e at i nfin ity, R i n tersec t ion of CD , F

“0

A a r e

coll i n ear. Hence i f the paral lel through P to B0 meet CD at R ,

AR gi ves the d i rection requ i red . We can now use the methodof Ar t. 75 to constru ct the asymptotes i n pos i tion and hence todr aw the hyperbola.

78 . P a r ab o la fr om fou r t a n ge n ts . S i n ce the l i ne ati nfin i ty i °° i s a tangent to the parabol a

,fou r ta ngents a

,b,c,d

define the curve. Let t be any requ i red tangent . Consider theBr ianchon hexagon i mabcdt (Fig . Let 1 , 2 , 3, 4, 5 , 6 be theverti ces z

,a,ab

,be

,ed

,dt

,i“in order . p or 1 4 i s then the

paral le l through cc?to a .

50 1 th is take any Br ian chon point B .

Joi n 2B meet i ng d at 5 : the paral lel through 5 to 3B i s thetangen t r equ i red .

By th i s method we can draw a tangent to a par abola paralle lto any r equ i red d i rection . For draw through 3 a paral lel r tothi s d irect ion to meet p at B ; B i s th e correspond i ng Br ianchonoi n t.pAlso we can construct at once the di r ect ion of the axi s . For

we have to find the poi nt of con tact of the l i ne at i nfin i ty. To

do thi s consi der the Br ian chon hexagon abcdiw

i”

. We have(ab, d i

m

) (be, (ed ,i°°

a ar e concu rren t. Hence i f theparal lels through ed to a an thr ough a b to d meet at E and


82 PROJ ECTIV E G EOM E'I‘

RY [ca

ax i s through B , R“ the poi n t at i nfin ity on D A, then P , Q, R

”

a r e col l i n ear or P Q i s paral lel to DA . Now P i s fixed, A , B , 0

be i ng g i ven P Q be i ng perpend icu lar to the ax is , Q i s fou nd and0Q meets the perpend icu lar to the axi s through A at the poi ntI) requ ired . The l i ne bi secti ng AD at r ight angles i s thereforethe ax is of the parabola .

Let V be the vertex . Cons ider the Pascal hexagonV0.

Then (AB , VI V0) , (I”

I CA )are coll i near. Let AB meet the ax i s at E , 0A meet the l ine at

Fig. 27.

i nfin i ty at V0 meet the paral lel to the axi s through B atThen F l ies on the paral lel to CA thr ough E and V l i es on F0.

Thus V i s known .

80. P a r a b o la fr om a ta n ge n t and i ts

c on ta c t , a n oth e r p o i n t a n d th e d i r e cti on of th e a xi s .

Let a ,A represent the tangen t and i ts poin t of con tact (Fig.

B the other poi nt, I the poi n t at i nfin i ty on the axi s,A" any

other poi n t 011 the curve .

[01v ] NON - FOCAL PROPERT IES o r TH E com e 83

Tw o cons tructions may be u sed , accord i ng as we prefer todescr ibe the curve by r ays through A or by r ays through B .

In the fi rst construction draw l i n es QR paral le l to the tangentat A to meet the paral lel to the ax is through B and A B at Q, 1 13respec t i vely . Then A Q meets the paral le l to the ax i s throughR at a po i n t M on the curve . The re sul t fol low s by cons ider i ngthe hexagon AAM I Q

IQ

B 1 4 i s P ”

, the poi n t at i nfin i ty on

the tangen t at A PQ

QR i s then the Pascal l i n e.I n the second construct ion draw a ar allel to AB to meet

AB at the tangent at A at Q'

aa the pa ral le l to the ax i s

Fig. 28 .

thr ough A at B '

. Joiu BR’meet i ng the par allel to the ax i s

thr ough Q’ at a poi n t A] . Then M i s on the parabola. For

Pm

Q'

B' i s the Pascal l i n e of the hexagon

8 1 . I n sc r i b e d Q u a d r a n gl e a n d c i r cum s c r i b e d

Q u a d r i la te r a l . I f AB OD be a quadrangle inscr i bed i n acon i c , consi der the Pascal hexagon AABO0D . Then

(AA . (AB ,CD ) . (BU, AD )

ar e col l i near .~ Hence the tangents at any tw o gi ven verti ces ofa quadrangle inscribed i n a con ic mee t on that s ide of i tsd iagonal tr i angle wh ich i s oppos i te to the d iagonal poi n t ly i ng onthe joi n of the tw o given vert i ces.

84 PROJECTIVE GEOMETRY [on

In l i ke mann er , fr om Br ianchon’s Theor em , i f a , b, c, d be fou r

tangents, cons ider ing the hexagon aabccd,then

(aa ,cc) , (ab , ed) , (be, ad)

ar e concu r rent. Hence the joi n of the poi nts of conta ct of anytw o given s ides of a quadr i lateral c i rcumscr ibed about a con i cpasses thr ough that ver tex of i ts diagonal tr iangle wh ich i sc

plposite to the d iagonal through the meet of the two given

8 ] es .I f a , b, c, 03 be the tangents at A , B , 0, D , (acaeg, (

bd) both lieon the j oi n of (AB , CD ) , (B0, AD . Hen ce the iagonal pointsof the complete quadrangl e forme by the four poin ts l i e on thed iagonal s of the complete quadr i later al formed by the tangen ts atthese points , or :The complete quadr i later al formed by fou r tangen ts to a con ic

and the complete quadrangle formed by the ir four poin ts ofcon tact have the same d iagonal tr iangle.

8 2 . I n sc r ib e d a n d c i r cum sc r i b e d tr i a n gle s . LetABO be a tr iangle i n scr ibed in a con ic.Fr om the Pascal hexagon AABB00 w e find

(AA , Boo, (AB , 009, (BE . 0A )ar e col l in ear , or the s ides of an in scr ibed tr iangle meet thetangents at the opposite verti ces at coll i n ear poin ts .I f abe be a tr iangle c i rcumscr ibed about a coni c, i t follows m

l ike mann er fr om the Br ianchon hexagon aa bbcc that the j oins ofthe ver tices to the points of contact of the Oppos i te s ides arecon cur r ent.

8 3 . New ton ’s T h e o r em on th e p r odu ct of s egm en ts of ch o r d s of a c on i c I fP OP ’

QOQ’

(Fig . 29) be anytwo chords of a c t 0

,whose d i rections are g i ven

,

the rat io of the r ectangl i s con stant . For suppose

the chord QOQ'

to move paral le l to i tsel f, wh il e the chord P OP’

r ema in s fixed . Let the l i n e through P parallel to QQ Q' meet

the con i c agai n at T. Then s i nce the d iam eter conj ugate toQQQ

' bi sects QQ' and P T and these a r e paral lel , i t i s obv ious

from Elementary Geometry that P Q’

, TQ meet at S on thi sd iameter . The penc i l s P T [Q] are then per spect i ve andtherefore proj ecti ve. Hence by the property of the con i c(Ar t. 37) the penc i l s P [Q] , P

'

[Q ] are pr oj ect ive . These

NON - FOCAL PROPERTIES OF THE CON IC 85

i l s determ i ne On any paral le l a to QQ’tw o proj ective ranges

[RC] Let a meet P P ' at U and let V “0 be the poi nt at

nfin ity O11 u . Then when Q i s at P’

, R i s at U and R' i s at

and when Q' i s at P , R I S at V °o and R ’

i s at U.

Hence i f R , , R ,

’R 2 , R 2

'

be pa i r s of correspond i ng poi nts ofthe r anges [R ] , [R

'

] w e have the set of fou r poi nts UR V”

R ,

correspond ing to the set Of four V”R

’UR z

'

. Equatirig thecr oss - r at ios

UR I 0 17n V l’

0 UR Q

'

Uf 132 . V°°

R 1

UR l UR Q

'

UR 2 UR 1

that i s 1 .

UR UR' const .

But, fr om Fig. 29 , OQ, UR bei ng paral le l and OQ'

,UR

' be i ngparal lel

HenceOQ . OQ UR . UR

86 PROJ ECTI V E GEOMETRY [ca

The r atio i s therefore not al ter ed by sh ift i ng the

chord QOQ' parallel to i tsel f. S im i l arly i t i s not al tered by

shi ft i ng the chord P OP ' paral lel to i tself. I t i s therefore i ndependent Of the pos i t ion of 0,

once the d irect ions of the chordsPP

'

, QQ' are fixed .

I f w e take 0 at the centre 0 the paral l el chords ar e b i sectedat the cen tr e.

Hence033

6

1

2; r a tio of the squares of the paral l el sem i

d iameters .If Q

’

Q,P

'

P we have two tangen ts to a con i c from anypoin t ar e in the r at io of the par allel sem i - diameters .

8 4 . Ob l i qu e or d i n a te a n d a b s c i s sa r e fe r r e d to

c onj u ga te d i am e te r s . I f the chor d P P (Fig. 29) coi n c idew ith the d iameter AA' conj ugate to the chord QQ

' the pr oper tyof the last ar ticle takes the form

0B bei ng the sem i - d iameter conj ugate to CA 5 or s ince

NQ'

N Q.

QN2

constOB ?

A N . NA’_ —

0A 2°

In Fig . 29 the con i c i s an el l ipse, and the d iamete r BOB’

meets the curve in real points . Therefor e 0B 2 i s pos i t i ve andAN . NA

' i s pos i t ive , so that N l i es between A and A’.

But i f the con ic be a hyperbola we know that i f ACA' meetthe curve i n real poin ts , i ts conj ugate B01)“ does not meet thecur ve. Hence there i s no real sem i - diameter 0B .

Nevertheless the theorem Of Ar t. 83 holds good and

621V2

AN . NA'

bu t N i s ou ts ide AA’and the constant i s negat ive . If we thencon str uct a l ength 0B 1 such that

0B ,2

QN2

0A 2 AN . NA

and lay i t Off along the di ameter conj ugate to AA '

,0B 1 may be

a constant,

v ] NON - FOCAL PROPERTIES OF THE come 87

spoken of as the rea l l ength , or s imply the leng th , Of the semid iameter conj ugate to CA . But i t should be carefu l ly rememberedtha t B , i s not a po i nt on the hyperbola .

Return i ng to the relat ion

QN” 0B 3

AN . NA'

CA 2

for the el l i pse, we have

QN2

(AO + ON ) (N 0+ 0A’

)0B2 0A 2

(0N 0A ) (0A ON )CA Z

For the hyperbola

QN 0A2 02V2

ON ?

QN0131

2

I n the case of the parabola A ' i s at i nfin i ty .

Take two chords Q, conj ugate to the same diameter,

Q1AT]2

Q2N 2

A N1 . N 1AI

A N2 . N2A

Q1N 1

2

Q2N 2

2 M A,

AN , AN 2 N 2A

now l et A’be taken at i nfi n i ty N 1A'

N QA'

1 .

Qe Q2N z

2

HenceA N , A IV,

if

}; constant for the parabol a .

The above relat ions lead to the wel l - k now n analy t ica lequa t ions Of the el l i pse

,hyperbol a and parabol a referred to

conjugate d i rect ions .

PROJECTIVE GEOMETRY [ca

8 5 . C a r n ot’s T h e o r em . I f the s ides B0, 0A , AB of

a tr i angle ABO meet a con ic at P , P’

Q, Q'

; R , R’

r especti vely,

thenBP . BP

'

0Q . 0Q'

AR . AR'

For l et OL,0A] , ON be the sem i - diameter s of the con ic

par al lel to B0,0A , AB respect i vely. Then by Ar t. 8 1

AR . AR'

ON 2

,

BP . BP'

OL 2

.

OQ . 0Q'

OM 2

AQ . AQ'

OAP’BR . BR

'

ON “ OP . 0P'

OL'“

Mu ltiply i ng these three r elat ions together the r esu lt fol lows .

8 6 . I n te r s e c t i on s of a c on i c a n d a c i r c le . Le t

a c i rcl e meet a con i c at four poi nts P , Q, R , S , and let P Q meetRS at 0.

Then i f CL , CM ar e the semi - diameters of the con i c paral lelto P Q, RS w e have

0L 2 OP OQ001 2

“

OR . OS

by the property of segmen ts of chords of a c ircle.Hence the sem i - diameters CL ,

CM are equal . Now theextremiti es of al l equal semi - diameters lie on a c ir cle con

centri c w i th the con ic . Thus there can be on ly four of themand

,by the symmetry of the con ic wi th regard to the axes

,they

l i e pa i r and pai r on tw o diameters equally incl in ed to the axes .Thus CL , 0111 are equal ly i ncl i ned to the axes, and the commonchor ds P Q,

R S of the con ic and c i rcle are equal ly incl ined to theaxes.The same holds of the other pa i rs Of common chords, v iz.

P S , R Q ; P R , QS .

In parti cu lar , i f a c i rcl e and con ic have three coi n c iden tin tersections P , Q, R , the common chord P S and the commontangent at P a r e equal ly incl i ned to the axes .The c i rcle i s then sa id to be the ci r cle of cur va tur e a t P : th i s

proper ty enables us to construct i t gr aph ically.

8 7 . E v e r y e l l i p s e c a n b e d e r i v e d fr om a c i r c l e bya n o r th ogon a l p r oj e ction . For cons ider the orthogonalproject ion of a c i rcle upon any plane through a d iameter a .

A l i ne perpend icu lar to thi s axi s Of coll i n eat ion x i s sti l lperpend icu lar to x after proj ect ion and rabatment about a intothe or ig inal plane, and i f P be any poin t of the origi nal figure

,

PX the per pend icu lar from P on x, P’ the corr espond i ng poi n t


90 PROJ ECTI V E G EOMETRY [CPL

P QRS (Fig . 30) of wh i ch the s ides P Q, RS ar e paral lel to thestretch ax i s and the s i des P S

, QR a r e paral lel to the d irection ofstretch . Let these meet the stre tch ax i s a t X,

Y r espectiv ely .

P QR S tran sforms i n to a paral le logram P’

Q'

R'

S’of which

P'

Q'

,R

’

S’ are par al le l to the stretch axi s . For i f A be the

str etch ratioP l t . QY = Q

'

Y and S ’l

'

: R'

Y r

Henceandpar allelogr am P ’

Q’R

'

S’ paral lelogr am P QRS S

'

P'

SP A.

Break i ng up any area i nto such elementary paral lelograms

X Y

Fig. 30.

and addi ng we see that A= r atio of tw o corr espondi ng ar eas .Hence ar ea of any conj ugate paral lelogram of the el l ipse

x area of cor r espondi ng c i rcumscr ibed square Of the aux i l i ary

4a24ab.

Thus the conj ugate paral lelogram Of an el l ipse i s of constan tar ea . Cal l i ng p the perpend i cu lar from the cen tre 011 any giventangent

,d the length Of the semi - diameter paral lel to th i s tangen t ,

the area Of the conjugate paral le logram of w hich the gi ven tangenti s a s ide i s cl ea rly 4p d, for 2d i s the base Of the paral lelogram and2p i s the height. 41703 4a b, i .e. p d ab.

8 9 . S um of squ a r e s of tw o c onj u ga te s em i

d i am ete r s . Let 0Q’

(Fig . 3 1 ) be two diameters at rightangles Of the auxi l iary c i rcle , 0P , 0Q the cor respond ing d iameter s

v ] NON - FOCAL PROPERTI ES OF THE CON IC 9 1

of the ell ipse . Then OP , 0Q are conj ugate . Let P'

P ZlI , Q’

QN,

be perpend icu lar to the major axi s .Then from the stretch property , i f qS be the angl e A0P

'

,

QN Tr Q’

N a cos (,b b cos

0fll z a cos d> ;0P 2

+ 0Q3 = 0111 2 + P ZlP + QN

'

3 0N

a2cos qt bi

’

Sin2 <1: b2 cos ct a?Sin2 gt

a2 bi .

Hence the sum of the squares of tw o conj ugate d iametersel l ipse i s constan t .

Fig. 3 1 .

9 0. P s e u d o - c onj u ga te p a r a l le log r am . Let AOA'

(Fig . 32) be a d iameter Of a hyperbola meeti ng the cu rve at realpoi nts A

, A’. Let the tangen t at A meet the asymptotes at

D , E and the tangent at A’meet them at F, G . Then , because

the ta ngen ts at A , A'

a r e parall el , DEFG i s a para l le logram ofw hich the asymptotes ar e d iagonals .I f P be a po i nt on the curve and P N the chor d through P

conj ugate to AOA’meeti ng A OA ' at N ,

PN2 0B

A N NA'

0A

0B , being the real l ength (see Ar t. 84) of the dianrete r conj ugateto 1 10A

7 2

AN . NA' (fr—ii?)

92 PROJECTIVE GEOMETRY [GEL

I f P moves Off to i nfin i ty on the hyperbol a i n the di rection of

the asymptote CD,AP becomes para l le l to CD . The tr iangles

2

CAD , ANP become s im i lar , and (fix) becomes equal toAD 2

.

AN h0A ) A

’Nappr oac es u n i ty.

Hence0B ]

? Ap e

0A 2 CAW

0B ,= AD .

Thus the i n ter cept of a tangen t between the asymptotesmeasures the real length Of the parall el d iameter . The paral lelogr am ther efore a p seudo- conj uga te paral lelogram . I ts

med ian l in es are conj ugate d iameters , but on ly one pa i r of s idestouches the curve .

We have seen (Ar t. 75) that the area of the tr iangle E ODcut off fr om the asymptotes by any tangent to the cu r ve i sconsta nt. The area Of the pseudo- conj ugate paral l elogr amDEFG i s four t imes the area of the tr iangle BOD and is ther efore also constan t.l f p =

per pendicu lar from 0 on tangen t at A , d : 0B ,= real

l ength Of d iameter conj ugate to 0A,DE = 2d ,

and area of

tr i angl e E OD = §p . 2d =pd . Hence in the hyperbola as i n theel l ipse p d z constant. Tak i ng the case where the s ides of the



9 2 . R e cta n gu la r h y p e r b o la I f the angle between theasymptotes i s a r ight angle the cu r ve i s cal led a rectangularhyperbola .

Conj ugate d iameter s of a rectangular hyperbola are equal :For i f DE (Fig . 33) be the tangent at A to such a hyperbola,meet i ng the asymptotes at D and E

,s i nce the angle at 0 1s a

r ight angle,the c i rcle on DE as d iameter passes

o

thr ough 0.

Hence A0= AD . Also 0A , AD , i .e . 0A , 0B , are then equal lyi ncl i ned

)to the a symptotes (0B , hav ing the same mean i ng as i n

Ar t 90

In part icu lar the tr an sverse and conj ugate sem i axes are equalfor a r ectangular hyperbola.

Aga i n con s ider the sem i - d iameter OH perpend icu lar to 0B , .

Becau se CE , 011 are perpend icu lar to CD,0B , r espect ively

,

the angl e EOH = angle D OB ,= angle AOD . 0A , 017 are thu s

equal ly in cl i ned to the axes 0X 0Y and OH IS ther efore r ealand equ al to CA , tha t Is to 0B , . Thus the r eal lengths of perpendicular sem i diameter s Of a r ectangu lar hyp erbola ar e equal .The diameter 00, conj ugate to OH makes the angle

G ,0E = angl e EOH = angle AOD . I t IS ther efor e per pendicu larto 0A . We have thus a set Of four d iameters equal i n r eall ength .

Notrce that th i s does not i nval i date the r esu lt ment ioned I nAr t. 86 that only on e d iameter ex ists equal to a g i ven d iameter ;and that th i s d iameter i s equal ly in cl i n ed to the axes wi th theg i ven diameter. For the lengths 0B , , 00, ar e not sem i d iameter sat al l

,but merely the analogues of sem i - diameters : they ar e only

cal l ed su ch by a conven t ion , B, , 0, not be i ng points on the cu r ve.

EXAMPLES V A.

1 . Show that by al tering the order of s ix poin ts on a conic, 60d ifferen t hexagons may be for nred, w i th 60 correspond ing Pas cal l ines .

Show that these 60 hexagon s have the i r Pascal l ines concurrent infours

,namely when they have a pair of oppos ite s ides common .

2. Show that i n the notat ion of Ar t. 71 for Pascal’s Theorem the

l ines ( 13, (35 , (5 1 , 24) ar e concurren t.3. Show that ( 13, (35 , (5 1 , 24 ) ar e poss ible Pascal l ines for

the s ix poi nts.4 . Sta te and prove the resul ts correspond ing to those of Exs. 1

,2,

3 for B r ianclron’s Theorem.

v ] EXAMPLES 95

5 . Provo that i f ABO, A’B O

’be two triangles i nscr ibed in a conic ,

the i r s ix s ides touch another con ic.

[The s ides B0,0A ,

AB ar e denoted by a , b, e B’O’

,C’A

’

,A

’B

’

by a’,b’

,e’.

We have

Cut the se by A’O

’

,AO. Then by the theorem of the cross - axis(bc

'

,b’c), (ca

’

,c'

a ) , (ab’

,a’b)

ar e concurren t. By the converse of Br ianclron’s Theorem the resul t

follows ]6. Deduce from Ex. 5 Ponce le t’s Theorem that i f there exist one

triangle inscribed in one conic and ci r cu ru scr ibed to another, there existan infin i ty of such triangles.

7. G iven five po in ts A,B

,0,D

,E on a con ic

,construct the tangen t

to the con ic at any one of them .

8 . G iven five tangents a , b, e, d, e to a con ic, construct the point ofcontact of any one of them.

9 . Deduce the results of Ar t. 75 from the property that a pair ofconjugate d iameters of a hyperbola ar e har nron ically conjugate wi thregard to the asymptotes, w i thout us ing Pascal

’s Theorem.

10. Obtain the theorem that a variab le tangent to a hyperbolacuts off from the asyrrrptotes a triangle of constant area by apply ingBr ianclrorr

’s Theorem to the hexagon aap bbq ,

’

a,b be ing the asymptotes,

p , q any tw o tangents .

1 1 . Obta i n the follow ing construct ion for a parabola by tangen ts ,g iven four tangents a , b, e, d . 011 (ab, ed ) take any point B. Drawth rough B a paralle l to a mee ting 0 at P

,and a paralle l to cl mee t ing b

at Q . P Q is a tangen t to the parabola.

1 2 . Show how to construct a parabola by tangen ts given threetangents and the d i rect ion of the axis . Construct also the vertex and

axis.

1 3 . Prove the follow ing con struc tion for a parabola, g iven a poi n t A ,

the tangen t a at A , a paralle l .r to the axis and another poin t B on the

curve . Construct a paralle logram ACBD w i th A B as d iag onal andBD ,

B 0 paralle l to a , a; respect ive ly . Draw LN paralle l to the

d iagonal CD to n reet B 0 at L ,B I) at Al . The mee t of AL and a

paral le l to .1: th rough M is a po in t on the curve .

96 PROJECTIVE GEOM BTRY [ca

1 4 . Deduce the construction of Ex. 13 from that of Ex. I V A,22 , by

tak ing B at infin i ty .

15 . From the construct ion s of Ar t. 80 deduce that M Q' i s pr o

port ional to (AQ’

)2.

1 6. I f the tangent at U mee t a pai r of conjugate d iameters atP , P

’, show that UP . UP

’= CD 2 where CD is the d iame ter parallel to

the tangen t at U.

1 7. Show how to find graph ically the d i rect ions of the axe s of anygiven con ic w i thout first find ing the cen tre .

18. Show that i f a c ircle touch a con ic and cut i t again in tw o realpoi nts the common chord and the common tangent a r e equally i ncl inedto the axes.

1 9 . I f a c ircle and con ic have double con tact the common chord of

con tact is paralle l to an axis.

20. Show that i f P,P

’ be points where a perpend icular to the

major axis of an ell ipse mee ts the curve and the auxil iary c i rcle r especti vely the tangen ts at P ,

P’mee t on the major axis.

2 1 . Prove that an e ll ipse can be obtained from the c ircle 011 i tsminor axis as d iameter by a stre tch parallel to the major axis.

22 . Show that the area of an ell ipse i s 7rab.

23. Show that the d iagonals of a conjugate parallelogram ar e themse lves conjugate d iame ters.

24 . Show how to con struct geome trically the equal conjugated iame ters Of an ell ipse

, g iven i ts axes .

25 . Show that i f a parallelogram be c i rcumscr ibed to or i nscribedi n a conic i ts d iagonals intersect at the centre of the conic.

26. The extremi tie s A, A’of a d iame ter of a rectangular hyperbola

a r e joined to a poin t P on the curve . Show that A P ,A

’P describe

tw o oppos i te ly equal penc ils. Show also that th i s i s not tru e unlessA

,A

’a r e extremi t ie s of a d iame ter.

27. Prove that the d iagonals of a comple te quadrilateral c i rcumscribed about a con ic ar e d ivided lrar nrorr ica lly by the d iagonal poin ts ofthe complete quadrangle formed by the i r four poi nts of contact.

28 . Show that the triangle formed by three tangents to a con ic is inplane perspect ive w i th the triangle formed by the i r th ree po ints ofcontact


CHAPTER V I .

FOCAL PROPERT IES OF THE CON IC .

9 3 . E v e r y c on i c c a n b e ob ta i n e d a s th e s e c t i on 0

a r e a l r igh t c i r cu la r c on e . Descr ibe a c i rcle (Fig . 34 a )touchi ng the con i c at the extremityX of a n axi sXA’

. Thi s ax i si s therefore a d i ameter of both con i c a nd c i rcle. The con ic andc i rcle may then be brought i nto plane perspect i ve by tak i ng a asax i s of col l i n eation (see Ar t. The pole O of perspect i ve i sthen by symmetry on the ax i s X'

A’. The van i sh i ng l in e 7 of

the c ircle is a paral le l to as, cutt i ng the ax i s of symmetry at I .

I f w e r otate the con i c about a , and jo i n corr espondi ng poi ntsof the rotated con i c and the c i rcl e, these j oi n s , by Ar t. 1 2 , wi l lpass through a vertex V w h i ch l ies on a c i rcle centre I and

rad i u s I O i n a plane through the axi s of symmetry perpend icu larto the plan e of the paper . Thi s c i rcle wi l l therefore i n generali ntersect the perpendicu lar to the plane Of the paper through thecentre 0 of the given c i rcle . Take for V on e of these i ntersections . Then the proj ect ion of the given ci rcl e from V upon aplan e through a paral le l to Vz

'

i s the gi ven con i c . The gi vencon i c i s the sect ion by th i s plane of a con e vertex V and

base the g iven c i rcle . But thi s i s a r ight c i rcular con e, Si nce Vi s on the per pend icu lar to the plan e of the gi ven c i rcle throughi ts centre .

I t remai ns to Show that the d imens ions of the gi ven c i rcl ecan be so adj usted as to make th i s constru ction real i n al l cases.

In order that the c ircl e locu s of V should cut the perpendicular th r ough 0 i n rea l poi n ts i t i s n ecessary and suffic ient that1 0 1 0.

E llip se. I n the case of the e l l ipse take X the extremityof the major ax i s a nd take for the c i rcl e a rad i u s i n termed iatebetween the major and the minor sem i - axes of the e l l ipse . Thefigure i s then as drawn i n Fig . 34 a . The common tangentsmu st necessar i ly i n tersect at O on the Side of 0 further from A’

.

CH . V I ] FOCAL PROPERTIES OF THE CON I C 99

Also i f I be the po i n t at i nfin i ty 011 the tangen t to theel l ipse from I meets the tangen t to the c ircle from I at a poi ntY on a

"

. .XY = semi - mi nor ax i s of the ell ipse,and thi s bei ng less

Fig. 34 .

than rad iu s of c i rcle,the tangent Y] to the c i rcle meets a t

I on the same s ide Of 0 as A’

. Thus 0 l ies be tween [ and Oand the construct ion leads to a r eal r ight c i rcular con e

100 PROJECTIVE G EOMETRY [ca

Hyper bola . Take X an extrem ity of the tr ansver se axi s(Fig . 34 b) . Let .7: meet one asymptote at Y ; l et J be thepoi n t of con tact of th i s asymp tote . Then J i s the poi ntof contact of the ta ngent fr om Y to the c i r cl e . 73 i s therefore the par al lel to a

' through J , meeting XC at I . O i s wherei .e . the parallel through J to the asymptote, meets X0.

I 0 I 0 i f angle OJ I > angle OJ I . Bu t, H be i ng cen tre of

hyperbola,angl e OJ I = angle II YX and becau se J I , J 0 are per

pendicula r to YI YJ respec t i vely,angle OJ I = angle J YI

Hence I O I 0 i f th e angle H YX > angl e Draw therefore YJ mak i ng with YI an angle l ess than XYH that c i r cl etouch i ng YJ and tou ch i ng XY at X,

whose centre fal ls on thesame s ide of .c as H , l eads to a real sol ut ion .

P a r abola . X be i ng the vertex,re the tangen t at X draw

any tangent t to the parabola meeti ng the ax is of the cu r ve at 0.

Descr ibe a c i rcl e touch ing t and touch ing a' at X'

,to fal l on the

same s i de of .7: as the parabola (Fig. 34 c) . Then I i s where thec ircle meets th e axi s of the parabola (for the van i sh i ng l i ne touchesthe c i rcle) . I , 0 fal l i n s i de the parabol a , O,

bei ng on a tangent tothe cur ve

,fall s outs ide i t. Hence 0 must l i e between I and O.

9 4 . Foc a l sp h e r e s . Cons ider a r ight c i rcu lar con e ver texV. Let i t be cut by any plan e i n a con i c s and let the plan e ofthe paper (Fig. 35) be the plane through V perpend icular to thepla ne of sect ion . By symmetry the l i n e AB in wh ich the plan eOf the paper meets the plan e of section i s an axi s of the sec t ion .

Constr uct a sphere touch ing the plan e of section at S and

touch ing the cone . I t w i l l touch the cone along a c i rcl e 3’of

whi ch B ’

A’

(Fig. 35) i s a diameter. The plane of th i s c i rc lemeets the plane of section i n a l i ne a' per pend icu lar to the plan eof sect ion and meet i ng the plan e of the paper at XCons i der i ng the c i rcu lar sect ion B ’

A’S of th i s Sphere by the

plan e Of the paper w e note that X l i es on the pol ar of V w i thr egard to th i s c i rcl e and X l ies on the polar of S (th i s bei ngthe tangen t at S ) . Hence X i s the pol e of S V or X, S

’

a r e

harmon i cal ly conj ugate w ith regard to B'

,A

’. Therefore a

: i sthe pol ar of S ' wi th regard to the c i rcl e 3’and by pr oj ection x i sal so the polar of S wi th regard to the con ic s . a

' be i ng an axi sof coll i n eat ion , i f on .7: we take tw o poi n ts P , Q which are conj ngate w i th r egard to 3

’ they are also conj ugate wi th regard to 3 .

But becau se they are conj ugate wi th r egard to s'

, by Ar t. 58 ,

PX XQ square of tangen t from X to c i rcle 3’ s uare of tangent from X to Sphere =A’192. I t fol lows by a well nown r esu lt(XS being per pend icu lar to P Q) that the angle P SQ i s a r ight


102 PROJECTIVE GEOMETRY [ca

meet .7: at R . Because SF, SR ,be i ng perpend icular

,are con

jugate l in es through S ,there fore the pol e of SF l i es on SR al so

i t l i es on the polar a' of S . R i s the pole of SF and ES , FRare conj ugate l i n es through P . Bu t these cannot be at r ightangles u nless R i s at i nfin i ty, that i s, u nless F i s on the perpendicu lar thr ough S to a

'

,i .e . on the axi s AB .

Al so ther e cannot be three foc i on the axi s AB . For cons idertw o poi nts U V

°° at i nfin i ty i n two d i rect ions at r ight anglesi n the plane. Let conj ugate rays through U V

°° meet theax i s AB at T, T

’. The penc i l s U [T] , V

°°

[T’

] are proj ect iveby Ar t. 55 and the ranges [T] , [T ar e proj ecti ve .But the l i n es joi n i ng a focu s to U bei ng perpend icu lar

lin es through a focus,ar e conj ugate l i nes through U

Hen ce at a focus the points T,T

’ coi nc ide . Foc i are ther efor eself - cor r espondi ng poin ts of the ranges [T] , and as theser anges by Ar t. 25 can not have more than tw o selfcorrespond ingpoin ts , ther e can be on ly tw o real foci and these are the poin tsS,H found i n the pr ecedi ng arti cle .

Al so fr om the symmetry of the curve abou t the axes i t fol lowsthat the two foc i S , H and the two correspond ing d irectr i ces .z

'

, 3/are symmetri cal ly s i tuated w ith r egard to the centre and the axi sperpendi cu la r to AB .

Al so the foc i ar e always inside the cu r ve, for i f they wer eoutside, the tangen ts from the foc i would be self- conj ugate, orper pend icu lar to themselves

,which i s imposs ible .

In the parabola the vertex B (Fig. 35) goes ofi to i nfin ity.

One foca l sphere then goes to i nfin i ty and there i s only one focu sat a fin i te d istan ce.I n what fol lows the axi s AB ,

on which the r eal foc i l i e, wi l lu sual ly be spoken of as the foca l axis of the curve. From theconstructions Of Arts . 93, 94 i t appears that the foca l axi s i s themajor ax i s in the el l ipse and the transverse ax i s in the hyperbola.

9 6 . E c c e n tr i c i ty . Let P , Q (Fig. 36) be tw o poi nts onthe con ic , S a focus, XY the correspond i ng d irectr ix . Let P Qmeet the d irectri x at Y and let P i ll , QN be perpend icu lars fromP , Q 011 the d i rectr i x . Then i f the ta ngents at P and Q n reet atT, T i s the pol e of P Q, S i s the pole of A

’Y, Y i s the pol e of

are conjugate l i n es through a focus , they are atr ight angles . Al so ST bei ng the polar of Y,

i t fol lows thatY, P , the poi n t where STmeets P Q, and Q are fou r harmon ic .

poi nts. Hence S YP ] ’Q) i s a harmon ic penc i l and the conjugaterays S Y, ST be i ng perpend icu lar , b i sect the angles between theother two (Ar t.

V I] FOCAL PROPERTIE S OF THE CON lC 103

IYP I=lYQ l= | SP =ISQ |But by s im ilar tr iangles

YP | =YQ I= IPM I=IQN

ISP I r lQN .

constan t for the con ic e say hence

The di stance of a poi n t on a con i c from a focus i s to i tsd i stance from the correspond i ng d irectri x i n a constant ra t io e,

which i s cal led the eccen tr i c i ty .

From the symmetry of the curve i t i s obv iou s that theeccentr ic i ty must be the same for the two foc i .

Fig. 36 .

9 7 . D i sta n c e s of foc i a n d d i r e ct r i c e s fr om th e

c e n t r e . I f the focal ax i s meet the cu rve at A,A

’ and S ,S

’ bethe foc i (Fig . then A and A ’ are the po i n ts whi ch d i v ideSA

? i n the ra t io e 1 . I f therefore e 1 , A’ i s at i nfin i ty and we

have a parabol a : i f e l,S (and therefore by sy rn rrretr y S

’

) i si n s ide AA ' as i n Fig . 36a

,hence the cen tre 0,

whi ch i s the midpoi n t of SS

’

,i s i ns ide the cu rve and the la t ter i s an el l i pse .

I f e > 1,S and S '

a r e ou ts ide AA’. The cen tre 0 i s on the

Oppos i te s ide of the cu rve to the foc i ; i t i s therefore outs idethe cu rve and the lat ter i s a hyperbo la .


We have, s ince A , A’d iv ide SX i n ternal ly and exter nal ly in

the r atio e 1

SA’

e . A’X= e AX’by symmetry.

Subtracti ng, AA’

e . A’

IX'

0A e . CXBut s i nce (A

’SAX) i s harmon i c,

O O O O O O O O O O O O O O O O O O O O O O O O

Hence usi ng ( 1 )9 8 . T a n ge n t s fr om a n exte r n a l p o i n t su b te n d

e qu a l o r su p p l em e n ta r y a n gl e s a t th e focu s . ByAr t. 96 S Y, ST (Fig . 36) bi sect the angles between SP , SQ.

I f P and Q are 011 the sam e bran ch of the curve (Fig . 36 a ) ,Y l i es outs ide P Q ST i s the i nternal bisector Of the angleP SQ and the tw o tangents TP , TQ subtend equal angles atthe focu s . I f P and Q ar e poi nts on Oppos i te branches of thecurve (Fig . 36b) , Y lies i n s ide P Q ; ST i s the external bi sectorof the angle P SQ and TSQ + TSP = tw ice (mean angle TS Y)2 r ight angles . Hence the tangents TP

,TQ subtend supple

mentary angles at the focus .In c iden ta l ly w e have proved the fol low i ng theorem : i f a

chord P Q meet the d i rectr i x correspond ing to a focus S at Y,

then S Y bi sects in ternally or external ly the angle P SQ accord ingas P

, Q l i e on oppos i te branches or on the same branch of thecome .

If P and Q coi nc ide , T coi nc ides w i th them . Hence thei ntercept on a tangen t to the curve between the poi n t of contactand a di rectr i x subtends a r ight angl e at the correspond i ng focus .I f a

,b be tw o fixed tangents to the con i c

,c a var iable ta ngent

meeti ng a,b at L

, M respec t i vely, S a focus , A , B ,0 the poi nts

Of contact of a,b,e,and i f a

,b, c al l touch the sa ri r e branch of

the cu rve,w e have by the above LSO= é ASO,

Hence by add ition LSM éASB . Therefore L llI subtends afixed angl e at S .

The student may prove for h imself that i f a, b, c touch

difier cnt branches of the curve the theorem sti l l holds, but that

LSM i s then equal to ” u se , ge rms or 3+ ,ASBaccor d ing to c ircumstances .

9 9 . S um a n d d i ffe r e n c e of foc a l d i sta n c e s . Through


106 PROJECTIVE G EOMETRY [CPL

SA’: 0A ( 1 — e)

A'

S . SA = 0A 2

( 1

OB 2 = ( 1

I f e > 1 , 0B 2 i s n egative . I f 0B , be the real l ength of theconj ugate S8 1D 1 - 2tX1S

0B ,

22 (e

21 ) 011

2 SA SA’0A SL

as before.

101 . T h e s em i - la tu s r e c tum i s a h a r m on i c m e a n

b e tw e e n th e s e gm e n ts o f a n y foc a l ch or d . Let P SQ(Fig . 37) be a focal chord , meeti ng the d irectr ix at Y. Then( YQSP ) i s harmon ic , we have by Ar t. 28

1 1 2

YP+Yo

“

YS'

YP : YQ : YS z P Ils R zSXSX

Hence1 i 2

P S SQ SL

Mu lt iply i ng upP S SQ

_

2_

P Q 2

P S . SQ SL P S . SQ—

SE

I f 0D be the d iameter paral lel to P Q, 0B d iameterparal l el to SL . Hence by Newton’s theorem

P S . SQ 0D 2

Thu s P Q 2 SL | . Ar t. 100.

Hence the lengths of focal chords are propor t ional to thesquares of the paralle l semi - d iameters .

102 . T h e ta n g e n t a n d n or m a l b i s e ct th e a n gl e s

b e tw e e n th e fo c a l d i sta n c e s . Let P (Fig. 37) be a pomton the con ic and let the tangen t at P meet the d irectri ces a

’

t

FOCAL PROPERTIES OF THE come 107

U Let the paral le l through P to the focal ax i s meet theectr ices at ll] , M

'

ThenP ill : I’llI

’

| P U P U11d 1 11 the tr iangles USP , U

’S’P the angles at S

, S'

ar e 1 ight

es . Hence these tr iarigles ar e s im i lar and the angle SP Ungle S P U’

Hence the tangent and normal at P bi sect the angles betweenfocal d i stances .The above proof holds formally for the hyperbola as wel l asthe ell i pse (the studen t , however , i s adv ised to draw the figurethe hyperbola and compare) .I n the ell ipse the poi nts Of the focal ax i s ou ts ide the cu r veouts ide SS ’

. Hence the po i n t T w here the tangen t at Pfocal ax i s

,be i ng a poi nt on a tangent, i s ou ts ide the

outs i de SS ’. Therefore in an ell ipse the tangent

ex ternal ly and the normal b i sects i nternally the angle

the other hand in the hyperbola the poi nts of the focaltside the curve l i e i n s ide SS ’

. T i s therefore i n s ide SS ’.

Therefore i n a hyperbol a the tangent bi sects i n ternal ly and thenormal b i sects exte rnal ly the angle SP S ’

.

103 . I n te r c ep t of th e n o rm a l on th e foc a l axi s .

Let the normal at P (Fig. 37 meet the focal ax i s at 0. BecauseP G , P T bi sect SP S

’

, S , 0,S’

, T are harmon ic,00 . 0T : OS 2

If P N be the per pend i cu lar from P on the focal ax i s, PN

goes through the poi n t of con tact of the symmetr ical ta irgen tthrough T. Thus P N I s the polar of T and A

,N

,A

’

, To

a 1 e

harmon ic ,0N . 0T= 0A 2

D iv id i ng,

CG OS—e“°

or ON

Also s i nce P G b i sects the angle SPS ’

,w e have

| S G | IGS'

I S G

lb’

P I P 15"

I 151) + 158"

AA’ 8 (for e l l ipse) ,

i i .= e (for hyperbol a) .


Thus104 . T h e fe e t of foc a l p e r p e n d i cu l a r s on a t

l i e on th e a u xi l i a r y c i r c l e . Let S Y,S

’Y’

(Fig .

perpend icu lars from S ,S

’ upon the tangent at P . L

meet S Y at F.

Con s ider the case where the con ic i s an el l ipse.The angles FP Y,

SP Y ar e equal , for PP Y = S’

P Y’SP

by Ar t. 102 . Hence the angles FYP , S YP bei ng r ight anglesand YP being common the tr iangles FYP

,S YP are congr uent.

Fig. 38.

FP‘ISP and : 2 0A . Also O, Y

bei ng m id- points of SS ’, SF r espect ively, I0Y l; IFS

’

I 0A

Hence Y l i es on the aux i l i ary c i rc le. S im i l arly Y’l i es 011 theaux i l iary c i rcle . The p r oof when the con ic i s a hyperbola i spreci sely s im i lar and may be left as a n exerc ise for the student.I f Z, Z

’ be the feet of perpend icu lars upon the tangentparal le l to the tangen t at P

,Z and Z’ al so l i e on the aux i l iary

c i rcl e .

Al so by symmetr y

| SZ | = S'

Y’

I; [S’Z

’= IS Y

Hence by the pr oper ty of

segments of chords of a c i rcle .

I f the con i c be an ell ipse S Y,S’Y’

ar e drawn in the samesense and A’

S . SA i s pos i t ive and equal to OB 2

(Ar t.


PROJECTIVE G EOMETRY

The theorem of Ar t. 104 now reads :The foot Y Of the perpe nd icular from the focu s upon a n

tangent to a parabola l ies On the tangent at the vertex .

Agai n i f i n the theorem stated i n the second par t of Ar t. 9we take e to be the l i n e at i nfin i ty , then L , IV are the poi ntsi nfin i ty on a

,b and the cons tant angle LS llI i s the angle betw e

the tw o tangents . Hence S and ab subtend theL

,AI L

,III

,S , ab are concycl i c or : the c ircl e

the tr i angle formed by three tangents to a parabola passesthe focus .Al so the angl e made by the tangent with the ax i s i s

angle made wi th the ax i s by the focal d i stance SP of i tscontact. Hence if TP

,TQ be two tangents the angl e

them i s é P SQ, or by Ar t. 98 i t i s TSP or

between tw o tangents i s equal to the anglethem at the focu s .If th i s angle be a r ight angle

,then the tangents ar e tangen ts

at the extrem iti es of a focal chord and T l ies on the d i rec tr i x .

Therefore the di rectr i x i s the locus of i ntersections of perpendicu lar tangents to a parabola .

107 . P a r am ete r of p a r a l le l c h o r d s of a p a r ab o la .

We have seen (Ar t. 84) that i f QQ’ be a chord of a parabola

b i sected at N by i ts conj ugate d iameter and i f th is d iameter meetthe cu r ve at P

,then

emPN

Thi s con stant i s cal led the parameter of the chords. To findi ts value take the chord QQ

'

to pass through the focu s S (Fig.

I f the d iameter conj ugate to QQ’meet the d i rectr i x atA] , the pol e

of QQ’l i es on th i s d i ameter and

(v QQ

’passes through S ) on thed i rectri x . Thus M i s the pole o QQ

'

. Ill Q, IVI Q’ are therefore

tangents to the parabola and,M bei ng on the di rectr ix , these

tangents are at right angles. The c i rcle on QQ’ as diameter

passes through M .

| QN | =

for M i s the pol e Of QN and MN meets the cu rve at P and ati nfin i ty ; .MN i s ther efore d iv ided harmon i cal ly by P and thepoi nt at i nfin i ty

,that i s

,bi sected at P .

QN2 PN 2

_HencePN

4W

_ 4 ' PN °

a con stan t.

v 1] FOCAL PROPERT IES or THE com e 1 1 1

But

The parameter i s therefore 4 . SP .

108 . T h e or th oc e n tr e o f th e t r i a n g le for m e d b yth r e e ta n ge n ts to a p a r a b o la l i e s on th e d i r e c t r ix .

Thi s i s read i ly proved from Br ianchon’s theorem .

Let a , b, c be the three tangen ts , b’

, c’ the tw o ta ngen ts per

pendicular to l) , c, and i”

the l i ne at i nfin i ty .

Cons ider the hexagon abb'

im

c'

o. Then (ab, f c'

) (bb'

,cc

'

)( if

-i”

, ca ) ar e concu rren t .But (a b, f c

'

) i s the para l le l to 0’ through ab

,i . e . the per

pendicula r from a b on 0.

S im i larly (a c, i s the perp end i cu lar from ac on 17.

Hence the Br ianchon poi n t i s the orthocentre of thetr iangle .

But (bb'

, cc'

) i s the d i rectrix , s i nce pe r pend icular tangentsb and b', c and c

' meet on the d i rectri x .

The orthocen tre therefore l ies on the di rectri x .

I Q S . C i r c l e of c u r v a tu r e . Suppose a c i rcle to tou cha come at P and to mee t i t aga i n at Q. I f any l i ne through Qmeet the 0011 10 and c i rcle agai n a t R , 18

' respec t i ve ly a nd the

1 1 2 PROJECTIVE G EOMETRY [CH

tangent at P at T,w e have by the wel l - known pr oper ty of the

c ir cl eTP 2

TQ TR’

and by Newton’s theorem for the con icTP 2 square of sem i - d iameter of con ic paral l el to TP

TQ TR square of sem i - diameter of con ic parall el to TRRor by d iv i s ion

TR’ squ are of sem i - diameter parallel to tangen t at P

TR square of sem i - diameter par al lel to TB

Fig. 40.

I f w e make Q coi nc ide w i th P ,the c i rc le becomes the c ircl e

of cu r vature at P and T coi n c ides wi th P . Hence i f any chordthrough P (Fig . 40) be drawn to meet the con ic at R and thec i rcl e of cu r vature at R ’

,

PR' squar e of semi - d iameter paral le l to tangen t at P

If the chor d P R be dr awn through a focu s,we have


1 1 4 PROJECTIVE GEOMETRY [OH .

EXAMPLES V I A.

1 . Show that the construct ion of Ar t. 93 never leads to a real righ tcircu lar cone i f the c ircle touches the e llipse at the extremi ty of a minor

2 . On the tran sverse axi s AB of a hyperbola as d iame ter a c i rcle i sd rawn (the auxil iary c i rcle of the hyperbola) . A ray th rough A mee tsthe c i rcle and hyperbola in P , P

’. Show that the tangents at P ,

P’

meet on the tangen t at B.

3 . Prove that the focal axis of a plane sect ion of a righ t c i rcularcone i s equal to the part of any generat ing l ine i ntercepted be tween itspoi nts of con tact wi th the focal spheres , and that the perpend icularaxis i s a mean proport ional be tween the d iame ters of the focal spheres.

4 . Prove that the latus rectum of a plane sect ion of a righ t c i rcularcone is proport ional to the perpend icular d istance of the plane of

sect ion from the vertex of the cone .

5 . Prove the follow i ng construct ion for the pole of any l ine q w i thregard to a con ic , given the two foc i S , S

’and the tw o d i rectrices s

,s’.

Let g meet 3 at P ,s’at P ’

. Through S ,S’draw perpend iculars to SP

,

S’P

' respectively : these mee t at the poin t Q requ i red .

6. Tw o poi n ts of a con ic be ing given and also one of the d irectrices,show that the locus of the correspond ing focus is a c i rcle .

7. Tw o conics have a common focus. Prove that tw o of the i rcommon chords pass through the i n tersect ion of the d i rectrice s corresponding to th is common focus.

8. A variable straigh t l ine mee ts two fixed straigh t l ines at P , Qand P Q subtends a fixed angle at a fixed po in t S . Show that P Qtouches a con ic of wh ich S is a focus .

9 . Through a given poi nt P tw o conics can be drawn having tw og iven points S , H for foc i . O f the se on e is an ell ipse

,the other a

hyperbola, and they cut at r igh t angles .10. Prove that the or thogonal projection of the normal P G upon

e i ther focal d istance is constan t and equal to the semi - latus rectum.

[I f K z foot of pe rpend icular from G on SP in Fig. 37, provetriangles PKG , USP s imilar and tr iangles SKG ,

UXS s imilar ; anduse S G = e SP .]

1 1 . I f the normal at P mee t the non - focal axis at G ’

,the project ion

of P G’upon e i ther focal d istance i s equal to the semi - focal axis.

v 1] EXAMPLE S 1 1 5

12 . Prove that i f the normal at P to a central con ic mee t the focalaxis at G and the non - focal axis at G ’

, then P G '

: P G : CA2: CB?

1 3. I f the tangen t and normal at P mee t the non - focal axi s at T’and G

’,prove that S , S ’

,P

,T’

,G

’ar e concycl ic and CG ’

. CT : OS 2.

1 4. I f from any point T on the tangen t at P there be drawn perpendicular s TL and TN to SP and the d i rectrix SL TN eccentric i ty.

1 5 . I f CD be the d iame ter conjugate to CP ,show that

SP . S’P CD 2

.

16 . Show that poi nts of con tact of tangents from the foc i to theauxil iary c ircle lie on the asymptotes.

1 7. Th e focus of a con ic sl ides on a fixed l ine,the coni c i tsel f sl id ing

on a fixed perpend icular line . F i nd the locus of the cen tre .

1 8 . A rectangular piece of paper ABCD is folded so that thecorner 0 falls on the oppos ite s ide AB . Show that the crease enve lopsa parabola of wh ich 0 i s the focus and AB the d i rectrix.

1 9 . The vertex of a con stan t angle move s on a fixed straigh t l ine ,wh i le one of its s ides pas ses through a fixed poi n t S . Show that th eother s ide enve lops a parabola, of wh ich S i s a focus.

20. Show that the tw o tangents from a poi nt to a con ic ar e equallyincl ined to the focal d istances ; and converse ly that i f S , H be fixedpoi nts and through a poi nt P l ines P T, P U be drawn so that theangles S P H ,

TP U have common b isectors,a conic can be descr ibed

w i th S,H as foc i to touch P T and P U.

2 1 . A focus and th ree tangents to a conic a r e given . Construct theaxes of the con i c in pos it ion and length .

22 . F i nd the locus of the focus of a parabola pass ing through tw ofixed poi nts A, B and the d i rect ion of whose axis i s given .

23. Prove that a l ine - pai r may be looked upon as the l imi ting caseof a hyperbola when the foc i coinc ide w i th the centre .

24 . Prove that a po in t- pai r may be looked upon as the l imi ting caseof a very flat e ll ipse or hyperbola

,the foc i be ing coinc iden t w i th the

vert ices . Show that the eccentric i ty of a pointf pa i r is un i ty.

25 . Prove that the pole of the tangen t at P to a central conic w i thregard to the auxil iary c i rcle l ies on the ord i nate of P .

26. I f S Y, S Z be perpend iculars from a focus S to tangen ts TP ,

TQ ,the perpend icular from T to YZ passe s through the othe r focus S ’

27. F ind the locus of the vert ices of the conjugate parallelogramsof an e ll ipse .

1 1 6 PROJECTIVE GEOMETRY [CH .

2 8. I f P P’

,DD

’be conjugate d iame ters of a hyperbola and Q any

poi nt on the curve , show that QP 2+ QP’2 d iffers from QD 2

+ QD’2 by a

constan t quant i ty.

29 . I f TP ,TQ be tangen ts from a poi n t T to a central conic, S , H

the foc i , show that th e b isectors of the angle P TQ mee t the non - focalaxis i n two fixed points when T describes the c i rcle S TH .

30. From a poin t P on a hyperbola PN is drawn perpend icular tothe transverse axis and from N a l ine i s drawn to touch the auxi l iarycircle at T. Prove that TN : P N = rat io of semi - transve rse to semiconjugate axis.

3 1 . Show that two parabolas wh ich have a common focus and the i raxes i n oppos ite d irect ions i ntersect at right angle s.

32. P QR be ing a tr iangle c i rcumscribed to a parabola, prove thatthe perpend iculars from P QR to SP , SQ ,

SR ar e concurren t.[Use the re sult of Ar t . 1 06 that the c i rcumc i rcle of such a triangle

passes through the focus ] .

33. I f a chord QQ ’of a parabola mee t a d iame te r P V at 0

,and i f

Q V , Q’V’be ord i nates to th is d iame te r

,prove that P V . P V P 02

34 . I f from a poi nt T outside a parabola a tangen t TP and achord TQQ ’

be drawn , and i f the d iame ter through P mee t QQ’at K,

show that TQ TQ’= TE Z

.

35 . Show that the chord of curvature to a parabola at P,drawn

parallel to the axis, is 4SP .

36. I f the normal at P to a parabola mee t the d irectrix at H,then

the rad ius of curvature at P = 2 . H P .

37. Show that any poi nt of a rectangu lar hyperbola is a point oftr isect ion of the intercept of the normal at the poin t between the centreof curvature and the poi n t where the normal mee ts the curve again.

38 . Prove that th rough any poin t P of a conic,three c i rcles of

curvature of the conic pass .[Let P Q , P Q

’be chords equally i ncl ined to axes, OR the d iame ter

conjugate to P Q ’mee t ing P Q at S . P [Q]T P [Q'

] (oppos i tely equal ) ;P (conjugate ) ; CUB] . The three points othe rthan P where the conic locus of S mee ts the orig inal conic have the irc i r ch s of curvature pass ing through P ,

for at such poin ts Q, R ,S

co inc ide and the tangen t at Q i s equal ly incl ined to the axe s w i th P Q .]

39 . Show that the central chord of curvature of a con ic atP = 2 . CD 2/0P ,

CD be ing the semi - d iame te r conjugate to UP .


CHAPTER V I I .

SELF - CORRESPOND ING ELEMENTS.

1 10. P r oj e cti v e r a n ge s a n d p e n c i ls of th e s e c on d

o r de r . The poi n ts of a con ic , l i ke the poi n ts of a stra ight l i ne ,may be spoken of as formi ng a ra nge , but such a r ange i s sa id tobe of the secon d order, the l i n ear range bei ng cons idered of thefirst order.S im i larly the tangents to a con i c are sa i d to form a penc i l of

the second order .These are termed forms of the second order, and the con i c to

wh ich they belong i s ca l led the i r base .

Ra nges and penc i ls of the second order wil l be denoted bywr i ti ng 2 as an index outs ide the bracket enclos i ng the typicale lement : thus [P

ER

Tw o r anges o t e second or der ar e sa i d to be proj ecti ve( i t w i l l be shown i n Chapter V I I I that they can actual ly beproj ected i nto on e another ) i f the penc i ls wh ich they determ ineat any poi nts of thei r r espect ive bases are proj ecti ve .

S im i larly tw o penc i ls of the second order are sa id to be pr ojectiv e i f the r anges wh ich they determi ne on any tangen ts tothei r respective bases ar e proj ective .

From the known proper t i es of proj ective penci ls and r anges ofthe first order g iven i n Chapter I I i t fol lows that two correspend i ng tr iads entirely determ i n e the r elation between tw o

proj ect i ve forms of the second or der . Al so i f we define thecross - ratio of four poi n ts on a con i c as the cross - rat io of thepenc i l which they determ i ne at any poi n t of the con i c and thecross - rat io of four tangents to a con ic as the cross - rat io of therange which they dete1mine on any tangent to the con ic , thenprojecti ve forms of the second or der are equ i - anharmon ic and

conversely.

Aga i n , as in Chapter I I, two cobasal forms of the second

CH . v r r] SELF - CORRESPOND ING ELEMENTS 1 1 9

order can not have more than tw o sel f- correspond ing elementsw 1thout bemg ent i re ly comcrden t.

l l l . C r os s - axi s a n d c r os s - c e n t r e of c ob a s a l p r o

j e ct i v e fo r m s of s e c on d o r d e r . Let P ,P

’

(Fig . 4 1 ) betw o correspond i ng poi n ts of tw o project i ve ranges [PR [P ly i ng011 the same con i c 8 .

Let A , A'

be any gi ven correspond ing po i n ts of these ranges .Projec t the range [P

'

]2 from A as ver tex , [P ]

2 from A ' as ve r tex .

The penc i l s A P'

] and A'

[P ] a r e proj ec t i ve and they havea sel f- cor r espon ing ray A

’

A . Hence they are perspective andrays AP '

,A

'

P meet at U 011 a fixed ax i s .r .

Thi s ax i s a: i s i ndependent of the choi ce of the poi nts A, A

'

Fig. 4 1 .

For let B,B

’ be any other pai r of correspond i ng poi n ts . Thenby the prev iou s resu l t AB ’

,A

'

B meet at V on x. Now con

s ider the Pascal hexagon We have (A'

B ,AB

'

)(PB

'

, P'

B ) a r e coll i near. a; i s therefore the Pascall i ne and P B '

, P’

B mee t a t IV 011 .r . The same l i ne a; i s therefore reached if we star t from A a nd A '

,or i f we start from

B and B '

.

There i s thus a fixed l i n e .1’on w hich mee t the cross - joi n s

AB'

, A'

B of any tw o correspondi ng pa i rs . Thi s w e shall cal l , asi n the case of l i near ranges , the cr oss - mz'is .

By rec i proca t ion , or by proceed ing i n a man ner s im i lar to theabove and us i ng Br ianchon’s theorem ,

w e reach the resul t thattwo proj ecti ve penc i ls of tangen ts to the same con i c have a cr oss

1 20 PROJECTIVE GEOMETRY [CH .

c entr e, thr ough which pass the joi ns of cr oss - meets (ab'

,a

'

b) ofcorrespond i ng pai rs .

1 12 . S e l f - c o r r e sp on d i n g e l em e n ts of c ob a s a l p r o

j e c t i v e for m s o f s e con d o r d e r . As i n the case of forms ofthe first order, tw o cobasal project iv e forms of the second ordercannot have more than two se l f- correspond ing elemen ts ; for i fthey have three, say A , B , 0 and i f P ,

P’

be any other pa i r of

correspond ing elements, {AB CP } {ABOP'

}and as in Ar t. 25

P’P .

These selfcorrespond i ng elements may be constructed asfol low s . I f the cross - ax i s of two proj ective ranges of the secondor der [P ] 2, [P

’

]2 ly ing on the same con ic 3 meet 3 at poi n ts

S, T (Fig . 4 1 ) the poi n ts S , T are sel f- cor r espond ing poi n ts of

the ranges [P ]2

,

For by the property of the cr oss - axi s AT, A’T meet 3 at

a pa ir of cor r espond i ng poi n ts . But they both meet 3 at T.

Hen ce a pa i r of correspond i ng t T, or T i ssel f- correspond i ng . S im i larly S i sI f the cross - ax i s 5c i s i tsel f a

spend i ng po i n ts S , T coi nc ide . I f xpoi n ts, there are no real se l f- correspReciprocati ng, we have the

l i nes of tw o proj ecti ve penc i ls of the second order bethe same con i c are the tangen ts from the cross - centre . There aretw o r eal sel f- correspond i ng l i nes i f the cross - centre i s ou ts ide thecon ic these coin c ide i f the cr oss - cen t

pre ss - centr e be i n s ide the con ic there are no realmes.

1 13 . Tw o c or r e sp on d i n g e l em e n ts c ob a s a l

p r oj e c t i v e for m s d e te r m i n e w i th th e se l f - cor r e sp on d

i n g e l em e n ts a c on sta n t c r os s - r a t i o . I t w i l l be suffic i entto prove th i s for two proj ec t i ve r anges 011 the same con ic , s i nceal l other cases can clearly be made to depend upon thi s. Now

fr om Fig. 4 1 , i f AP’

, A'

P be tw o chords meeti ng 011 ST,then

P , P’ are corr espond i ng poi nts of the ranges determ i ned by

the tr iads S ,T, A S , T, A

’. 011 the other hand i t i s obv ious

from symmetry that A ’

,P

’

a r e correspond i ng poi n ts i n the rangesdeterm i ned by the tr i ads S , T, A S

,T, P . Hence the cross

r at io of the fou r poi nts STAA ’ i s equ al to the cross - ratio of thefou r poi nts STPP ’

,which proves the theorem .

1 14 . C on st r u c t i on of s e l f - cor r e sp on d i n g p o i n ts .

The r esu lts of -Ar t. 1 1 2 prov ide u s with a construction for deter


1 22 PROJECTIVE GEOMETRY

1 1 6 . D i r e c t i on s of a sym p tote s of a c on i c g i

b y fi v e p o i n ts . I f i n the constru ction of the preced i ngAr ti cle the l in e a be the l i ne at i nfin i ty A I B IOI , i 4n 02are at i nfin ity. Proj ect i ng from 0 we have tw oproj ect i ve penc i ls hav i ng 0 for a common vertex . The rays0A; 002

”ar e paral lel to G

'

A g

m

, UP i .e. to

O’

A , O'

B , 0 0. Thu s the penci l i s ob ta i ned bydrawi ng through 0 paral lels to the rays of Fi ndthe self- correspondi ng rays of the penc i l O (A Q

Q

B Q

Q

CZQ

) and thepenci l 0 (ABC) these rays lead to the poi nts S where thel in e at i nfin i ty meets the con i c . They g ive therefore the d irect ions

Fig. 42 .

of the asymptotes . The asymptotes are then constructed i npos i t ion by the method of Ar t. 76.

1 17 . C on st r u ct i on of th e p a r a b o la s th r ou gh fou r

g iv e n p o i n ts . Let 0, A , B be the fou r g i ven poi nts (Fig .

Through 0 draw any ci rcle meeti ng 0A , GB at A ,B , a ndthe ar allels through 0 to GA ,

G B at A 2B2 .

Then i f P i s any poi n t 011 the parabola and P P 2 a r e

the poi n ts where OP and the paral le l through 0 to O'

P meetthe c i rcle

, [P 1]2

, [P 2]2 are tw o projecti ve ranges 011 the c i rcle

whose selfcorrespond ing poi nts ar e the poi n ts correspond ing to

v r r] SELF - CORRESPOND ING ELEMENTS 1 23

the poi nts at i nfin i ty on the cu r ve, s i nce when P i s at i nfin i tyOP , O

’P are paral le l .

In the case of the parabola the poi n ts at i nfin ity are coi n c iden tbeca use the l i n e at i nfin i ty touches the curve . Hence the se lfcorrespond i ng poi n ts of the ranges [PH] , [P a r e coi nc ident orthe cross ax1s touches the c i rcle (Ar t.But we know one po i n t on the cross - ax is , namely the poi n t U

where A l B2 meets The cross - ax i s i s therefore ei ther of

the tw o ta ngents from U to the c i rcle . The joi n of O and thepoi nt of con tact of the cross - ax is w i th the c i rcl e g ive the d i rec t ionof the poi n t at i nfin i ty on the parabola

,or the di rect ion of the

axi s . Hav i ng the di rection of the ax i s and four poi nts 011 thecurve we may construct the parabola by the me thod of Ar t. 79 , or

more d i rec tly as fol lows . Take any poi n t Q on the cross - ax is .Joi n QB2 mee t i ng the ci rcl e at P QBl meeting the c ircl e atP .. The paralle l through 0'

to OP 2 meets OP I at a poi nt P011 the parabola .

S i nce two tangents can be drawn to a c ircle from U,the

problem 1s i n general capable of tw o solu tions . These sol utionsare coinciden t

b

i f U be 011 the c i rcle . I n th i s case e i ther A I a ndB 1 (or A . a nd B.) coi n c ide , that i s, three of the given poi nts arecol l i n ear and the con ic then degener ates i n to tw o paral le l stra igh tl i n es , wh ich 1s a spec i al case of a

.par abola , or el se Al and A. (orB 1 air d B ) coi nc ide ,

A (or B ) 1s then at i nfin i ty, so that threepoi n ts and the d irec t ion of the ax is ar e gi ven a nd the parabolacan be draw n by Pascal’s theorem . I f ,.U be w i th in the c i rclethere are no r eal solut ions to the problem .

1 1 8 . R e cta n gu la r h y p e r b ola th r ou gh fou r p o i n ts .

C a s e of fa i lu r e . The same pri nc iple wi ll enable u s to con

s truct the rec tangular hyperbola through fou r gi ven poi nts 0 O'

,

A , B . Draw a c i rcle through O and find the poi n ts A I , B

B. and the poi n t U on the cross - axi s by the same construct ion asbefore . Now s i nce i n the rectangular hyperbol a the asymptotesare to be at r igh t angles the se lf correspond ing rays of the penc i ls

O [P .]b

are at r igh t angles,that 1s

,they mee t the c i r cle at

the ex trem i ties of a d iame ter ; or the cross ax i s of [P , i] i sa d iame ter . Hence joi n U to the cen tre of the c ircle , and wehave the cros s - ax i s requ i r ed . The joi n s of O to i ts i n tersec t ionswi th the c i rcle gi ve the di rect ions of the asy mptotes . Hav i ngthese and fou r poi n ts on the cu rve w e can constr uc t the cu r ve byAr t. 76 or d i rec tly from the present corrstr uctiou as expla i ned 1 11

the last art icle .

I f U be at the centre of the c i rcle , any diameter may be

1 24 PROJECTIVE G EOMETRY [ca

taken as the cr oss—ax i s and an i nfin ite n umber of rectangularhyperbolas may be drawn through the fou r poin ts . I n th i s caseA 2 8 1 , A 1 8 2 be i ng d iameters, OA ,

OB a r e perpend icu lar to OB 2 ,

0A., that i s , to O'

B, O

'

A ,or O i s the orthocen tre of the tr iangle

O'

AB . I t i s easy to prove that when thi s i s so any one of thefour given poi n ts i s the orthocentre of the triangle formed bythe other three .

1 1 9 . T a n ge n ts fr om a n y p o i n t to a c on i c giv e n

b y fi v e ta n ge n ts . Let t, t’

,a, b, c be five tangents to a con i c .

Let A B, 0 be the poi n ts where t meets a ,

b, c,

A , B’

, 0'

t'

a.b. a

Let 0 be any poi n t i n the plane .

If p be any tangent to the con i c meeti ng t at P and t’ at P ’

The r anges [P ] , [P’

] are proj ect i ve : hence the penc i ls 0[P ] ,O [P

’

] are proj ect i ve .

I f 19 passes through 0, OP and OP' are coi n c ident.

Therefore the tangents to the con ic thr ough 0 ar e the sel fcor r esporrding rays of the penc i ls 0[P ],Determi ne these se lfcorrespond i ng rays fr om the triads

O (ABC) , by the me thod of Ar t . 1 1 4 and these gi vethe tangents requ i re

EXAMPLES V I I A.

1 . A con ic passes through fi ve points 0,A , B, C . Show how

to con struct graph ically i ts i n tersect ions w i th any c i rcle through 0, 0’

w ithout draw ing the conic. Prove that the conrmon chord not pas s ingthrough 00’ i s always real , even when the c i rcle does not mee t thecon ic agai n in real points.

2 . A con ic is given by fiv e po ints. W i thou t draw ing the curve finda test to determine wh e ther i t i s an ellipse

,hyperbola or parabola.

3 . Prove that tw o con ics can be d rawn through four given po in tssuch that the i r asymptote s make an angle a w i th one another : and

show how to construct them.

[ In the construct ion ofAr t. 1 17 the cross - axis must cut off a constan tar e from the c i rcle through 0 and therefore touches a c i rcle concen tricw i th th is c i rcle ]

4 . I nve st igate the nature of the s imple quadrilate ral formed byfour poin ts i f i t i s imposs ible to draw a real parabola through them.

5 . D i rectly equal range s on a c i rcle may be defined as range s inwh ich tw o d i rectly equal penc i ls whose vertices ar e 011 the c i rcle mee tthe c i rcle . Show that the cross - axi s of two such ranges is at i nfin i ty.


CHAPTER V I I I .

IMAG INAR IES AND HOMOGRAPHY.

1 20. P o i n t a n d l i n e coor d i n a te s i n a p la n e . Thepos i t ion of a poi n t P i n a plan e may be defined by tw o coor din

ates x, y gi ven by the i ntercepts cut off, on two fixed axes,

between the ir i ntersect ion or origi n and the paral lel s through Pto the axes . I n th i s system of coord i nates the coor di nates of thepoi n ts of any stra ight l i n e sat i sfy an equation of the first degr ee

Ax By O 0.

If w e d iv ide th i s equat ion by 0 i t takes the formla: my 1 0.

A stra ight l i n e i s therefore completely defined when w e k now thetw o coeffic i ents 1, m . These may then be spoken of as thecoord i nates of the l i n e .

The coord i nates of the poi n ts of a curve sati sfy a r elat ion wh ichi s cal led the Cartes ian equation of the curve .

I n l ike man ner the coordi nates of the tangents to a curvesati sfy a r elat ion whi ch i s cal led the tangential equat ion of thecurve .

If the coord i nates l, m of a l i ne sati sfy a relation of the firstdegr ee, th i s can be put into the form

la + mb+

and th is shows that the l ine whose coordi nates are l,m passes

thr ough the poin t whose coord i nates ar e a, I) .

An equa t ion of the first degree i n l, m i s ther efore thetangenti al equation of a poi n t and the l i nes whose coordi natessati sfy th i s equation are rays of a penc i l .I f l : 0

,m =0, x or y or both must be i nfin i te i f la: my i s to

be equal to the fi n i te quanti ty 1 . Hence I 0,m 0 a r e the

coordi nates of the l i n e at infin i ty. S im ilarly i f a) : 0, y=0 the

l i nes thr ough the or ig in must have l or m or both i nfin i te.

CH . v r r r] IMAG INAR IES AND HOMOG RAPHY 1 27

Not ice the dual i ty impl i ed by th i s arrangement of poi n t andl i n e coord i nates . By giv i ng the symbol s a d i fferent i nterpre tat ion and taki ng 1

,m as coord i na tes of a po i n t

,.r, y as coordi nates

of a l i n e and bear i ng i n mi nd the symmetry of the relat ion of

i nc idence my 1 0 i n .r, y and l, m respect i ve ly, we see that

to any geome tr ical theorem corresponds another i n wh ich poi ntsand l i nes ar e i nterchanged . Thi s i s the pri nc ipl e of du al i ty wh ichw e have already deduced from the theory of rec iprocal pol ars inAr t. 65 . The presen t resu lt show s that th i s pr i nc ipl e i s enti relyi ndependen t of the theory of rec iprocal polars .

12 1 . P o i n t a n d p la n e c oo r d i n a te s i n sp a c e . I nl ike manner the pos i t ion of a poi n t P i n space may be defin ed bytak i ng three axes OPP , OY, OZ through an or igi n 0 and drawi ng through 0 planes paral lel to YOZ , Z OX, XOY to meetOA

’

, OY, OZ respect i vely at L ,A]

, N . Then the segmen ts OL ,

Oi l] , ON taken wi th proper s ign a r e denoted by x, y , z and

cal led the coord i nates of the po i n t . I t i s shown i n treat i ses orr

analy t i ca l geometry (see Salmon , G eometr y of TIzr ee D imen

s ions , or C. Smith , Solid G eometr y ) that i n thi s system of coord i nates a plane i s represented by an equation of the fir st degr eei n the coordi nates wh ich may be put i nto the form

and conversely that every su ch equation defines a plan e .

( I, m ,72) may be cal led the coord i nates of the plane and the

above equat ion expresses that the plane (1, m,n ) and the poi n t

(.c ,1, z ) ar e i nc i dent.The coord i nates of a poi nt on a surface sati sfy a s i ngle

equat ion i n .c, y , z wh ich i s cal led the Car tes i an equat ion of the

su rface.The coordi nates of a plan e tangent to a surface sat i sfy a

s i ngl e equat ion i n l, m ,n wh ich i s cal led the ta ngent ial equation

of the su rface .

The equa t ion ( 1 ) expresses , when .r, y , z are treated as con

stan ts and l, m, n as var i ables , that the coord i nates of the planespass i ng through y , z sati sfy the equat ion ( 1 ) of the firstdegree i n l, m , n .

Conversely such an equat ion of the firs t degree i n l,m

,n

represents a set of planes through a po i n t . Such a set of planesi s cal led a sheaf of planes and the poi n t through which theypass i s cal led the vertex of the sheaf.

An equa t ion of the fi rs t degr ee i n l,m

, n i s therefore thetangent ial equat ion of a poin t .

1 28 PROJECTIVE G EOMETRY [OH .

As in Ar t:1 20, l = o, m = 0, n = 0 ar e the coord i nates of the

plan e at r rrfirn ty , whereas a : 0, y 0, z 0 correspond to i nfin i te

plane - coord i nates .12 2 . P r i n c ip le of D u a l i ty i n sp a c e . The symmetr ical

form of the equat ion

impl i es that i f the poin t (a', y , z ) a nd the plan e ( I, m,n ) are

in c ident, so are the plane (x, y , z ) and the poi n t l, m ,

Thus to any theorem con nect i ng po i n ts and planes, t ere cor

responds a r ec iprocal theorem con nect i ng planes and po in ts ,obta i n ed from the first by i n terchangi ng the i nterpreta tion s ofx, y ,

z and l,m

,n . I n th i s tran slation the joi n of tw o poin ts

corresponds to the meet of tw o plan es . Hence a stra ight l i necorresponds to a straight l i n e. To the set of l i nes through apoi nt, wh ich i s cal led a sizeaf of lines, corr esponds the set of

l i nes i n a plan e, w hich i s cal l ed a p la ne of l ines . To a sheafof planes through a point corresponds the set of poi n ts of a plan e ,which i s cal led a p la ne of poin ts . To a range of points on al i n e corresponds a set of planes through a l i ne or axis, which i scal led an Axia l P encil. To a set of l i nes through a poi n t andly i ng in a plan e (a flat penc i l) corresponds a set of l i nes lyi ng i na plan e and passr ng through a poi n t (another flat penci l) . Toa poi nt on a surfa ce corresponds a tangen t plan e to the cor

r espondi ng surface . To the tangent plan e at a poi n t correspondsthe po i n t of con ta ct of a tangen t plan e .

To the poi n ts where a stra ight l i n e cuts a surface correspond thetangent planes drawn through a l i n e to the correspondi ng surface.The degree of a surface be i ng defined as the number of poin ts

i n w hi ch i t i s cut by any l ine and the class of a su rface as then umber of tangent planes wh i ch can be drawn to i t through anyl ine

,i t fol lows that the degree of a surface i s equal to the class

of i ts reciprocal surface .

1 2 3 . C r os s - r a t i o of a n a xi a l p e n c i l . A11 ax ial penc i lof four plan es a , ,8 , y , 8 thr ough a l i n e x, has a defin i te cross - rat io.

For cut i t by any tw o stra ight l ines a , , a 2 . These meet aBySi n r anges respec t i vely. On a; take twopoi n ts V V, . The planes a , V, , a .V2 meet in a l i ne «

a s w h ichcu ts aByS i n a rang e A3B 303D 3 . Then the rangesA

,.B , O3D 3 a r e perspect i ve from V, ; and the ranges A ,B , On ,

are perspective from V , . Hence w e have{A ,B , O, {A ,,B 3O3

{A ,BaOs } {A 2B 2020 2}.



the sol u t ion of certa i n algebra i c equat ions,invol v ing the data.

I f by alter ing the n umeri cal values of these data, Wi thou t al teri ng thei r nature, these e lements d i sappear fr om the geometr icaltheorem, they wi l l not d i sappear from the algebra i c theorem ,

foran algebra i c equat ion con t i n u es to have solu t ions

,even when i ts

constants are such that these sol ut ions are not real . The algebra i e sol u tion wi l l therefore sti l l g i ve v alues for the coord i natesof those el emen ts whi ch have d isappeared fr om the geometr icalsolu tion , but these coord i nates w i l l be complex , that i s of theform a + r

°

b,where i : — 1 and a

,b are real . The poi nts,

stra ight l i nes or plan es defined by such coordi nates have no

v isual ex i stence ; n evertheless all ana lyt i ca l theorems r emai ntr ue of them and ther efore al l geome tr i cal operat ions , whichare i nter pr etable by means of analys i s, wi l l conti nue to hold forsuch imagi nary e l ements . And th i s i s tru e not only of poi nts,stra ight l in es and planes, but of all cu rves and surfaces of h igherdegr ee .

Thus the locu sa“?

y2

a

i s not a r eal c i rcle : n evertheless i t possesses , analytical ly, a ll

the proper ti es of a c ircle and, i f w e admi t imagi nary el ements,w e may perform with i t the Oper at ions whi ch w e can per formwith an ordi nary c i rcl e .We wil l therefore

,fr om thi s poi n t onwards, assume the

ex isten ce of such imagi nary elements , so that i f a con struct ionwhi ch leads to certa i n elements in one case fa i ls to l ead geometr i cal ly to such e lemen ts i n another case, w e shal l say thatthose elemen ts ar e still there, but are imagi nary.

Thu s w e know that tw o proj ect ive col l i near ranges wi l lgenerally have two self- corr espond i ng po i nts . Thi s shows thatthe problem of determ i n i ng the self- correspondi ng poi n ts of twosuch ranges i s analyti cal ly capable of two sol utions . Hence i tw i l l have tw o analyti cal sol utions i n a ll cases . We shal l thensay that tw o such ranges have a lw ays two selfcorrespond i ngpoi nts , but that these may be r ea l or imagina r y .

I n the same w ay a stra ight l i n e wi l l be conce ived as alwayscu tting a con i c at two poi n ts, r eal or imagi nary ; and from apoi nt

.

two tangents , r eal or imagi nar y, can always be drawn toa conrc.

Agai n w e know that, i n gener al , tw o d i sti nct con ics w i l lin tersect in fou r poi nts . The pr oblem of find ing the i n tersections of tw o con i cs has therefore fou r analyti cal sol u tions .We shal l say that i t has always fou r geometr i cal solu tions, that

v r r r] IMAG INAR IES AND HOMOG RAPHY 1 31

i s , every two con ics have fou r poi nts of i ntersection , r eal orimagi nary.

The student may obj ect that th i s i ntroduction of imagi naryelements i s real ly

,from the geometr ical oi nt of v iew

,a mere

verbal delus ion , for i n what way can we d)

er i ve help i n practic efrom a construc t ion i n wh ich on e or more steps are imagi nary ?The answer i s that these imagi nary elements can not, i ndeed , beu sed i n dr awi ng - board constr uctions

,but i t may

,and does ,

happen that a demonstr a tion , involvi ng such imagi nary el emen ts ,l eads to a resu lt wh ich i s fr ee fr om them . Thu s by means ofimagi nary poi n ts and l ines w e can obta i n r eal theorems , prec i selyas we ca n

,by means of poi n ts and l i n es at i nfin i ty, obta i n

theorems rela t i ng to figures at a fin i te d istance.

I t i s tru e tha t i n al l cases such theorems m ight be obtai nedby r eason i ng wi th pu rely real el ements . But such proofs are oftenexceed i ngly compl i cated ; also tw o theorems whi ch , when we useimagi nary elements

,are only part icular cases of the same theor em ,

r equ ire, i f we restr ict oursel ves to real el ements, proofs wh ich ar e

not i n frequen tly qu i te d i ss im i lar . The s impl i c i ty and un i tyobta i n ed by the i ntroduct ion of imagi nar y e l ements add verygr eatly i n power to the methods of geometry.

12 5 . C onj u ga te Im a g i n a r i e s . If the coordi nates of anelement ar e of the form a to, the element whose coord i nates ar eobta i n ed fr om those of the first by changi ng the Sign of z

' i s sa i dto be a conjugate imagi nary to the first element .Thus the poi n t (0, — z

'

,1 i ) i s the conj ugate imagi nary poin t

to (0, i , 1 — i ) .I f tw o elements are i nc ident , thei r conjugate imagi nary

elements are al so i n c iden t .For any equation i nvolv i ng imagi nar ies may be reduced to

the form U+ i V ==0, where U and V are real . We have therefore U= 0, V =0, and therefore U — i V : 0, that i s , the equationobta i ned by changi ng the Sign of z

'

every where i s also sat i sfied.

I t fol lows s im i larly that i f a real and an imagi nary elemen ta r e i nc iden t , the real elemen t and the conj ugate imagi nar yelemen t a r e also i nc ident . For a real elemen t may be lookedupon as i ts ow n conj ugate imagi nary .

I f an element A of any na ture i s determ i ned by two otherelements P , Q (poi n ts, plan es or i ntersecti ng l in es) , i ts conj ugateimagi nary element A ' i s determ i ned by the conj ugate imagi naryelements P ’

, Q'

. For s i nce A , P are in c iden t A’

, P’ are

i nc ident ' and s i nce A , Q are i nc iden t A'

, Q' are i nc iden t .

HenceA I n part icular Q’z P or

9— 2

1 32 PROJECT IV E G EOMETRY [CPL

Hence the element ( i f any) determi ned by tw o conj ugate imagi naryelements rs always real .I n par ti cular the j oi n of tw o conj ugate poi nts or the meet of

two conj ugate planes are rea l l i nes . Two conj ugate l in es wh ichi n tersect determi n e a real poi n t of i n tersect ion and a r eal plan e .The elemen ts determ i ned by a real el emen t A and two con

j ugate imagi nary elements P , P'

ar e conj ugate imagi nary. For Abe i ng i ts ow n conj ugate imagi nar y, AP i s the conj ugate imaginar yto AP

’

.

Al so, i f S be any locu s or envelope which i s r eal or i nto whoseanalyt ical equat ion on ly real coeffic i ents enter, and P be anyimagi nary elemen t inc iden t w ith S ( i .e . ly i ng on or tangent to S ) ,the r elation of inc idence i s expressed by an equa t ion

U i V : O.

Thi s impl i es U i V : 0.

Bu t the latter i s what we obta i n i f we change the Sign of z' i n

the coordi nates of P , s i nce the coeffic i en ts of the equation for Sdo n ot con tai n 2. Hence P ' i s also i n c ident w ith S .

I t fol lows that i f tw o real loc i have on e imag i nar y i nter sect ion P

,the conj ugate imagi nary poi n t P’ i s also an i n tersection ,

s i nce i t mu st l i e on both curves The correspond ing chord P P '

,

bei ng determi ned by two conj ugate elements,i s real .

12 6 . Num b e r of r e a l e l em e n ts i n c i d e n t w i th a n

im a g i n a r y e l em e n t . An imagi nary poi n t has on ly one reall i ne through i t, namely the one joi n i ng i t to i ts conj ugateimagi n ary poi n t. For i f i t had two i t wou ld be the i ntersec t ionof two real li n es and therefore a real poi nt.Sim i larly an imagi nary plane has on ly one real l i ne ly i ng i n i t,

n amely i ts in tersection wi th i ts conj ugate imagi nary plan e . Fora plane through tw o r eal l i nes i s a real plan e .

An imagi nary l i ne,for the same reason , cannot have tw o real

po i n ts on i t . But imagi nary l i nes may be of tw o ki nds . A l i n eof the first k i nd has one real poi nt 011 i t. A l i n e of the secondk i nd has no r eal poi n t on i t.By the last Ar ti cle the conj ugate imagi nar y l i n e p

’to a l i n e p

of the first k i nd passes through the real po i n t on p . p , p’therefore

i ntersect and,be i ng conju a te , determ i ne a real plane .

Thus a l ine of the first i nd has one real plan e pass ing tlr r oi t . I t cannot have a second, for i t would then be the meet ofr eal pla nes and so be a real l i ne .

Conver sely, i f an imagi nar y l i n e 1) has one r eal plane pas

rm]


1 34 PROJECTIVE G EOME'

I‘

RY [OH.

u ; and a ,’x2

’

,.r ,

’ the x s of the four correspondi ng poin tsP I 1 P 2 ) P S , P 4

ThenP 3

’P 4

'

(at;P 3

'

P 2

'

(a s’

D Bx, D

A rc. 0 Am, O

(AD — B0) (c . a x)(Ax2 O) (Ax, 0)

Hence(w" w. me

’— a s

’

) (c'

.— x1) (ma— xa)

’

the other factors al l cancel l i ng . Therefor e

or the cross - r at io of four poin ts of a r ange i s equal to the crossr at io of the fou r cor responding poi n ts of a homograph ic r ange

.

I t fol lows from the above that homograph i c ranges are pr oj ect i ve. For gi ven tw o homograph ic ranges construct tw o proj ect i ver anges hav i ng tw o correspond i ng tr iads the same as in the tw ohomogr aph i c ranges . Then s i nce both the proj ective and thehomograph i c relat ion are equ i a nharmon ic, to any fou rth poi n t ofon e range w i l l cor r espond the same fourth poi n t of the other

,

whether pr oj ecti vely or homogr aph ically . The two given homograph ic r anges are therefor e pr oj ective r anges .

12 9 . H om og r a p h i c fl a t p e n c i l s . I f the r ays of tw oflat penc i l s are con nected by a on e - one correspondence such thati f m be any parameter i n terms of which the coordi n ates of anyr ay of one penc i l can be expressed l i n early (and , conversely, whichi s un i qu ely determ i ned when th i s ray i s g i ven ) , and i f m

’ be as im i lar par ameter for the other penc i l , then m and m

' are re latedby a rat ion al a lgebra i c equation : then th i s equat ion wi l l be ofthe form

and the tw o flat penc i ls are sa id to be homograph ic .Usual ly m,

m’ar e the tangen ts of the angles made by the

r ays w i th fixed l i nes in the planes of the penc i ls .I t rs clear that the r anges i n wh ich tw o such penc i ls wi l l out

any transver sals u,u’ are l ikew i se homograph ic . For the d i s

tan ces a', a“of the poi nts of sect ion measu r ed along a

,u’ar e

conn ected w ith m, mm’

(and ther efore wi th each other ) by r ational

v r r r] IMAG INAR IES AND HOMOGRAPHY 1 35

algebra i c relat ion s , and al so the correspondence between a,a

"

i s seen to be one - one.

S i nce these homograph ic ranges a r e equ i - anharmon ic and pr ojectiv e, the tw o homographi c penci ls which stand on these rangesare also equ i - anharmon i c and proj ect i ve .

Conversely proj ec t i ve penc i ls are homograph ic , s i nce thecorrespondence between the rays i s one - one and the relat ionbetween the coord i nates of correspond i ng rays must clearly beboth algebra i c and rat ional .

130. H om og r ap h i c axi a l p en c i ls . In l i ke ma nnertwo axial penc i ls whose planes correspond u n iquely whi l e thecoordi nates of correspond i ng planes are conn ected by an algebra i crelation are sa id to be homograph ic .

The flat penc i ls i n wh i ch two homograph i c ax ial penc i l s meetany tw o gi ven planes ar e themsel ves homograph i c and proj ect i ve .

The ranges i n wh i ch two homograph i c ax ial penc i ls ar e metby any tw o given stra ight l i nes a r e homographi c and proj ecti ve .

Note that we can not use the term proj ect i ve of homograph i cax i al penc i ls , s i nce these are n ot plan e forms and cannot thereforebe proj ected i nto one another .Tw o homograph ic axial penc i ls are ent irely determined by

tw o correspondi ng tr iads . For take two stra ight l i nes meeti ngthe axial penc i ls i n projec t i ve ranges, tw o correspond i ng triads ofp lanes of the axial penc i ls determ i ne on the l i nes tw o correspondi ng triads of poi nts of the ranges . These determi ne the relat ionbetw e

l

en the ranges and therefore the relat ion between the axi alpener s .Notice that i f tw o homographi c axial penc i l s have a common

axi s they have tw o self- correspondi ng planes, which cor respondto the two self- corr espondi ng po i nts of the proj ect i ve r anges i nwh ich the axi al penc i ls are cu t by any stra ight l i n e .

1 3 1 . H om og r a p h i c u n l i k e for m s . I f there be a oneone algebra i c correspondence between the rays of a flat penc i la nd the poi n ts of a range

,the two forms wi ll st i l l be spoken of

as homograph ic .

S imi larly a range and an axi al penc i l, or an axi al penc i l and

a flat penc i l,may be homograph ic .

Clearly from tw o u nl ike homographi c forms may be der i ved,by proj ec t ion or sec t ion , tw o l ike homogr aph i c forms .A particular case of homographi c u n l i ke forms i s furn i shed by

the pr i nc iple of dual i ty,the correspondence between any element

and i ts rec iprocal bei ng obv iously one - one and algebra ic .

1 36 PROJECTIVE GEOMETR Y [e rr

13 2 . H om og r ap h i c r a n ge s a n d p e n c i ls of th e

s e c on d o r de r . I f there be a one one correspondence betweenthe elements of two forms of the second or der (ranges or penci ls)whi ch i s express ible by an algebra i c rel ation be tween the co

ord i nates of the e lements the forms ar e sa id to be lronrogr aph ic.

A form of the second or der may also be homogr aph i c w i th a formof the first order .I t i s easy to show that i f the forms of the second or der are

both r anges,or both penc i l s, such homograph i c forms are pr o

jectiv e forms of the second order as defined i n Ar t. 1 10.

For example,i f w e join two homogr aph i c ranges [P 2]

2

to ver t i ces O,S ly i ng on the i r respect ive bases

,w e ob ta in tw o

penc i ls r elated by a on e one algebra i c corr espondence . Thesepenc i l s are accord i ngly homograph i c and proj ecti ve and ther anges [P 2]

2ar e proj ect i ve .

1 3 3 . G e om et r i c a l e v i d e n c e o f H om og r a p h y . I tmay be asked : when may w e assert, fr om purely geome tr i calev idence , that the correspondence between tw o forms i s homographic ? For i f we had to have recou rse to ana lys i s every t imei n order to apply the tes t whether the con nec ti ng relation i s ofthe homogr aphi c type, the labour of calcu la t ion wou ld i n manycases be con s i derable

,and the pr i nc iple wou ld be of l i ttl e val u e

in pure geometry.

We shal l therefore suppose that our attention i s to be con

fi ned to what are called algebrai c curves or su rfaces,that i s

,

curves or surfaces whose equat ions are rational and i ntegral i nthe coord in ates . The cond i tions (a ) that a poin t shal l l i e 011

such a locu s, (b) that a straight l i n e or plane shal l touch such

an en velope, are rational i ntegral algebra i c in the coord i nates ofthe poi n t , l i n e, or plane . Therefore i f a correspondence beestabl i shed by means of the fol low i ng processes ( 1 ) tak i ng joi n sof poi nts or meets of planes, or planes through po i nts and l i n esor meets of planes and l ines (2 ) find ing i ntersect ions of algebra i ccu rves or su rfaces w i th straight l i nes or wi th other algebra i ccu r ves or surfaces (3) drawing tangent l i nes or planes to suchalgebra i c curves or surfaces

,or findi ng poi n ts of con tact of

such tangent l i nes or planes (note that this i ncl udes find i ngcommon tangents to two curves or su rfaces and also constructi ng polars) : at each step, p r ov ided w e n ow/zer e intr oduce a n

a r bi tr a r y r estr iction on ou r clioice of a lter na tives, an algebra i ccond it ion i s brough t i n , which i s r at ional and i ntegral . I n theprocess of el im i nation no r ad ical s and no transcendental functionscan be i ntroduced (for the complete elirn i rrarrt of two algebra i c


1 38 PROJECTIVE GEOMETRY [CE

i s a fixed point on the con ic, are not homograph ic . For the g i vencond it ion i s equ i valent geometr ical ly to stat i ng that B i s thei ntersect ion with the con i c of a c i rcle of fixed radi u s and cen treA . Thi s c i rcl e has four i ntersections wi th the con ic, any one of

which may be taken for B . Therefore to one r ay OA shou ldcorrespond fou r r ays OB , and i t i s on ly by an a r bitr a ry con

v ention (to secure conti nu ity of sl id i ng motion) that th is n umberis r educed to u n ity.

Note tha t th i s does not hold i f AB sl ides on a fixed ci r cle 3 .

For then we may r estate the problem as fol lows, s ince AB subtends a fixed angle at the c i rcumference . Take a second fixedc i rcle 3’equal to the first. I n i t place a fixed chord ED equalto AB . G iven any pos i t ion of OA ,

draw E Q paral le l to OA to

meet 3’at Q and OB i s then paral lel to QB . The cor r espondencei s now cl early one - on e .

The above w i ll g ive the r eader some not ion of the l im itsw i th i n wh i ch the appl i cat ion of the pr i ncipl e of on e - one cor respondence i s val id, but rapid i ty and certa i nty in r ecogn izingthese geometr i cal ly w i l l be best en sured by the con s ider ation of

examples .

134 . E v e r y cu r v e of th e s e c on d d eg r e e i s a c on i c .

For let 0, 0’ be tw o poi nts on a cu rve of the second degree .

Dr aw any ray OP through 0 : i t meets the curve at one otherpoi n t P ,

s i nce 0 i s already on the cu rve . Jo in O’P . Then i fw e start from OP , O

’P i s u n iquely determ i ned . Conversely

i f we start fr om O'

P,Si nce 0’ i s already on the curve, O

'

P meetsthe curve aga in at on e poi nt only, hence OP is u n ique lydetermined. 0 [P ] , O

'

[P ] are therefore homograph ic penc i ls .Hen ce they ar e proj ect ive . Therefore by Ar t. 4 1 the locus of Pi s a con ic .I n l ike mann er w e can show that every plane curve of the

second class i s a con ic . For l et t, t’ be tw o ta ngents to the

curve. On t take any poi n t T. Through T one ta ngent p canbe drawn to the curve and on e on ly, meeti ng if at T

'

. T, T ar e

seen to correspond u n iqu ely . Hence [T] , [T’

] are homogr aph ic and therefore proj ective : by Ar t. 42, TT

’ envelops acon rc.

1 3 5 . Nota t ion for h om og r a p h y . The nota t ion A

wh ich was i ntroduced in Ar t. 35 for “ i s proj ec tive wi th ” wi llnow be extended to homograph ic forms and be read “ i s homograph ic w i th . Thi s notat ion w i ll not contrad ict the prev ious ,s i nce proj ect ive forms are homogr aph ic.

v r r r] IMAG INAR IES AND HOMoc RAPHY 1 39

13 6 . H om og r a p h i c p la n e fi gu r e s . Let a one - one

algebra i c correspondence be establ i shed between the poi n ts of aplane figure d) and the poi n ts of another plane figu re I f i nadd i t ion the transformat ion be such that to a straight l i ne of <1)cor responds a s tr aight l i ne of oS

’ and conversely , the two plan efigures are sa id to

o

be d irec tly homographi c or , mor e s imply,homograph i c .The r elation between such figur es w i ll be cal led a bomogr apby .

Let .r, y be the coordinates

o'

m the plane of <f> of a poi n t POf

Let a", y

’ be the coor d i nates i n the plan e of ct'

of the cor

r espond ing poi n t P ’

of

Then i f the cor r espondence between .r, y and a

”

, y’ i s to be

one one, a) , y' when Sol ved for

,must not i nvolve radi ca l s contai n

i ng .r, y , that 1s, they must be rat ional fu nct ions of a', y . Re

ducing them to the same denominator we haveP QR y R

where P , Q, B ar e polynomial s m a, y .

To the stra ight l i n el’x'

my 1 0

of the figure q‘> corr esponds the locusP Ql —

E+ m

224- 1 2 0

of the figure qt.

This locus (2 ) 1s not a stra ight l i n e u nless P , Q,B ei ther

r educe to express ions of the firs t degree m .r , y or else havea common factor , such that when i t i s d iv ided ou t of P , Q, B ,

therema i n i ng factor i s of the first degree .

I n e i ther case equations ( 1 ) r educe to the form

A 3x + B 3y + Os’ J A 3x + B 3y + O3

and then the locu s (2) becomes the stra ight l i n e1’

(A , .r B ,y m’

(A..r B gy O.) A , .r B ,,y 03 0,

which bei ng redu ced to the formix my 1 0,

-+ O3

1 40 PROJ ECTIV E GEOMETRY [e rr

show i ng that the l i n e coord i nates tran sform accordi ngs imi lar law.

The equations (3) can be w r i t ten(A 2x

’A ,) x (B 3 x

'

B ,) y 0,

(A2y'

A 2) x (B 2y'

B 2) y (Osy’02) 0.

Solv ing these for a; we find

(Call,

02) (B 2x'

B 1 ) 01) (Ball,

B 2)

(A fgx,

A 1) (B 3y’B ?) (A ggy

,

A 2) (n,

B l )

(B 203 B 302) at,

(8 301 B 1 03) y,

(B I 02 B 201)(A 2B 3 A 2B 2) x

’

(A ,B ,- A ,B 2) y

’

(A ,B 2 A 2B ,)and s im i larly

(02A 2 02A2

'

) x + (O2A , + (O, -A2 02A’J(A 2B 2 A 2 2

’B ) .r + (A 2B ,

- A + (A B2 A 2B , )(5)

Equat ions (5) Show that the tran sformation fr om $ 3] to x’

y’ i s

of the same type as the transformat ion from a’

y’to my . We

deduce that to a stra ight l i ne of 45 corresponds a stra ight l i n e ofos’

,and on e on ly

,wh ich can be otherwi se establ i shed by solv i ng

back equat ions (4 ) for l’

,m

I t i s clear from the defin i t ion that corr espond i ng r anges andcorrespond i ng penc i ls i n two homographic figures are themselveshomographi c .

13 7 . A p la n e h om og r a p h y i s d e te r m i n e d b y tw o

c or r e sp on d i n g te t r a d s . Let A ,B ,O,D A 2 B2O2D 2 be tw o

tetr ads or sets of four poi n ts in the plane figures ¢2 . Thesetetrads may be arb i trar i ly gi ven , wi th the one res tr iction that nothree poi n ts i n e ither tetrad are to be col l i near . The n a lremograph ic correspondence can be establ i shed between qt, and <1S2 asfol lows .

Let P , be any pornt of Draw through A 2 a ray A 2P 2

such thatA2 {B2O2D P 2} A ,

There i s only on e su ch r ay by Ar t. 25 .

Al so draw through B a ray B 2P 2 such thatB 2 {A202 2D P 2}: B , {A ,

P 2 , bei ng the i ntersect ion of A 2P 2 , B 2P 2 , i s determ inedu n iqu ely when P , i s given , and conversely. Thi s constr uctionthen establ i shes between qS, and ¢2 a one one poi n t to poi n tcorrespondence , wh ich 1s eas i ly verified to be algebra i c .


1 42 PROJECTIVE GEOMETRY [CPL

1 3 8 . V a n i sh i n g l i n e s . The equations of the van i shi ngl in es of the homography are eas i ly wr i tten down from equat ions(3) and (5) of Ar t. 1 36 . For i f x'

, y’are to be i nfin i te we must

haveA 2a + B 2 y +

Th i s then i s th e van i sh i ng l in e of the figur e I f .cc, y ar e to

be infin ite,then

(A 28 3 A2B 2) x'

(A 2B , y'

A,B 2 A 2B , O,

and th i s gi ves the van i sh ing l i n e of the figure1 3 9 . R e c ip r oc a l t r a n s fo rm a t ion o r c or r e la t ion .

I f m the equat ions (4) and (5) of Ar t. 1 36 w e interchangel'

, m' and x’

, y’we fin wr it i ng for shor tn ess

3 203““ 3 302 , a s

‘ B 301 — B 1 03 ; as B 1 02 3 201 ,

wi th cor respondi ng mean i ngs for ,B’S and 7

’s

O,

A 2x + B 2y + O2’

A 2x + B 2y + O2

B , .z"

+ B 2y’

- 1- B 3

a e + ay + a’

Q B + Q y + Q’

a , l’+ a 2m

’+ a 2 ,

8 , l’+ B2m

'

+ fl2

a , l’

,8 ,m

'

y , a 2 l’

,32m

’

y2

d ,, l'

B,,m'

73’

a 2 l'

,8 2m

'

72°

These equat ions may be Shown as i n Ar t. 1 36 to be the n ecessar yequat ions of tran sformation in any one - one algebrai c correspondence of plan e figures i n wh ich l i nes cor respond to poi ntsand poi nts to l i nes . Cl early any penc i l i s homogr aphi c , andther efore equ i - anharmon ic, with the cor r espond ing range. Thi stransforma t ion i s therefore of the ty pe d i scu ssed i n Ar t. 63 .

I t i s,how ever, much more general than th is transformation ;

for the tran sformation by rec iprocal polars i s l im i ted to figuresin the same plan e, whereas the present tran sformat ion i s for anyplane figures . Al so i n the transformat ion by rec iprocal polar sthe same l i n e p corr esponds to the same poin t P whether P becons idered as belong i ng to one figure or to the other . Whereashere

,i f the figures be taken coplanar and the axes of coord i nates

ident ical,i f w e pu t y

’=y w e do not in gener al obta in

l'

l or m’m .

The pr esen t tran sformation i s the most gen er al case of a plan er ec iprocal tr ansformat ion .

v r r r] IMAG INAR IES AND HOMOGRAPHY 1 43

An obviou s modification of the r eason i ng of Ar t. 1 37 w i l lShow that a correlation i s determ i ned when fou r poi nts A , , B , ,

D , of on e figure , no three of wh ich are col l i n ear , are made tocorrespond to four l i nes a, , b, , c, , d, of the other figu re

, no threeof which are concurrent .For i f P , correspond to p 2 we have

A r {B 1010 1P 1}= a s

B 1 {A ror p rp r i z 1’s

which determ i n e a 2p 2 and b2p 2 , and therefore p 2 .

And i t i s easy to show that i f P , descr ibes a stra ight l i ne,p 2 passes through a poi nt .

Tw o su ch figures may be said to be rec iprocal ly homogr aph i c ,or

, more s imply , rec iprocal or correlat i ve . The relation betweenthem may be Spoken of as a rec iprocal homogr aphy or a correlat ion .

EXAMPLES V I I I .

1 . Prove that i f an imag i nary l i ne l do not intersect i ts conjugateimagi nary l ’, the l ine drawn from a real po in t P to mee t 1 and l’ i salways real.

2 . Show that the rec iprocal elemen ts of tw o conj ugate imaginarye lemen ts ar e themse lves conjugate imagi nary when the rec iprocalelenr e rrts of real e lements a r e real .

3 . Show that conjugate imaginary elemen ts project i nto conjugateimaginary elemen ts when the project ion i s real .

4 . Show that the square of the d istance be tween two conjugateimagi nary poin ts i s essen t ial ly negat ive .

5 . A variable c i rcle cuts a fixed c i rcle at a constant angle a and

passes through a fixed poi n t 0. I f the poi nts of i ntersect ion of th isc i rcle w i th the fixed c i rcle be P

,P

’

,Show that the ranges [P ]2, [P

’

]2 ar ehonrogr aphic.

6 . The coord inates of tw o poin ts on a straigh t l ine ar e connectedby the relat ion

1 l la: .r

’

f'

Show that the poin ts descr ibe proj ect ive ranges .

1 44 PROJECTIVE G EOMETRY [e rr v r r

7. The angles 6, 6’ wh ich tw o l ines th rough a fixed origi n mak

w ith an in i t ial l i ne ar e connected by the equat ion

_

A0+B_

Co+D’

Explain care fully w hy the two l ines do not describe homogr aph ipenc ils.

8 . I f the angles 6, 6’ i n the last quest ion be connected by threlat ion

A sin 6 B0 sin 6+ D

show that the l ines do not descr ibe honrogr aph ie penc i ls.9 . A con ic th rough four fixed points

,two of wh ich lie on a fixe

conic s, meets s at P ,P’. O i s a fixed poi nt 011 s . Prove that O

OP’describe homograph ic penc i ls.

10. Through the vertex of a flat penc i l planes ar e drawn pe r pendicu lar to the rays of th e penc i l. Show that the axial penc i l so formei s homograph i c w i th the given fiat penc il .

1 1 . A coni c through four poin ts A ,B

,O, D mee ts fixed l ines th r ougl

A and B at P and Q . Show that P, Q describe homograph ic ranges.

1 2 . I f i n a homograph ic relat ion the points (0, (0, ( 1 , li n the plane of (w , y ) correspond re spect ively to the points (0, ( l , 3)

in th e plane of Show that the po i nt (0, 2) i n “I tplane of (x, y ) corresponds to the poi nt (3, 4 ) i n the plane of (x

’

,


1 46 PROJ ECTIVE G EOMETRY [err

14 1 . E v e r y p la n e h om og r a p h y i s a p r oj e c t i v et r a n s for m a t i on a n d c on v e r s e ly . For cons ider any plan ehomogr aphy. Take tw o corr espond i ng tetrads such that no

thr ee poi n ts of each are col l i near, and constru ct a proj ecti vetransformat ion i n wh ich these a r e corr espond i ng tetrads . S i nceboth homography and proj ection pr eserve cross - ratio consta nt,the construct ion gi ven i n Ar t. 1 37 for fi nd ing the poi nt P 2 correspe nd i ng to any gi ven poi nt P , appl i es to both the project i ve andhomogr aph ic transformations . These tw o transforma t ions therefore determ i ne the same correspondence between the two figures ,that i s

,the gi ven homogr aphy rs i dentical w i th the proj ect i ve

transformat ion .

Conver sely every proj ect i ve transformation i s hornogr aphical,for i t i s a on e - on e algebra i c transforma t ion i n wh ich poi nts correspe nd to poi nts and stra ight l i nes to straight l i nes .I t fol lows from Ar t. 1 37 that tw o correspondi ng tetrads of

poi nts or l i nes entirely determ i ne the proj ec t i ve correspondencebetween two planes .

14 2 . D e du c t i on s fr om th e a b ov e . I f w e ar e gi venthr ee poi nts A , , B , , O, on a con i c s, and three poi n ts A 2 , B 2 , 02on a con ic 32 the con i c s , can always be proj ected i nto the con ic s2and at the same t ime the thr ee po i n ts A B , , 0, i nto the threepoi nts A2 , B 2 , 02 .

Draw the tangents to s, at A , , B , meeti ng at D , and thetangents to 32 at A 2 , B 2 meeti ng at D 2 . Proj ect the four po i n tsA , , B , , B , i n to the

2

four poi nts A 2 , B 2 , O2 , D 2 Then 3 , pr ojects i n to a con i c wh ich tou ches D 2A 2 at A D 2 B2 at B . andpasses thr ough 02 . But th i s con i c m u st be 32 for tw o pa irs ofcoin c ident poi n ts and another poi n t determ i ne a con i c u n iquely.

In l ike man ner i f a , , b, , c, be three tangents to a con i c s, ;a 2 , b2 , 02 three tangents to a con i c 32 , l et d, be the chord of contactof a ,b, , 032 the chord of con tact of a 2b2 . Projec t i n toa 2b2 c2d2 . Then 3 , projects i n to a con i c touch i ng a at a 2d2 , b2alt1

b2al2 and touch i ng also (12 . And th i s con ic can be non e othert an 32 .

These tw o r esul ts Show that tw o ranges or penc i l s of thesecond order can always be ac tual ly proj ec ted i n to on e anotherso that any tw o gi ven tr i ads correspond . Equ i - a nharmon i c rangesof the second order are ther efore actu ally proj ecti ve, which j ustifiesthe name given to them 1 11 Chapter V I I .

Not ice that the cond i t ion tha t two g iven r anges of the secondorder, or two gi ven penci l s of the second order

,shal l correspond

deter in ines en t irely the pr oj ective r elat ion between the two plan es .

1x] TRANSFORMATION o r PLAN E F IGURES 1 47

14 3 . S e lf - c o r r e sp on d in g e lem e n ts o f tw o c op la n a r

p r oj e ct iv e fi gu r e s . Clearly two coplanar proj ecti ve figuresca nnot have rrror e than three n on - col l i near selflcor r espondingpoi n ts or more tha n three non - concu rren t selfl cor r'esponding l i nes ;or else i t would fol low from Ar t. 1 4 1 that they coi n c ided altogether .Cons i der any poi n t O of the plane . To 0 cons idered as a

poi n t of the firs t figure let O2 correspond ; to O cons idered as apo i nt of the se cond figure let 0, correspond . The correspondi ngrays of the first and second figures through 0, 02 respect i ve lysweep ou t two project i ve penc i ls : hence thei r i ntersec t ion descr i besa con ic a . The correspond i ng rays of the first and secon d figuresthrough 0 respec t i ve ly sweep ou t tw o proj ect i ve penc i ls :hence the ir i ntersect ion descr ibes a con i c c . a

, e hav e an i ntersec t ion 0 : in general they w i l l have three other i ntersectionsP , Q,

B .

Now OP of the first figure corresponds to O2P of the secondand O,P of the firs t figure corresponds to OP of the second .

Hence (OP , O,P ) , i .e . P , of the first figure , corr esponds to(02 P , OP ) , i .e . P

,of the second figu re. Thus P

,and therefore

also Q, B ,a r e the three sel f- corresponding poi nts . On e of these

i s always r eal,s i nce of the four i ntersec t ions O

,P

, Q, R of n,c,

O i s always real .The three sel f- correspond i ng l i n es a r e clearly P Q, QB ,

BP .

They may al so be ob ta i ned by cons ider i ng a lur e a: and i ts tw o

correspondents .r , , a: Jo i n i ng correspond i ng poi n ts 011 .r , , a

: and011 .r

,.r 2 w e obtai n two con i cs touch i ng .r . The i r three other

common tangen ts p , q, r a r e selfcorrespond i ng l i nes . S i nce x i sa real corrrmorr tangent, a second common tangent m u st also bereal

, so tha t one selfcorrespond i ng l i n e i s real . Clearly i f P i sthe on ly real se l f- corre spond ing poi nt, QB i s the only rea l selfcorrespond i ng l i ne , although Q, It themse l ves a r e not real .I n general 011 a sel f- correspondi ng l i ne p there are only two

sel f- correspondi ng po i n ts , Q, I t be i ng the seltlcor r esponding poi n tsof the proj ect i ve ranges for nr ed by correspond ing poi n ts 011 p .

I f however a th i rd sel f- correspond i ng poi n t on p exis ts , then everypoi n t of p i s sel f- corre spondi ng. We have then the case of figu resi n plane perspect i ve , p i s the axi s of col l i nea t ion and the se l fcorrespond i ng po i n t P n ot 011 p i s the pole of perspec t i ve .

The con i cs a and r then co i nc ide, each of them break i ng upi nto the l i nes p and OP . All the poi n ts of p are se lf- corresponding : bu t the on ly sel f- correspond i ng po i n ts of OP are Pand the poi n t where

10— 2

PROJECTIVE G EOMETRY

14 4 . Tw o r e c i p r o ca l t r a n s fo r m a t i on s a r e e qu i v a

l e n t to a p r oj e c t iv e t r a n s for m a t i on . Cons i der two plan er eci

pjr ocal figures <I>2 .

ow take a figure O2 rec iprocal w i th ¢2 . and ¢3 now

correspond poi n t by poi nt and l i ne by l i n e and s i nce the correspondence between elements of qt, and 41 2 i s one - one and algebra i c,and that between elements Of qb2 a nd ci>2 i s on e- one and algebrai c,the corr espondence between elements of ct, and 982 i s al so one - on e

and algebra i c .Therefor e the figures qt, and ¢2 ar e homogr aph i c and ther efore

proj ect ive.

14 5 . A n y r e c i p r oc a l t r a n sfo r m a t ion i s e qu iv a l e n t

to a p r oj e ct i v e t r a n s for m a t i on a n d a t r a n s for m a t i on

by r e c i p r oc a l p ol a r s . For let 95, and 32 be g iven rec iprocalfigur es . Let qb2 be the rec iprocal polar figure of o52 with regard toany coni c . Then by the last Arti cl e d) , and <l>3 a r e proj ect i ve.Thu s a proj ective transformat ion transforms to 952 and thetransformat ion by r ec iprocal polars tran sforms <f>2 to 952 .

14 6 . L ocu s of i n c i de n t p o i n ts a n d e n v e lop e of

i n c i d e n t l i n e s O f tw o c op la n a r r e c ip r oc a l fi gu r e s .

I f two r ec ipr ocal figures ct, qt’ be coplanar, w e pr oceed to find

the cond it ion that a poi n t and i ts corresponding l in e shal l bein c ident.I f P be a poin t on i ts corr esponding l i n e p

'

, P bei ngcons idered as belongi ng to figure qt, i t al so l i es on its corresponding l i n e when consi dered as belonging to figu r e For l etP : Q

’. Then

,s in ce P

,i .e. Q

’

, l i es on p’

, q passes through P ,i .e .

thr ough Q'

.

In l i ke man ner i f a l i n e passes thr ough i ts cor respond ing poi nti n one figure

,i t passes also through i ts cor r espondi ng poi n t i n the

other figure.Such poi n ts and l ines may be cal l ed i nc iden t poin ts and l i nes .Let u be any l i n e of one figure, U

’ i ts cor r espondi ng poi n t ofthe other . Let P be any poi n t of a , i ts corr espond i ng l i n e p

’

passes through U’ and meets a at P ’. The range [P ] 1s homo

graph i c w i th the pen ci l [p ] and therefor e proj ect i ve wi th therange [P

’

. The tw o ranges [P ] , [P’

] have therefore tw o sel fcor r esporr i ng poi nts S , T which are such that they l i e on thei rcorrespond ing l i nes .The locus of such poi nts has therefor e two poi nts of

i nter section wi th any stra ight l in e. Hence i t i s a con ic s, byAr t. 1 34.


1 50 PROJECTIVE G EOMETRY IX

2 . G iven three pairs of correspond ing poi nts of tw o homograph icplane figures and one of the sel f- corre spond ing l ine s , construct thein tersect ion of the other tw o se l f- corre spond ing l ines.

3. Show how to set up a one - on e correspondence of a plane i ntoi tself such that a con i c in th e plane is trans formed i nto i tsel f and threeassigned points of i t i nto three othe r ass igned poin ts of i t.

4 . Show that the rec iprocal polar figure of a c ircle 3 w i th regard toanothe r c i rcle 0 i s a conic

,one of whose foc i is the centre of c

,and find

a construct ion for the other focus. Show also that the con ic is ane ll ipse , parabola, or hyperbola accord ing as the centre of c l ies ins ide

,

on , or outs ide 3 .

[The studen t should note that th is furn ishes a me thod of d iscoveringfocal propert ies of the conic from propertie s of the circle . I t w il l be aninstructive exercise for h im to deduce the results of Arts. 9 8 , 102, 1 04in th i s w ay ]

5 . The polar rec iprocal of a c i rcle,taken w ith regard to a rectangular

hyperbola,is a con ic of wh ich the centre of the rectangular hyperbola is

a focus.6. Prove that a coni c is the polar reciprocal of i ts auxil iary circle

w i th regard to a c i rcle of imaginary rad ius whose cen tre i s a focus.7. I f one conic s i s i ts own polar rec iprocal for another conic t then

th e conic t is i ts ow n polar re c iprocal for the conic 3 .

[Show that the conics have double contact at A ,B and that i f 0 be

the common pole of AB then i f a ray th rough 0 mee t t at U,T the

tangent at U is the polar of T w i th regard to a ]

8. Show that given two points A ,B of a conic s

,a con ic t can be

found having double con tact w i th s at A, B such that s is i ts ow n

rec iprocal w i th regard to t.

CHAPTER X.

INVOLUT ION .

14 7 . I f i n tw o c ob a sa l h om og r a p h i c l i k e for m s

on e p a i r of e lem e n ts c o r r e sp on d dou b ly ,a l l p a i r s

c o r r e sp on d d ou b ly . Let P ,P

’ be tw o corr espond i ng elements(denoted by Roman capitals , bu t not restr i cted to mean poi nts)of tw o homographi c l ike forms (I) , <f>

' hav ing the same base.Then 1 11 general i f P be cons idered as an e l ement of cb

’ theelemen t of d) which then corresponds to P i s not P

’

,but some

other poi nt .It may , however , happen that P

’ corresponds to P ,whether

P be cons idered as be longi ng to gt or as belongi ng to P andP

’ are the n sa id to corr espond doubly .

I n th i s case ever y other pa i r of cor r espond i ng e lements Q , Q'

al so correspond doubly I or srnec by Ar t. 2 1 a cross - ratio i s notal tered i f w e i n ter change tw o of i ts elements , prov ided the othertw o he also i n terchanged ,

But by hy pothes i s P P ’

Q ,P

'

P Q’

ar e correspond i ng tr i adsof (b

' respec t i ve ly . Hence the above equation expresses thefact tha t to Q

’

of (it corresponds Q of qb’

or Q , Q' correspond

doubly .

14 8 . I n v o lu t i on . Tw o coba sal homograph ic l i ke forms ,i n whi ch every elemen t corresponds doubly

,ar e sa id to be i n

in volution,or to form an i n vol u t ion on the i r base . The corre

spe nd ing elerrren ts a r e spoken of as mates i n the i n vo l u t ion .

14 9 . Tw o p a i r s of m a te s d e te r m in e a n i n v o lu t i on .

Let (P , (Q, Q’

) be the two pa i rs of ma tes . Then the triadsP P

'

Q, P’

P Q' define tw o homogr aphic forms which a r e in

i n vol u t ion s i nce one pa i r of e lemen ts , name ly P ,P

'

, corresponddoubly . The i n vol u tion i s therefore de term i ned .

Note tha t one pai r of ma tes i s i n su ffic ien t ; for two pa i rs of


cor r espondi ng poi n ts (P , (P’

, P ) are not enough to determ i netw o homogr aph ic forms .

150. D ou b le e l em e n ts . S i nce tw o homogr aphic cobasall ike forms have tw o self correspond i ng elements, an i nvolut ionwi l l have two sel f- corresponding mates , wh ich may be r eal orimagi nary.

These ar e cal led the double e l ements of the i n vol ut ion . S i n cea double element i s equ ivalent to a pa i r of mates , an i nvolut ioni s en t i rely gi ven by i ts double el ements .An i nvolut ion whose double e lemen ts are real i s sa id to be

byp er bollc ; one whose double e l ements ar e imagi nary i s'sai d to

be ellip tic.

1 5 1 . A n y p a i r of m a te s a r e h a r m on i c a l ly c onj u ga tew ith r e ga r d to th e d ou b l e e lem e n ts . For l e t (P ,

P'

) be apa ir of mates , A ,

B the double elements Then the elementsAPBP

’ cor respond to AP BP or

{APBP’

The set APBP ’ar e therefore equ i - anh armon i c wi th themselves,

P and P ’ be ing i nterchanged : ther efore (Ar t. 27) P and P’are

harmon i cal ly conj ugate wi th regard to A , B .

1 5 2 . I n v olu t i on on a st r a i gh t l in e . C e n t r e of i n

v olu t i on . Cons ider now the case of an i nvol ut ion on a straightl i n e. Let 0 be the mate of the poi nt at i nfin i ty on thestra ight l i n e. 0 i s called the cen tre of i n vol ution . I f (P ,

(Q, Q’

) be two pa i rs of mates , we have

OP . 0 Q OQ’

OP OQ’

—

O Q’

. OP” '

OQ*

OP

therefore OP . OP’

OQ . OQ’ constant for the involu t ion .

I f Q, Q’ coi nc ide wi th one of the double poin ts A ,

’

B we haveOP . OP

’= GA 2 = OB 2

.

In a hyper bol i c involu t ion A , B ar e r eal , thu s OA"

, OB2ar e

pos i t i v e and OP . OP’ i s pos i t ive . Conversely, i f OP . OP

’ i spos i t ive

,A

,B ar e r eal . Therefor e in an el l ipt i c i n volution

OP . OP’ i s n egative and conversely.

S ince OA 2: OB 2

,O i s m idway between the double poi nts .

1 5 3 . R e la ti on b e tw e e n th e m u tu a l d i sta n c e s of s ix

p o in ts i n i n v o lu t ion . Let (A , , (B , , (01 , 02) bethr ee pa i rs of mates of an i nvol ution .


1 54 PROJECTIVE G EOM ETRY [on

c ircles can always be taken of so large a r ad i us that they i nter sectin r eal poi nts 0

,D . OD then meets a; at the centre O of

invol ution and a c i rcle w i th centre 0 and rad iu s equal to thetangen t from O to e i ther c i rcle meets x at the two double poi ntsA ,B . For OA2 = 00 . OD : OP . OP

’= OQ . OQ

’

.

Fig. 43.

1 5 7 . A n i n v o lu t i on i s e l l i pt i c o r h y p e r b ol i c a o

c o r d in g a s a p a i r of m a te s a r e,o r a r e n ot

,s ep a r a te d

b y a n y oth e r p a i r of m a te s . Cons ider firs t a n i nvolu t ion ona straight l i ne . Let (P , (Q, Q

'

) be any two pa irs of mates;

x] INVOLUTION 1 55

Construct the double poi nts of the i n vol u t ion by the method ofthe las t Ar t i cle .

Then i f (Fig . 43 a ) the segments P P’

, QQ’

do not overlap ,tha t i s, i f the ma tes (P , P

'

) a r e not separated by the rrr ates (Q,the tw o c i rcles i n tersect a t poi nts 0,

D on the same s ide of .r .

O i s outs ide OD and therefore ou ts ide both c i r cles . The tangentsfrom O to the c i rcles , and therefore the double po i n ts , a r e realand the i n vol u tion i s hyperbol i c .

I f one of the segmen ts P P ’ l i es en t i rely i n s ide the otherQQ

’

the same resu lt follows (see Fig . 43 b) . I n th i s case alsoone pair of mates a r e not separated by the other . For Q, Q

’ arenot then looked upon as separa ted by P and P '

Q and Q’may

be con nec ted by a con t i n uou s sequ ence of poi n ts of the l i ne ,pass i ng through the po i n t at i nfin i ty bu t not i nc l ud ing P or P

'

.

For ma tes to be cons i dered “ separated ” they m u st be i n an

or der such as P QP’

Q'

.

Bi r t i f the segmen ts P P , QQ’

overlap (Fig . 43 c) , then(P ,

P ) ar e separa ted by (Q, The c i rcl es through P , P’

and Q, Q’ i n tersec t at poi n ts 0

,D on oppos i te s ides of J) . O li es

i ns ide both c i rcles . N0 real tangents can be drawn from O tothe c i rcles, and the i n volu t ion is

o

el l ipt i c .

I f w e now cons ider an i nvolu t ion flat penc i l, or a n i n vol ut ion

ax ial penc i l , these de term i ne on any s traight l i n e wh ich meetsthem an inv olutron range (s i nce tw o cobasal horr rogr aphic penc i ls ,fla t or axial , de termi ne homogr aph ic ranges on any tran sversaland elements correspond i ng doubly i n the penc i l s g i ve elemen tscorr espond ing doubly r n the r anges) . Also i f ma tes ar e separatedi n the penc i l , they a r e so i n the range wh ich rs a sect ion of the

penci l , and the double rays or planes of the one pass throughthe dorrble poi n ts of the other . Tire i nvol u t ion penc i l and thei nvol u t ion range are therefore ell i pt i c and hyperbol i c toge ther.I t fol lows tha t i f i n an i n vol ut ion err cil (flat or ax ial) apai r of ma tes are separa ted by any otlieer pai r of ur ates the 1 n

vol u t ion i s el l ipt i c . I f they ar e not so separated the i nvol u tioni s hyperbol i c .

The student shou ld have no difficu lty in prov i ng, by prec i selys im ilar reason i ng, that the same theorem holds also for a n

i n vol ut ion of poi nts 011 a con ic and for a n i n vol ution of tangentsto a con ic.

15 8 . I n v o lu t i on fi a t p e n c i l . In a n i n volu t ion penc i lthere rs no specra l ray correspond i ng to the cen tre of a n r rrvolu

t ron range , for no ray rs the analogue of the porn t a t r rrfi rn ty .

I f OA,OB a r e the double rays, (OP , OP

'

) a pai r of mates ,

1 56 PROJECTIVE GEOMETRY [e rr

then cutting the penci l by a stra ight l i n e par al lel to OP ’

,which

meets the doubl e rays at A and B (Fig. AB i s b i sected atP by OP , s ince OP , OP

’are harmon i c conjugates wi th r egard toOA , OB and therefore P and the po i n t at i nfin i ty on OP

’ areharmon i c conj ugates wi th regard to A , B . Hence i f theparallelogram whose s ides a r e OA , OB be completed, i ts d iagonal sare paral lel to a pai r of mates .I f the double rays are at r ight angles, every such par al lelo

gram is a rectangle . I ts d iagon als ar e equally in cl i n ed to thes i des of the r ectangle, therefore i f the double rays are at rightangles

,they b isect the angles between any pa i r of mates .

Fig. 44.

15 9 . R e la t i on b e tw e en s ix r a y s of a n i n v olu t i on .

Proceed i ng as in Ar t. 1 53,we have, i f (OA , , (0B , ,

002) are thr ee pa i rs of mates of an i nvol ut ion penc i l ,O O {A 2A 1 3 202},

and,u s i ng the express ion for the cr oss - rat io of a penc i l in terms

of the angles made by the r ays (Ar t.

s i n A , GA 2 s i n B , OO’, s i n AgOA , s i n B 2002

s i n A ,OO’, sin B , OA , sin A ., OO2 . s i n B gOA ,

when cesin B , OC

’1 sin 020A2 sin A , OB2

s i n B zOOz . sin C ,OA, sin AgOB , ,

and i n ter changi ng suffixes as in Ar t. 153 we have the gen er alresu l t

(sin BOO . sin OOA sin AOB) ,

(sin BOO s i n 00A sin


1 58 PROJECTIVE c s orrrnr rr r [011

Let A , , B , be the double poi nts of one i nvol ut ion ,A B2 the

double poi nts of the other . Then i f P ,P

' are mates in bothi n vol ut ions, (P ,

P'

) a r e harmon i cal ly conj ugate w i th regard toboth (A , , B ,) and (A , , B ) that i s , they are the double poi nts ofthe i n vol u t ion determ i ned by the tw o pairs (A (A , , B 2) .They can therefore be fou nd geometri cal ly by the constructionof Ar t. 1 56, prov ided the tw o given i nvol utions are hyperbol i c .

I n th i s case the pai r of common mates are imagi nary or r eal ,accord i ng as the double po i n ts of on e i n vol u tion are

, or ar e not,separated by the double poi nts of the other.I f

,however, one in volut ion i s hyperbol i c and the other el l ipti c ,

Fig. 45 .

l et (A , , B ,) (Fig . 45 a ) be the dor rble poi n ts of the hyperbol i ci nvolu t ion and 02 , D 2 the tw o poi nts from which the el l ip t i ci n volu tion i s r ectangularly proj ec ted . Then the po i n ts w herethe base x i s cu t by any c i rcle through D 2 a r e rrrates of the

e l l ipt i c i nvol u t ion , for by symmetry su ch a c i rcle ha s x for ad i ameter and i ts poi nts of i n tersec t ion wi th .r therefore subtenda r igh t angle atConstru ct the c i rcl e wh i ch touches .r at B , (or A ,) and passes

through 02 . Let O be the m iddle poi n t of A ,B , . O i s thecentre of the hyp erbol i c i nvol ut ion . Join 002 , cutti ng th i s

x] INVOLUTION 1 59

c i rcle aga i n at E . Through E,D 2 draw a c ircle meeti ng a' at

P ,P

’

. Then P,P

’

a r e the poi nts requ i red .

For P , I” bei ng on a c ircle through D 2 ar e mates i n the

e ll ipt i c i n volu t ion .

Also OP OP'

OE . OU.. OB , P,P

’

ar e mates i n thehyperbol i c i n vol u tion .

Fi nally i f both i rrv ol rrtions ar e ell ipti c let D , and D 2 bethe poi n ts from w hi ch they ar e r ectarrgrrlar ly projec ted (Fig 45Then by sy rnnretr y a c i rcl e w i l l pass through the four po i n ts

D D The poi nts P , P' where thi s c i rcle meets x are

mates i n both i n volu t ions .Note that i n the last tw o constructions the poi nts found are

always r ea l .I f new i t be requ i red to find the mates common to tw o con

cen tr i c i rrv olution penc i ls of vertex 0, w e cu t the penci l s bya transversal . We obta i n tw o i n volution r anges and find the i rcommon mates as above . The rays joi n i ng these common matesto O are the common ma tes of the gi ven i n vol u t ion penc i ls .I n l i ke mann er the problem of findi ng the common mates of

tw o cobasal i nvol ut ion s of any k i nd i s always reduc ibl e to thesame problem for i n volu tion ranges 011 a s tra ight l i n e .

The problem has a real sol u t ion i n al l cases except when thetw o i n volu t ions ar e both hyperbol i c and the double elemen ts ofone a r e separated by the double e lemen ts of the other .I n part i cu lar every i n volu tion fiat penci l has always one pai r

of r eal rrr ates at r igh t angles and one only, these bei ng the

common mates of the gi ven i nvol u t ion w i th the rectangulari n volut ion through the same v ertex . These must be real

,s i nce

the recta ngular i nvol u tion i s e l l ip t i c .

1 6 3 . I n v o lu t i on s of c onj u ga te e l em e n ts w ith

r ega r d to a c on i c . The tw o coll i near proj ecti ve ranges formedby assoc ia t i ng w i th each poi n t of a l i ne i ts conj ugate poi nt w i thregard to a con ic (Ar t. 55 ) define an i n vol u tion , s i nce , from thesymme try of the conj uga te re lation , tw o correspond i ng poi n tscorrespond to each other doubly . The double poi nts of th i s i rrvol u t ion a r e the poi n t s w here the s traigh t l i ne mee ts the con i c .

S im i larly conjuga te l i nes through a poi n t form an in vol u t ionof wh ich the double rays a r e the tangen ts from the po i n t .In par t ic u l ar conj uga te d iame ters form an i n vol u t ion , of

wh ich the dorrble rays ar e the asymp totes .S i nce the i n volu t ion of conj uga te d iame te rs ha s one real pa i r

ofma tes a t r igh t angles and one only,w e obta i n a new proof of the

theorem ofAr t. 68 that a con i c has on e, and only one, pa i r of axes .

1 60 PROJECTIVE GEOMETRY [011

164 . T h e c i r cu la r p o i n ts a t i n fi n ity . Cons ider thetw o ( imagi nary) poin ts in which the l i n e at infi n i ty i

°°

in a plan emeets any c i rcle i n the plane . S i n ce the pol e of i i s the centr e0 of the c ircl e the i nvolu tion of conj ugate poi nts on i °° i s g ivenby the i ntersect ion of i w ith the i nvol ution of conjugate raysthr ough 0. But s i nce conj ugate d iameters of a c i rcle ar e atr ight angles (Ar t. 5 7) the latter i n vol ution i s the r ecta ngularinvol u tion through 0.

The tw o i n tersect ions Q ,Q’of i wi th the c i rcle ar e ther efor e

the dorrble poi n ts of the in vol ution in wh ich the r ectangular involu t ion through 0 meets iBut i f we take any other poi nt 0 in the plan e and joi n O to

the poi n ts of the involu t ion on i °° w e obta in an i nvol ut ion through0 whose r ays are para ll el to the corresponding r ays of the in volut ion thr ough 0. The i n vol ution thr ough 0 i s therefore al sorectangu lar . Thus the double rays of al l rectangular i nvol ut ionspass through the same tw o poi nts 9 ,

Q’at i nfin i ty. These poi n ts9

, Q’

ar e therefore determi ned qu i te i ndependen tly of the par ticular c i rcl e chosen . Hen ce a ll c ircles pass through the same twopoi nts 9 ,

Q’.Con versely every con ic wh ich passes through 9 ,

Q’ i s‘

a c i r cle.For l et 3 be such a con i c and let A ,

B, 0 be any three other poi nts

on 3 . Descr i be the c i rcl e c through A ,B

,0. Then i t passes

through 0,Q’

. o and 3 have five poi nts A , B , 0, Q, 0’common

and therefore coi nc ide .

For these reason s the poi n ts 0 ,Q’ are cal led the ci r cu la r

p oin ts a t infin i ty . Be i ng the i ntersections of a real l ine (the l i n eat i nfin i ty) with a r eal curve, they are conj ugate imagi nary poi n tsby Ar t. 1 25 .

Two i n terest i ng cases of c i rcles ar i se when the con ic through0

, Q’degenerates i nto a l i ne - pa i r . I f 9

,Q’ are on the same com

ponent of the pai r,the latter consi sts of the l i n e at i nfin i ty and a

straight l i n e at a fi n i te di stance . Thu s any stra ight l i n e, togetherw i th the l i n e at i nfin i ty

,may be regarded as form i ng a c i rcle of

i nfin i te r ad iu s .I f 9

,Q

' be on d ifferent components of the pai r we see that anypa ir of l i nes through ( 2, Q

’ form a c ircl e .

I f thei r poi nt of i ntersection P be r eal , every l i n e through P i sa tangent to the curve at P (see Ar t. 44) and the c i rcle i s cal led ap oin t

- ci r cle.

1 6 5 . T h e c i r cu la r l i n e s . The l i nes joi n ing any poi n tof the plan e to Q,

Q’are cal led the cir cula r lines through the poi n t.I f the poi n t be r eal, the c ircu lar l i nes thr ough i t ar e conj ugate


1 62 PROJECTIVE G EOME'

I‘

RY [OH .

1 6 8 . T h e o r th opti c c i r c le . Con s ider the penc i l s of

conj ugate rays wi th r espect to any con ic 3 through the c i rcularpoi nts Q ,

Q’

. These penc i ls a r e proj ect i ve by Ar t. 55 . Thei rproduct i s ther efore a con ic pass i ng through Q ,

Q’

,that rs, a

c i rcle .

Let P be any poi n t on th i s c i rcle . Then P Q , P Q’bei ng l i n es

through P conjugate w i th regard to s are harmoriically conjugatewith regard to the two tangents from P to 3 (Ar t. Therefore these tw o tangents are

c

at r ight angles (Ar t.

Conversely i f these tw o tangents are at r ight angles P Q, P Q

’

a r e mates i n the involu t ion of conj ugate r ays through P,and P

l i es on the product of the conj ugate penci ls through Q,Q’. We

have then the theorem .

The locu s of in tersect ions of tangents to a con i c at r ightangles i s a c ircle .The c i rcl e i s cal led the or thoptic ci r cle

.

of the con ic , fr omthe property that at any poi nt of i t the con i c subtends a r ightangle . I t i s al so cal led the di r ector cir cle, by analogy withi ts degen erate case when the con i c i s a parabola , when the locusof i ntersect ions of tangents at r ight angles i s the di rectr ix (Ar t.

The explanat ion of thi s from our poin t of v i ew i s that mthe case of the parabola Q Q ’touches the cu r v e and 1s ther efor e aself- corr espond ing r ay of the conj ugate pen c i ls thr ough Q

,Q’.

These are accor di ngly per spect ive and the locu s breaks up in toQ Q

’

(the l i n e at infin i ty) and another stra ight l i ne, wh ich i s thedi r ectr ix.

The orthopti c c i rcle i s con centr i c wi th the con i c . For thetangent at Q to the or thopti c c i rcl e i s the l i ne thr ough Q con

j ugate to Q Q ’with r egard to the con i c (Ar t . I t must ther efore pass through the pole of Q Q

’

,i . e . thr ough the centre of

the con ic . S im i larly the tangent at Q’to the or thoptic c ircle

passes through the cen tre of the con ic . The pol e of Q Q ’ wi thr egard to the c i rc le ( i . e . the centre of the c i rcl e

,Q Q

’ bei ng thel i ne at infi n i ty) 1s thu s the centre of the con ic .

The r ad i us of the orthopti c c i rcl e i s immedi ately fou nd bydrawing the (pe endicula r ) tangents at the extremiti es of theaxes . The semiEfiiagonal of the rectangle so formed is the rad iusr equ i red I t i s J ea n 03 2 I n the hyperbola 0132 = so

the radiu s of the orthoptic oithe orthoptic c i rcle 1s imagi nar y . If OB ,

=2 01 1 2, or the hyperbola 1s r ectangu l ar, i t shri nks i n to a poi nt at the cen tre . Thusthe on ly real perpend icu lar tangents to a r ectangu lar hyperbolaare the asymptote s .

x] INVOLUTION 1 63

1 6 9 . T h e fou r foc i of a c on i c . By the defin i tionof a focu s the i n vol u tion of conjugate l i n es through i t i s rectangular . Thu s the tangents from a focu s to the con ic

,be i ng the

dorrble r ays of such an i n volu t ion , a r e the c i rcu lar l i nes throughthe focu s and pass thr ough Q , Q

’

. Con versely a poi n t F w hich i sthe i ntersect ion of tangents from Q ,

Q’must be a focus,for the

double rays of the i n volu tion of conjugate rays through F w i l l bethe tangen ts from F

,namely FQ , FQ

’

. But these be i ng thec i rcular l i nes, the i nvolut ion defined by them must be recta ngular,or F is a focus .S ince two tangents t, , t2 can be drawn to a con i c from Q and

two tangents t,’ can be drawn fr om Q’

,a con i c w i l l have four

foc i , namely t, t, tgtz’. Of these two are real and tw o

imaginary, as fol low s . Take on e tangent t, from Q . Thi sbe i ng an imagi nary l i ne i n a r eal plane, has a real poi n t F,

on i t (Ar t. The other tangent from F1m rrst be a con

j ugate imagi n ary l i n e to t, , for two imagi nary tangents from area l poin t to a real con i c must be conj ugate imagi nar i es , as canbe shown fr om reason i ng sim i lar to that used i n Ar t. 1 25 to provethat i ntersect ions of a real l i ne and a r eal con i c ar e conj ugateimaginar i es .Thi s other tangent from F, , be i ng a conj ugate imagi nary to

t, , i . s . to F,Q

,must be F, Q

’. Cal l i t then t, Let t2 be the

other ta ngen t fr om Q . I f F2 be the rea l poi nt on i t , thenand t, , t,

’ are conj ugate imagi nary l i nes . F, , F, arethe tw o r eal foc i of the curve . t, t2

’

,which we may cal l F3

and F, , a r e the i n tersection s of non - conjugate imaginary l i nesand ar e imagi nary po i nts . They are, however, themselves conj ugate imaginar y poi n ts be i ng i ntersect ions of two conjugateimaginary pa i rs (Ar t. Hence F,,F,1 i s a real l i n e .

But by Ar t. 61 the d iagonal tr iangle of the complete quadr ilater al c i rcumscri bed to the con i c i s sel f- polar w i thregard to the con i c . But the s i des of th i s d i agonal tr iangle areF,F, , F3F4 , Q Q

’

. The mee t of F,F, , F,, F, i s therefore the pol eof Q Q

’

,i . e . the centre 0 of the con ic ; F3F4 , F,F2 a r e then con

j uga te d iameters . By the harmon ic property of the completequadrangle F the two s ides of the diagonal tr iangl ethrough 0,

v iz. OQ,0Q

’

,ar e harmon ical ly conj ugate to the tw o

s ides of the qu adrangle through 0,namely OQ

,

CQ’ be i ng c i rcu lar l i nes a r e perpend i cu lar and so

must be axes . The tw o imagi nary foc i therefore lie 011 wha t w ehave cal l ed h i therto the non - focal axi s of the cu rve .

1 70. C on foc a l c on i c s . If two foc i F, , F,, of a con ic be1 1— 2

1 64 PROJECTIVE G EOMETRY [e rr

given , the other foc i F,, a r e determi ned . For they ar e ther ema i n i ng vertices of the complete quadr i lateral formed by thefour l i nes F,Q, F,Q

’

F2Q ,F, Q

’

In par t i cul ,ar coriics which have the same tw o real foc i haveal l their foc i the same . Such con ics are cal led confocal con ics .They touch four fixed l i nes

,namely the s ides of the quadri lateral

mentioned above .

171 . T h e c i r cu l a r p o i n ts a r e foc i of a p a r a b ola .

In the case of a parabola the l ine at i nfin i ty Q Q ’ i s a tangent .Thus t, , t,

’ coi n c i de with Q Q ’

. The quadri lateral of tangentsfrom Q ,

Q’ reduces therefore to a tr iangle . F, , 1 . e . (t, t, r ema i nsas the only rea l focus of the cu r ve at a fin i te d istance , F2 i s thepoi n t of contact of the l i ne at i nfin i ty , re the poi n t at i nfin i tyon the axi s . F3 and F,, become i ntersect ions of t, and t,

’wi th thel i n e at i nfi n ity, that i s, they coi nc ide wi th Q ,

Q’ wh ich are thu sfoc i of the curve .We have ther efore an exception to the theorem of the last

Ar t icle, for the givi ng of Q, Q’ does not here determin e the

other foc i .

172 . Im a gi n a r y p r oj e c t i on s . By means of the c ircu larpoin ts a n umber of impor tan t theoret i ca l results i n proj ectioncan be deduced .

Thus any tw o con ics can alway s be proj ected s imul tan eouslyin to c i rcles .For let A, B be any two of the i ntersections of such con i cs .

Then by p roj ect i ng A , B into the c i rcular poi n ts i n any plan e,the con i cs ar e proj ected i n to c i rcles .Thi s r esu lt 1s of great importance, s ince i t enables rrs to

apply to a pa i r of con ics any proj ecti ve theorem proved for apai r of c i rcles.Thi s proj ection of two given poi nts i n to the c i rcu lar poi nts

i s of course imagi nary i f the two grverr points are real . If thetw o gi ven po i nts are conj ugate imag inary poi nts

,they wi l l 1 11

general be gi ven as the i ntersections of a stra ight l i ne a; wi tha con ic s, when x and 3 do not cu t i n r eal po i nts . Take 0 thepol e of a; wi th r egard to s and two pai rs (OP , (OQ, OQ

’

) ofconj ugate l i nes through 0 wi th r egard to 3 . Project .r to i nfin i tyand the angles P OP ’

, QOQ i nto r ight angles (Ar t 0 proj ectsi n to the centr e of the con ic and (OP , OP ) , (OQ, 0Q

’

) i n to pa i r sof conj ugate d iame ters at r ight angles , i . .e i n to axes . Bu t s i ncea con ic w i th more than on e pa i r of axes must be a c i rcle, 3

proj ects in to a c ircl e and i ts i ntersections wi th .r i n to the inter


1 66 PROJECT IVE G EOMETRY [err

struct the con i c hav i ng S, O for foc i and the c i rcl e w ith centre

0 and radi us CA for aux il i ary c i rcle . Thi s con ic touches thetangent at A to the c i r cle centr e S ,

for th i s tangent i s per pendicu la r to SA and A i s a poi n t on the aux i l iary c ircl e of thecon i c see Ar t. Si rrr ilar ly the con ic touches the tangent atB to t e c i rc le cen tre S and the tangents at A , B to the c irclecen tre 0Cons ider now the other i n ter section s of the two gi ven c ircles ,

n amely Q,Q’. The tangents to the c i rcle centre S at Q ,

Q’passthrough S s i nce

'

S i s the pol e of Q Q ’with regard to the c i rcl e .They are therefore S Q ,

S Q'

. But these are a l so tangents to thecon ic, s i nce S i s a focus .I n l ike manner the ta ngents at Q

,Q

'

to the c i rcle centre 0are tangen ts to the con ic .Hence the e ight tangen ts at the fou r common poi n ts of two

c i rcles touch a COIl l C. Proj ecti ng the c ir cles back into any tw ocon i cs w e obta i n the r esultThe eight tangen ts to tw o con ics at their four common points

touch a con i c .

Reciprocat ing th i s theorem w e obta i n the fol lowi ngThe e ight poin ts of contact of the four common tangents to

two con i cs l i e on a con ic .

1 74 . I n v olu t i on r a n ge on a c on i c . Let S be a con i c,0 any poi n t i n , i ts plane . Let P be any poi n t on the con ic.Joi n OP meeti ng the con ic aga i n at P ’

. The correspondencebetween P and P ’

is on e - one and algebra i c , the ranges[P

2, [P

'

]2 are homograph ic . Also i n th i s constru ct ion P

an P’ may be i ntercha nged ; hence P ,

P’ cor respond doubly .

The r ays thr ough 0 therefore determ in e an invol u tion range onthe con ic , of wh ich the doubl e poi nts are clearly the poi nts ofcon ta ct of tangents from O to the con ic .Con verse ly let there be an i nvol ution r ange on the con ic ,

of wh ich ( .P , (Q, Q’

) are two pai rs of mates . Let PP’

, QQ'

“

meet at O. Compare the given i nvol u tion wi th the one obta inedfr om the i ntersect ion s of rays through 0 wi th the con i c . Thesetwo in volu t ion s have two common pai rs of mates, namely (P ,

(Q, they are therefore i denti cal by Ar t. 1 49 . Hence join sof mates of an i n volu tion r ange on a con i c pass through a fixedpoi n t whi ch rs cal led the centr e of the i nvolu t ion on the corr rc.

Al so by the property of the cross axi s of tw o p r oj ecti ver anges on a con i c (Ar t. 1 1 1 ) the meets of cross - joi n s (P Q, P

’

Q’

)and (P Q

'

,P

’

Q) l i e on a fixed l i ne pass ing through the doubl e

x] INVOLUTION 1 67

poi nts . Thi s rs cal led the ax is of the involut ion and rs clear lythe polar of the centre of i n vol u tion .

By means of thi s property the double po i n ts of an i nvolut ionrange on a con ic can be i

(

mrnediately corrstr rrcted as soon as tw opai rs of mates (P , ( ,Q Q

’

) are gi ven . For the axi s of 1 11v olrrtion 1s the j e r r i of (P Q,

P’

Q’

) and (P Q'

, P Q) and thi s meetsthe con ic at the double points requ ired .

175 . C on s t r u c t i on of d ou b l e r a y s of a n i n v o lu ti on

fl a t p e n c i l . The above property of the i n vol ut ion range on acon ic may be u sed to constr uct the double r ays of a flat penc i li n i nvol u tion of which tw o pai rs of mates (p , p ) , (q, q

’

) are g iven .

Descr ibe any c i rcle (or con ic) through the ver tex of the penci l .The i nvolu t ion penc i l determ i nes 011 the c i rcle an i n volu t ionr angefiln wh ich ( P , P ) , (Q, Q

'

) ar e pairs of mates ; P , P , Q, Q'

bei ng the poi nts where p , p’

, q, q’ respect i vely meet the c ircle .

Dete rm i ne the doubfl

le poi nts of th i s i nvolu tion range on the

c i rcle as above and j o in them to the ver tex of the flat penc i l .The joi n s gi ve the dou ble rays r equ i red .

176 . 1 n v o lu t i on of ta n ge n ts to a c on i c . By r ecipr ocati ng the theorems of Ar t. 1 74 we obtai n the r esu lts : matesi n an i nvol ution of tangents to a con ic meet on a fixed l i n e

,

which we cal l the i n volution axi s . Also j oi ns of cross - meets( p g, p

”

,q ) (p q'

, p'

q) pass thr ough a fixed poi nt,which we call

the i nvolu tion centre . I n th i s,as i n other theorems orr r ecipr o

cat ion , the reader w i l l find it a usefu l exerc i se to construct theproof of the rec iprocal theorem from that of the gi ven theorem ,

by rec iprocat i ng each step .

The double tangen ts of the i n vol ut ion are clearly the tangentsat the poi nts where the i nvol ut ion ax i s meets the con ic . Al so,as i n the case of the range, the centre and ax is of involu tion arepol e and polar w i th regard to the con ic .

From two pai rs of mates (qq'

) the cen tr e and ax i s ofi n vol u t ion are at once cons tructed and ei ther of these w i l l g i vethe double tangents .

177 . T h e Fr é gi e r p o i n t . An i n volu t ion flat penc i lwhose vertex i s on the con i c determ i nes a n i n vol u t ion of poi n tson the con ic . I n particu lar, i f the i nvolu t ion penc i l be rec tangu lar,we reach the fol low i ng theorem . I f 0 be any poi nt on a con ic ,OP ,

OP’

two perpend icular chords , mee t i ng the con i c a t P , P

respecti vely , P P passes th r ou r r lr a fi xed poi n t F. Tak i ng Pcoi nc iden t wi th O, OP ,

OP are the ta ngen t and norma l a t O and

1 68 PROJECTlV E c so rr rr r n r [e rr

PP’ coi nc ides wi th the normal at O. The fixed poi nt F there

fore l ies on the normal at O. The poi nt i s called the Fr e’gierp oin t fr om its d iscoverer .I f the con ic be a recta ngular hyperbol a and OP , OP

' bedrawn parallel to i ts asymptotes, P P

’

, and therefore the Fregi erpoi n t

,i s at i nfin i ty . In any other pos i tion , therefore, P P

' i sparallel to the normal at 0. Thus if on any chord PP '

of arectangular hyperbola as d iameter , a c i rcl e be constructedmeet i ng the curve at O

,O’the normal s at O, O

’ are paral lelto P P

178 . I n v olu t i on a xi a l p e n c i l . The propert i es of an

i n vol ut ion of planes through an ax i s are closely s im i lar to thoseof an i nvolu t ion of copla nar r ays through a po i nt. Such an

i nvol ution determin es cor r espondi ng involu t ion s , of poi nts on

any stra ight l i n e wh ich cu ts i t, of r ays on any plane which cu ts

i t. By con stru ct i ng the double el ements of e i ther of these thedouble planes of the axi al penc i l may be fou nd . As befor e , i ftw o i n volu t ions of planes ha ve the same ax i s they have one pairof common mates, which i s always real un l ess the two giveninvol u tion s have tw o pa i rs of r eal double planes wh ich ar e

separated by on e another .The r elation between the d ihedral angles of six planes in

invol ution i s found by tak i ng a sect ion by a plan e per pend icularax i s . The angl es of the flat pen c i l so found mea su r e the d ihedr ala ngles of the axi al penc i l . These are therefore conn ected by theformulae of Ar t. 1 59 .

Also planes at r ight angles form an i n vol ution of which thedouble planes pass through the c ircular poi nts at infin i ty in theplan e perpendicular to the axi s .Prec i sely as at the end of Ar t. 1 62 we can show that every

i nvol ution of planes through an ax i s has one pa ir of perpendicular elemen ts .

EXAMPLES X .

l . Prove that any i nvolut ion penci l can be projected in to arectangular involut ion .

2 . Show that the tangents at the po in ts of an involut ion on a conicform an i nvolut ion of tangen ts having the same axis and cen tre as thegiven involut ion of po ints.


1 70 PROJECTIVE G EOMETRY [err

1 3. G iven two poin ts A , B on a con ic,find two other points P , Q

on the con ic such that A and B shall lie on a circle of wh ich P Q is ad iame te r.[P , Q ar e the i n tersections w i th the con ic of the l ine join ing the

Fr égie r points correspond ing to A and B .]

1 4. I f (A , A’

) be a fixed pai r of mate s in an involut ion on a s traigh t

and find the value of th is constan t in terms of the d istance s ofA ,A

’

from the centre of the i nvolut ion .

15 . Show that any l ine through the cross - centre of tw o project ivepenc ils nreets the tw o penc i ls in an i nvolution . F i nd the nrate of the

cross - cen tre : - find also the cen tre of th is i nvolution when the givenpenc i ls ar e penc i ls of paralle l rays .

1 6. Fr om the first resu lt of Ex. 1 5 prove that the three pai rs ofoppos i te s ides of a comple te quadrangle mee t any straigh t l ine i n threepairs of an involution.

17. The s ides B C, CA ,AB of a tr iangle AB C mee t a straigh t l ine

at P, Q , B . I f P

’

, Q’, R ’ar e mates of P

, Q, R i n an involution , provethat P ’

A, Q

’B , R

’

C ar e concurren t.1 8. I f through the vertices of one tr iangle l ine s a , , b, , c, be drawn

paralle l to the s ide s of another triangle, and through the vertice s of thelatter triangle l ines a , , b2 , 02 be drawn paralle l to the s ides of the fi rsttriangle , prove that i f a, , b, , c, ar e concurrent, so a r e a , , c, .

1 9. A,B ar e fixed poi nts on a fixed tangen t a to a con ic s. P

,P

’

a r e harmon ically conjugate w i th regard toA ,B. I fp , p

'be the tangen ts

from P , P’to 3 , show that pp

’ l ies on a fixed straigh t l ine .

20. I f a figure be i nverted w i th rega rd to any orig in , show that ani nvolut ion on a c i rcle i nve rts i nto an involut ion on the correspond ingc i rcle.

2 1 . Show that in any con ic i f G ,G

'

be the poi n ts where the normalat P mee t the axes, F the Fr égie r poin t correspond i ng to P ,

then P,F

ar e harmonically conjugate w i th regard to G ,G’.

22. Show that in a parabola the locus of the Fr égier poi nt is anotherparabola, equal to the g iven one .

23. A system of con ics through four points A , B,0, D de termine s

an i nvolut ion 011 any con ic through two of them, A and B.

I n part icular a sy stem of coaxial c i rcles de te rmines an involut ion 011

any given c ircle.

x] EXAMPLES 1 71

24. Three chords A ,A, , of a c i rcle a r e concurren t. I f

0 be the cen tre of the c ircle , prove the relat ions in $ B ,OC sr rr s i n 5A OB

sin $ B .OC , s in § OzOA srn 5A ,OB ,

and s imilar relat ions .

25 . Show that i f P ,

-P’be a variable pai r of points on a l ine

,

symme trically s i tuated wi th regard to a poi n t 0 i n the l ine , P , P’ar e

urates i n an involut ion : and find the double points of th is i nvolut ion .

26. P is a poin t on a fixed straigh t l ine a,wh ich mee ts a con ic s at

A ,B. The tangents from P to 3 mee t the tangent t to 3 parallel to the

tangent at A or B at P P I f C be any fixed poin t on t, prove thatCP , O

’P 2 constan t.

27. Show that any tw o concentric project ive penc i ls i n a plane canalways be proje cted i n to d irectly equal penc i ls .

28. Show,by cons idering the c i rcle as the product of tw o d irectly

equal penc ils and apply ing the construct ion of Ar t. 1 16 for its asynrptotes,that each of the circular lines through a point may be looked upon as

mak ing any g iven angle w i th i tself.

29 . S irow an alyt ically that the c ircular l ines ar e parallel to the l inesy : i i t and ver ify that they make the same angles i tan" i wi th everystraigh t l ine i n the plane .

30. D iscuss the form assumed by the anharmonic property of fourfixed poin ts and one var iable poi n t on a con ic

,when two of the fixed

poi nts ar e the c ircu lar poi nts .

3 1 . Prove that if t be the product of conjugate penc i ls w i th regardto a con ic 3 through two poin ts A ,

B,then AB has the same pole w i th

regard to s and t.

32. Show that, i f t be the produc t of conjugate ranges w i th regardto a conic s 011 tw o straight l ines a , b, then the poi nt U where a

,b mee t

has the sanre polar w i th regard to s and t.

Prove al so that i f a tangen t to t mee t 3 at P , Q the l ines UP , UQar e harmon ically conjugate w ith regard to a

,b.

33. Deduce from Ex. 32 that i f mates in an involut ion penc il mee ta conic at P

,P

’

,then P P ’ touches a fixed conic.

34. Prove that chords of a con ic s wh ich subtend a right angle at afixed po in t 0 enve lop a conic of wh ich 0 is a focus and the polar of 0w i th regard to 3 is a d irectrix.

1 72 PROJECTIVE G EOMETRY [011 x

35. Show that i f a s imple quadrilateral exist wh ich is i n scribed i na con ic s and c i rcumscribed to a con ic there exist an infinite numberof such s imple qrradr i late r als and they have the same i ntersect ion of

d iagonals.

36 . Prove that i f OA,OB be two l ines inte rsect ing at O the cross

rat io 0 {AOB Q'

} e” ,where 6 = angle AOB .



wh ich are pr oj ect ion s of the same range, and such forms a re

cer ta rnly not 1nc1dent.

180. C on st r u c ti on of c op l a n a r h om og r a ph i c fo rm s

of th e s e c on d or d e r . Let tw o ranges of the second order ontw o con i cs s, and 32 be given by two correspond i ng tr iads

A ,B , O2 (Fig . Correspond ing poi n ts of the tworanges may be constructed as fol lows .Join A,A 2 meeti ng 3 , at U and 32 at V. Let UB , , VB ,

meet at B3 and U V02 meet at 03 . Joi n a , meet ing

Fig. 48.

UV at A3 . Then i f P , be any poi n t on s, and UP l meets a ,, atP 3 and VP 3 meets 32 at P , , the ranges of the second order

[P 2

2ar e homogr aph i c and they have A ,B ,O, for

cor r espon i ng tr iads . They are therefor e the r anges requ i red .

I f one of the ranges,say i s of the fi rst or der, a

s im i lar constru ct ion holds, but th is t ime V may be taken anypoi n t on A ,A , .

S im i larly i f tw o penci l s of second order about con ics s, , s,be given by the cor respond i ng triads a 2b2c2 (Fig.

XI ] HOMOG RAPH IC PLANE FORMS OF SECOND ORDER 1 75

then from A a , a ,) dr aw the tw o ta ngents n,v to a, , s, . Let

ab, = B , , ac, ob = B , , cc,= Let B , B ,

z b

and let 0 be the i r i ntersection let OA a 3 .

'l‘heu i fp , be any

ta ngen t to 3 , meeti ng a at P , , and if OP , be joi n ed to meet v atP , and p , be drawn fr om P , to touch s, , the penc i ls Of the secondorder [p ,] [p ,] ar e homogr aph i c and have a , b, c, for

correspond i ng tr iads .A s im i lar construction holds i f the pen c i l [p ,] i s of the first

order, on ly now a may be taken any l i n e through A and p , i sjoi n ed to the vertex of the pen c i l [p ,] i n stead of be i ng draw nta ngen t to a con i c .I f the gi ven forms are u nl ike

,say a range and a penci l of

second order,w e can correlate as above the gi ven range of the

second order w i th the range formed by the poi n ts of con tact ofthe gi ven penc i l of second order . In th i s way the tw o Or igi nalforms are geometr ical ly connected .

1 8 1 . Num b e r of s e l f - c or r e sp on d in g e l em en ts of

h om og r a p h i c form s of fi r st a n d s e c on d o r d e r,n ot on

th e s am e b a s e . Clearly a range of the first or der and one of

Fig. 4 9 .

the second order cannot have more than tw o self- correspondi ngelemen ts s i nce a stra ight l i ne meets a con i c i n two poi n ts only .

They may have tw o sel f- correspond ing e lemen ts, for i f we takea fiat penc i l whose vertex i s on a con ic , the ranges determ i nedby thi s penc i l on the con ic and on any s tra igh t l i ne have thei ntersect ion s of the s traight l i n e and con i c for se lfcorrespond ingoi nts .pConversely i f such ranges have two sel f- correspond i ng poi n ts ,

sayA , B ,the l i n es joi n i ng the i r correspond i ng poi n ts pass through

a ver tex 0 ly i ng 011 the base Of the r ange of second order . For

1 76 PROJECTIVE G EOM ETRY [e rr

l et 0, 0’be corr espond ing poi nts 011 the straight l i n e and con i c

r especti ve ly (Fig. Join CO" meeti ng the con i c at 0. Theni f P , P

’ are on a l i n e through 0,the ranges [P ] , [l

P’

]2 are

proj ect ive . But they are determi ned by the same tr ia 8 ABC,

ABO’as the or iginal ranges . They are therefore iden ti cal w i th

these ranges .In l ike man n er a penci l of the first or der and one of the

second order may have tw o self- correspondi ng elements,namely

the tangen ts from the vertex of the first penc i l to the con i cwh i ch i s the base of the second penc i l , but they cannot havemore.Al so

,when they do have two self- corresponding l i n es

, w e

Fig. 50.

can show, by reason i ng s im i lar to that u sed for the ranges , thatcorrespondi ng l i n es i ntersect on a tangent to the con ic wh i ch i sthe base of the penc i l of second order .

1 8 2 . Tw o h om og r a p h i c fo rm s of th e s e c on d or d e r

c a n n ot h a v e m or e th a n th r e e s e l f - c o r r e sp on d i n ge l em e n ts . Two ranges of the second order have the i r bases3 , and s, i n tersecti ng i n fou r po i n ts, but they can not havemor e than three se lfcorrespond i ng poi nts .

In the fi rst place we wi ll show that they may have threeself- cor respondi ng poi nts . For l et 0, A , B , 0 (Fig . 50a ) be the


1 78 PROJECTIVE GEOMETRY [e rr

longer appl i es, because the r ange determ i ned by a penc i l of thesecond order upon any stra igh t l i ne i s not homogr aphic w ith thepenc i l , u nless the straight l i n e happen to be a tangent to thecon i c wh ich i s the base of the penci l . The student can eas i lyconv i nce h imself of thi s by r evers i ng the process , when he w i l lfi nd that

,although to each ta ngent to the con i c corresponds only

one poi nt of the strai ght l i n e, to each poi nt of the stra ight l in ecorrespond two tangen ts to the con ic . The cor respondence i stherefore not on e—one.

Con s ider a. range of the second order [P ]2on a con i c s and

a homogr aphic penc i l [p’

] of the first order whose vertex i s U.

Take a vertex 0 on s and joi n OP p . The pen c i l [p ] i s of thefirst order and The locu s of Q :

pp’ i s a

con i c t pass i ng through 0. The con ies s, t have ther efor e thr eeother i n tersec t ions bes ides O, of which at least on e i s r eal , s i nceby Ar t. 1 25 imagi nary in ter sections occur in pa i rs and O i sal ready one r eal i ntersection . But at an in ter sect ion of s, t ther ay p

’passes thr ough i ts corr esponding poi nt P , and, conver sely,i fp

’pass through P , P i s an i nter sect ion of s, t. Hence ther e a r ethree pa i rs of cor r espond i ng elements inc iden t

,of w hich on e pa i r

is always r eal .This holds even i f 0 be on i ts cor r espond ing r ay , for then , i f P

be at O, OP i s tangen t to the con ic it (s ince i t cor respondsto the join of the ver ti ces) . But OP i s al so tangen t to thecon i c 3 . Thu s 3

,t tou ch at O and have on ly two other i nter

sect ion s .If then there were a fourth pa i r of corr espondi ng elemen ts

in c i den t, the con ics s, t wou ld have five points common and wou ldcoi n c ide every pa i r of cor respondi ng elements wou ld be inc ident

,

and, s i nce the penc i l and the r ange are homogr aphi c , th i s can bethe case on ly i f the vertex of the penc i l l i es on the con ic wh i ch i sthe base of the r ange . Otherwi se to each r ay of the penc i l wouldcor r espond the two poi nts i n wh ich i t cuts the con i c .Rec iprocati ng thi s theorem w e obta i n the corr espondi ng

theorem for a range of the first order and a penc i l of the second .

The proof i s prec ise ly s im i lar to the one above i f we i n terchangeterms accordi ng to the r ules gi ven i n Ar t. 65 . The resu ltr u ns :I f a penc i l of the second order and a r ange of the first order

be homograph ic, then , i n general , three pai r s of corr espond i ngelements are i nc iden t, of wh ich at least one pa i r are r eal . If morethan three pa irs are i nc ident, the tw o forms are al together i nc identand the base of the r ange touches the base of the penc i l.

Hor rocRAPH rc PLANE FORMS OF SECO ND ORDER 179

18 4 . T h e p r odu c t of tw o h om og r a p h i c p e n c i ls of

th e fi r s t a n d of th e s e c on d or d e r r e sp e ct i v e ly i s a

cu b i c . I f we for rrr the produc t of tw o coplanar homograph icpenci ls [p ] , w e obta i n a cu rve i n the plane. Draw anystra igh t l i ne a m the plane . Thi s meets [p ] i n a homogr aph icrange of the fi rs t order [P ] . Then

,i f Q :

pp’

, a poi n t P of u i son the locus of Q i f i t l res on i ts correspond i ng l i n e p

’. By the

last Article there are three such poi n ts P on every l i n e a, of

which one at least i s real . The locus i s ther efore one of the th i rddegr ee, or , as i t i s cal led shortly, a cubic .

18 5 . T h e v e r tex of th e p e n c i l of th e fi r st o r d e r i s

a dou b le p o in t on th e cu b i c . Let 0 be the vertex of thepenc i l of the first order

,s the con i c which i s the base of the

penc i l of the second order . Let a’

, c’be the tw o tangents from O

to the con ic, a , 5 thei r correspond i ng rays through 0.

Then 0 appears tw ice on the locus , once a s a n’and once as cv’.

Also correspondi ng to these two i nterpretat ions of 0 there i s ad i fferent ta ngent to the cu rve . For i f p

’ approaches a ’, p ap

pr oaches n and the point s p’ approaches 0 so that OQ

approaches a . a i s therefore on e tangent to the curve thr ough 0.

S im i larly i f p’approaches r ’, Q approaches 0 as OQ approaches a .

So that v i s another ta ngen t to the curve at O. The cu r ve hastwo branches wh ich i ntersect at 0.

Such a poi n t 0 i s cal led a double point on the curve and everyl i ne thr ough 0 i s cons idered as meeti ng the curve i n tw o coine ident poi n ts at O. The on ly case of a double poi n t wh ich wehave met wi th h itherto i s that of the degenerate con ic or l in e- pa i r,where the i nter section of the tw o l i nes i s a double poi n t .Thi s we can ver i fy by not i ng that a r ay through 0 meets the

locus aga i n at on e oi n t on ly, as i t shou ld , to w it, at the poi n twhere i t i s met by t e correspond i ng r ay of the penc i l of secondorder . There are tw o exceptions , however , name ly the rays a ,

These meet the crr r ve i n three coi nc ident poi n ts at O and areknow n as the proper tangen ts to the cu rve a t 0.

I t may be Shown that,i n order that a cubi c may have a

double poi nt, a certa i n condi t ion m rrst be sa t i sfied . Hence thecubic of the presen t Arti cl e i s not of the most general type .

18 6 . C on st r u c ti on of d i r e c ti on s o f th e a s ym p tote s

o f th i s c u b i c . The poi n ts where the cubic mee ts the l i ne a ti nfin i ty i may be constructed as follows . Draw a ta ngent a to

s (Fig. 5 1 ) from 0. Let a'

be the ta ngent correspond i ng to1 2.— 2


the r ay a thr ough 0. Let 0’ be the tangent paral lel to a , c i ts

cor r espond ing ray. Let b,b' be any other pa i r of cor r espoii ding

l i nes . Let a,b, c meet z at A

°°

B°° and a '

,b’

, 6’meet a at

(0m

: A“D I f p , p be any pa i r of cor respondi ng

l in es (not shown 111 Fig. 5 1 ) meeti ng”

2 , a a t P°°

, P' respect i vely

the r anges [P°°

[P’ are proj ect i ve , and A

”

B”

0” are

two correspond ing tri ads . The parall el P ”

P’ through P ’

to ptherefore en ve lops a parabola which touches a and t at A’

, C”

Fig. 5 1 .

respecti vely,these bei ng the correspondents to a c C — A

°°

Thi s parabola al so touches B 'B,i e . the parallel through B ’

to b.

We are ther efore given one tangent B ’

B , another tangent a andi ts poi n t of con tact A '

,and the d i rection of the ax is 0. The

parabola can then be draw n by Br ian chon’s 'l heor em . The thr eecommon tangents to the parabola and to 3 , other than a ,

then gi vethe d irections of the three asymptotes of the cubic . Fany on e of these common tangents

,meeti ng a at T , and


1 82 PROJECTIVE G EOMETRY [OH.

as two, s i nce 0 i s a double poin t. There a r e accor d ingly fou rothers A , B , U, D . Cor respondi ng to each such i ntersect ion wehave a poi nt P on i ts cor respond i ng l i n e ( s i nce P and Q are thenco i nc ident) .Or , i f it be des ired to avoid the above a peal to analyti ca l

cons iderations, we may proceed as fol lows . T e locu s of Q mustmeet 32 at some poi n t A (real or imagi nary) . Through such apoi n t A the correspond ing l i n e a passes . Take for 0 the poi n twhere a meets agai n . The penc i l s 0 [P ] , [p ]

2 have now a for

a selfcorrespond i ng ray. The locus of Q now r educes to a con i c

Fig. 52 .

through 0 (a be i ng i rrelevant) . Th i s con i c 1) (shown by thedotted l i n e i n Fig. 52) cuts 32 at three other poin ts B ,

C, D ,

wh ich a r e i n c ident wi th the i r correspond i ng l i n es . 0 i s not

inc iden t w i th i ts correspond i ng l i ne , u nless c tou ches at 0 ;for i f p passes through 0, Q i s at 0 and P i s where OQ, i .e . thetangent at 0 to cuts tha t i s, P i s not at 0 i f v , 32 do not

touch at 0. But i f they do touch at 0,then 32 have only two

other poi n ts of i ntersec t ion and there a r e s t i l l only fou r poi n ts onthei r correspond i ng l i n es .

Hence i f there be a fi fth po i n t B through which passes i tscorrespond i ng l i n e 6 , the con i cs v , 32 have five points in common

HOMOG RA PH I C PLAN E FORMS OF SECOND ORDER 1 83

and coi nc i de enti rely . Thus every poi nt l i es on i ts correspond i nghue and the tw o g i ven forms are me lden t.

1 90. P r odu c t of c ob a s a l h om ogr a p h i c for m s of th e

s e c on d or d e r . Let [P 2]? be tw o homographi c ranges Of

the second order on the Same con ic 3 . Let A A ? be a gi venpa i r Of correspond i ng poi n ts of these ranges and STthe cross - axi s(Fig . 53 Then A ,P z , A QP l meet at U on the cross - axi s .Projec t S

,T i n to the c i rcular po i nts at i nfin i ty Q ,

Q’

. 3 proj ectsi n to a c ircle s

'

(Fig. 53 U i nto a po i n t U Therefore A I

’P Q

’

,

A Q

’

P I

' are paral lel , and the arcs A I’P I

’

,A Q

’P Q

'

a r e d i rectly equal .ranges on the c i rcle are thus determined by

Fig. 53.

d irectly equal flat penc i ls whose vertex i s on the c i rcle . The a r eP I

’P Q

' sub tends a fixed angle a t the c i rcumference and thereforeat the centr e . Hence the chord P l

’P g

’ touches a fixed concen t r icc i rcle t'.Now tw o concentr i c c i rcles touch on e another at Q

,Q’

. Fori f 0 be the i r common cen tre the tangents a t Q ,

( 2' are CO , 09

’

a nd are the same for both . Projec t i ng back i n to the or igi nalfigure

,P 1 P 2 touches a fixed con ic t wh ich has double contact

w i th the or igi nal con ic at S and T.

Rec i proca t i ng thi s theorem ,w e see that the product of tw o

homogr aph i c penc i ls Of tangen ts to the same con ic s i s a con ic

hav i ng double con tact w i th the Or igi nal con i c along the tw o

tangen ts to th i s con i c from the cross - centre Of the penc i ls .

1 84 PROJ ECTI V E G EOMETRY [OH .

In the spec ial case where these homogr aphi c forms a r e ini n vol ution , the above envelope and locus degenerate i n to a poi n tand a stra ight l i n e r espec t i vely

,the poi n t appear i ng as the

i n tersec t ion of the tw o componen ts of a l i n e - pa1r and the l i n e asthe joi n of the componen ts of a poi n t pa i r . The l i n e - pai r i n thefirst case i s the pa i r of tangen ts from the centre of

.

i n vol ut ion tothe coni c and the poi nt pa i r in the second case i s the pai r ofpoi n ts at wh i ch the ax 1s of in vol u tion meets the con ic .The above theorems a r e of cons i derable importance and a

number Of i n teresti ng part i cu lar deductions flow from them .

I n part i cu lar let there be tw o ranges on the l i n e at i nfin ity2 defined by the i ntersect ions wi th i ” Of tw o d irectly equalpen c i ls m which correspond i ng rays make an angle a with on e

a nother;and let Q

”

, Q be correspondi ng poi nts Of these r anges .S i n ce i ” touches every parabola, the ta ngents from Q , Q to

a parabola defin e two homographic penc i ls of tangents to theparabola . The i r product 13 a con i c meeti ng 2 at the tw o pointscorr espond i ng to the poi n t of contact of i ” wi th the parabola.

Hence w e have the theorem . the locus of i ntersections of twota ngen ts to a parabola wh ich make an angle a (other than r ight)with on e another 1s a hyperbola whose asymptotes make an anglea with the axi s of the parabola.

1 9 1 . H om og r a ph i c i n v o lu t i on s . We may tr ea t a pai rof mates i n an i nvol ution as a s i ngle en tity and establ i sh a one

one algebra i c corr espondence between the pa i rs of one i nvolut ionand the pai rs of another ; su ch correspondence does not establ i shany one - one r elation between the i nd iv idual mates, but onlybetween the pai rs as a whol e . Tw o i n volu tions cor r elated inth i s w ay wi l l be cal led homograph ic .

We wi l l first show how to der i ve tw o homograph ic i nvol ut ionranges on the same con ic 3 one from the 0

‘

.tl18 1 Let (P I ,

(P 0 ,P 2

’

) (Fig . 54) be tw o correspondi ng pai rs ; l et 0 02 be thecorr espondi ng i n vol ut ion centres and 191 , p . the rays through0 O2 determ i n i ng the correspond ing pa i rs 011 3 . Then by hy pothes i s the rays p , , p 2 a r e connected by a one - one algebrai ccorrespondence. '

l he penc i ls [p ] are homographic : i fQ : p un .

) the locu s of Q 1s a COIl l C wh ich meets s at four poi ntsA , B ,

0,J) .

When Q i s at A , one poi n t of a pai r (P 1 , P l

’

) of one i nvolu tioncoi nc ides w i th on e Oi nt of the cor i esponding pa i r (P 2 , P .

’

) Of theother i nvol u tion , t 10ugl1 i t should be noted carefu l ly that thepai rs as a whol e do not 1 11 gen er al coi nc ide . Such a poi n t as A


1 86 PROJ ECT IVE G EOMETRY [OH .

We noti ce first that, s i nce , from the last Arti cle , the r elationbetween tw o homogr aph ic i nvol utions i s determ i ned by a r elationbetween tw o homograph ic simple forms

,which lat ter i s i tself

determi ned by tw o cor respond i ng triads , tw o correspondi ngtr iads of pai rs can be arb i trar i ly assumed and complete ly de term i ne the r e la t ion between two homograph i c in vol ut ions .Suppose now that A B the double elements of one

i n vol u t ion,and A B 2 , the double elemen ts of another i n vol ution ,

correspond . Let the pai r (01 , Cl’

) correspond to the pai r (02 ,Then by the property of invol u tions that any pa i r are harmon i cal ly conj ugate wi th regard to the double elements

,w e hav e

{A IOI B ICI’

}and the sets of four e lemen ts A ICIB IOI

’

, A 2028 202’ can be

brought i nto homograph ic correspondence . Now the homegr aphic correspondence thus defined w i l l transform the inv olut ion (P I , P l

’

) i nto a homographi c i nvol ut ion (P 3 , P s

’

) cobasal w i th(P 2 , P g

'

) a nd homograph i c wi th i t . But the cobasal homograph ici nvol utions (P 3 , (P 2 , P 2

’

) have three self- correspondingpai rs, namely the double elements A B 2 and the pa i r (02 ,Therefore they must be iden tical . Hence th i s homogr aph i ccorrespondence conn ects P I with P 2 and P I

’with P g

'

,i .e. there

i s a homograph i c correspondence between the i nd i v idual compon eu ts of the pai rs . S im i larly a homograph ic correspondenceexi sts wh ich con nects P I with P 2

’ and P I

’ with P 2 .

19 3 . P r odu ct of tw o h om og r a p h i c in v olu t i on s of

th e fi r st or d e r . I f we form the product Of tw o homogr aphi ci n vol ution pen c i ls of vertices 01 , 02 , that i s , find the i ntersec t ion s171 11 2 , p i p a

'

, p l'

p z , p i p; where ( P2 , p") are two corresponding pa i r s, these i n tersect ions l i e 011 a cer tai n locus .

As i n Ar t. 1 1 5 w e proceed to find the i ntersections Of th i slocu s w ith any stra ight l i ne x. The two i n volu tion penci lsdeterm i ne on x two col l i n ear homograph i c i n vol u t ions Of which(P , , P l

’

) (P 2 , P g

’

) a r e correspond i ng pa i rs , where P etc. l f

ei ther Of the poi n ts P , , P l

’ coi nc i de w ith e i ther of the poin tsP 2 , P the po i n t of coi nc idence l i es 011 a ray of each Of twocorrespond i ng pai rs of the gi ven homographi c i n volution penc i l s

,

that i s , i t l i es O11 the i r prod uct.But

, by the last Arti cle , there are fou r such sel f- correspondi ngpoi n ts Of the col l i near homographi c i nvol ut ion ranges (P , ,

(P 2 , The locus therefore meets any straigh t l ine i n fourpoi n ts , that i s, i t i s a cu rve Of the fou rth degree .

Al so the ver tices 02 a r e double po i n ts 011 th is cu r ve. For

HOMOG RAPH IC PLANE FORMS O F SECOND ORDER 1 87

let (a , , be the pai r of the penc i l vertex 01 cor respond i ng tothe pai r Of the penc i l ver tex 0, Of which 0201 i s a component .As the ray p 2 approaches 0201 , (p , , approach ( a , , andtw o poi n ts 011 the locus coi nc ide at mov i ng u l t ima tely alonga , , 01 i s thu s a double poi n t, (a , , be i ng the properta ngents at In l ike manner 0; i s a double poi nt and , i f

i s the pai r Of the penc i l through 02 correspond i ng to thepa i r of the penc i l through 01 Of wh ich i s a componen t

,then

1 12 ,”a; ar e the proper tangents at 02 .

I f 01 happens to be a sel f- correspondi ng ray of the penc i l s,

the locus breaks up i n to 0102 a nd a cubic curve passi ng through02 . In th i s case , however, 01 , 02 ar e not doubl e poi nts 011

the cub ic,for the two mates to 0102 ar e now the on ly ta ngents

at 0 02 .

Al so i f i t so happen that the correspondence between the tw oi n vol u t ions i s of su ch a nature that ind i v idu al mates can bebrought i n to one - one correspondence (Ar t. the locus of thefou r th degr ee breaks up i nto tw o con i cs , these be i ng the productsof the tw o pa irs of homograph ic penc i ls formed by the i nd iv i dualmates .Recipr ocat i ng the above theorems or proceed i ng d ir ectly in a

s im i lar man ner, w e have the resu l t that the product of tw o homegraphi c i nvolu tion r anges of the first or der i s a cu rve Of thefourth class, to which th e bases Of the given in vol ut ions are doubletangen ts . If the ranges have a selfcorrespond ing poi n t th i senvelope breaks up i nto a poi nt and a curve of the thi rd class .I f the i nd iv idual mates can themselves be homogr aphically correlated

,i t breaks up i nto tw o con ics .

19 4 . I n v olu ti on h om ogr a p h i c w i th a s im p le for m .

We can extend th i s method and defi ne i n an analogou sman ner homography between an i n vol u t ion and a s imple form .

I n order to es tabl i sh the rela t ions between these , w e proceed as i nAr t. 1 9 1 , and cons ide r a r ange on a con i c s and a homograph i ci nvol ution range 011 the same con i c . The range may be defi nedby a penc i l [p ,] through a vertex 0, 011 s and the i n vol u t ion by ahomograph ic penc i l through a vertex 02 not 011 the con i c .

We Obta i n the figu re i f i n Fig . 54 w e take 01 011 the con ic . Allpoi n ts P ,

’ then coi nc ide w i th so that th i s is real ly a spec ia lcase Of the last. The produc t t Of the penc i l s [1 13]

now c u ts3 a t 0. and at t/mv e other poi nts , and i t i s easy to s 10W , as i nArts . 1 82 , 189 , tha t Ul is not a sel f- corres 10 11di 1 1g po i n t u nles s t

, S

touch at 01 . Hence an involu t ion anda homographic s imple


form on the same con ic have three self- correspondi ng poin ts ;and the r esu lt can be ex tended to cobasal in volu tions and formsOf any type as in Ar t. 1 9 1 .

1 9 5 . P r odu c t of a n i n v olu t i on a n d a h om ogr a ph i c

s im p le fo rm . Proceedi ng as i n Ar t. 1 93 we can show that theproduct of an i nvol ut ion penc i l Of vertex 01 and a homograph ics imple pen c i l Of ver tex 02 i s a cubi c hav i ng 01 for a double poin tand pass i ng through 02 . I f 0102 be a self- corr espond ing elementthe locu s breaks up i nto a s traight l i n e and a con ic .

S im i larly the product Of an involu t ion range on a l i ne i t, anda s imple r ange on a l i n e 11 2 i s a curve Of the thi rd class hav i ng n1for a double tangent and u2 for an ord i nary tangent . If up ? bea selfcorrespond i ng poi n t the envelope breaks up i nto a poin tand a con i c .

Fig. 55 .

19 6 . T h e p r odu c t of tw o h om og r a ph i c p e n c i ls of

th e se c on d or d e r i s a cu r v e of th e fou r th d e g r e e .

Cons i der two homograph ic penc il s of tangents about tw o con icss , and 32 . Let u be any straight l i n e in the plane . Take anypoi n t P 0 on u (Fig. and draw from P 0 pa i rs of tangents tos, and 32 touch i ng these con ics at P , , P 1

’and P 2 , P 2

’respect i vely.

Then the i nvol utions (P , , (P 2 , P g

’

) are homogr aph ic . Let

P 3 , P 3

’be the poi nts of con tact Of the tangents to 32 which

correspond to the tangents at P , , P ,

’to 3 1 . Then , owing to the

homography between the tangents to S I and the tangents to s ) ,

the pai rs form an i n volu tion homograph i c w 1th thatformed by the pa i r s (P , , and therefor e w i th the one formed


1 90 PROJECTIV E GEOMETRY [CH .

5 . P, Q a r e two poin ts 011 a tangen t to a con ic s . From P , Q

tangen ts p , q ar e drawn to 8,meet ing at R . I f P Q be of constan t

length , find the locus of R .

6 . Th rough a fixed poi n t 0 a ray is drawn to meet a given c i rcle atP . F i nd the enve lope of a straigh t l ine through P wh ich makes a cons tan t angle w i th OP .

7. Show that i f two con ics have double contact, any tangent toe i ther de te rmines on the othe r ranges homograph ic W i th each otherand w i th the range descr ibed by the po in t of con tact.

8. Tw o conics have double con tact at A ,B. A chord P Q of on e

conic sl ides on the othe r. Show that the cross - rat io Of the four poi n tsA , B,

P, Q i s cons tan t.

9 . P i s a po in t 011 a con ic 3 ; from P a tangen t i s drawn to a conict wh ich has double con tact w i th s to mee t 8 again at P 1 ; from P I

another tangen t i s drawn to t to mee t 3 again at P 2 : and so on . Afte r noperat ions w e reach a po i nt P ,, by a chai n of tangents. If the chain oftangents sl ide round t, prove that the range s [P ] 2, [P 2]

2,

ar e al l proje ct ive and have common se lfcorrespond ing points. Provealso that i f for one pos i tion of P (other than a po int of con tact of s, t)P ,, coinc ides w ith P ,

i t w ill do so for all pos it ions of P .

Deduce that i f a polygon of n s ides exist wh ich can be i nscribed i n aconic s and c i rcumscr ibed to a con ic t having double contact w ith s, aninfin i te number of such polygon s exist .

10. A straigh t l ine mee ts a conic at A,B. On AB poin ts P , Q ar e

taken so that the cross - rat io {AB P Q } is cons tan t. From P and Qtangents ar e drawn to the con ic mee ting at R . Show that R l ies one i ther of tw o fixed con ics having double con tact w i th the original con icat A and B.

[Proj ect A ,B in to the c i rcular poin ts ]

1 1 . Prove that a variable c ircle wh ich cu ts two fixed circles atr igh t angles de termines on these c i rcle s tw o homograph ic i nvolut ions.

12 . a, , 3 2 a r e tw o con ics, u a fixed tangen t to 3 1 . From a poi n t Q0of u tangen ts QOQ” QOQ 2 ar e drawn to a, , 32 . 02 ar e fixed pointson a, , a, respect ively : OIQ I 02Q2 mee t at Q. Show that the locus of Qis a cubic having 02 for a double po i n t and construct the prope r tangen tsto the cubic at 02 .

13. I f in tw o homograph ic i nvolut ion penc ils of the fi rst order thejoin of the vertices 0, 0

’ i s a double ray of each penc i l and self- corresponding, prove that the remainder of the product is a con ic w ith regardto wh ich 0,

0’ar e conjugate poi nts.

x1] EXAMPLES 1 9 1

[For the other double ray through 0 mee ts i ts correspond ing pair inthe poin ts Of con tact of that pair and so is the polar of

1 4. I f S be a double poin t 011 a cub ic , 0 another fixed poi nt on thecub ic, OP Q a ray through 0 cutt ing the cub ic again at P , Q ,

show thatSP

,SQ ar e mate s i n an i nvolut ion .

Hence Show that i f 0,A

,B a r e col l i near , and i f 0,

C,D ar e coll inear

,

the locus of the poin ts of con tact of tangents from 0 to all cubics havinga common double poi n t and pass ing through 0,

A,B

, C, D cons ists of

tw o straigh t lines through the double poi n t.

1 5 . Show that the hyperbola wh ich i s th e locus of in te rsections oftangen ts to a parabola mak ing a cons tan t angle a w i th each other hasthe same focus and d i rectr ix as the origi nal parabola.

[Show that the poi nts of con tact Of the tangents from Q,Q’to the

parabola lie on the hyperbola ]

CHAPTER XI I .

SYSTEMS OF CON ICS.

19 7 . R a n ge s a n d p e n c i ls of c on i c s . A set of con i cspass i ng through fou r fixed poi nts A , B ,

C,D are sa i d to form a

p enci l of 0072563 .

Through any fifth point E of the plan e there passes on e con i cof the pen c i l and one only, s ince fiv e poin ts determi ne a con i c .A set of con i cs touch i ng four fixed l i nes a , b, c, d are sa i d to

form a r a nge of conz’

cs.

Ther e ex is ts on e con ic Of the range,and one on ly, wh ich

touches any given l i n e e of the plan e .

19 8 . I n v olu t i on d e te rm in e d b y a p e n c i l of c on i c s

on a n y st r a i gh t l in e . Con sider any stra ight l in e a . Let P

be any point of a . The con i c of the pen c i l thr ough P meets uaga i n at one poi nt P ’

, which is therefor e u n iquely determ inedi f P be given . Conver sely i f P ' be given P i s known . Also,s ince P and P ’determi ne the same con i c of the pen c i l , when Pi s taken at P ’

, P’ i s at P . The ranges [P ], [P

’

] on u ar e ther efore connected by a one - on e correspondence i n which the e lementscorrespond doubly. Hence they form an i nvol ution upon n .

The double poi nts S , T of th is i nvol u tion a r e the poi n ts ofcontact Of the con ics Of the penci l wh ich touch a . Thi s enablesu s to solve the problem : to draw a con ic through fou r gi venpoi n ts A , B ,

C, I ) and touchi ng a gi ven stra ight l i ne i t. We see

that th i s problem has i n general two sol ut ions wh i ch are rea lonly i f the i nvolu t ion determ in ed upon a by the penci l of con icsi s hyperbol i c .

Three Of the con i cs Of the penc i l degenerate i nto the l i nepa i rs formed by Oppos i te s ides of the quadrangle ABUD . Theyare (AB ; CD ) , (AO DB ) , (AB ; BC) . We thu s Obta in thetheoremThe three pai rs of Oppos i te s ides of a complete quadrangle


1 94 PROJECT IVE G EOMETRY [CPL

conj ugate wi th regard to the pai r of mates (P , P’

) m wh ich a i scu t by any con ic of the penc i l . S

,T are ther efor e conjugate

points w i th r egard to every con i c of the penc i l .Conversely i f two poi nts S , T are conj ugate w ith r egard to

ever y con i c of the penc i l , they mus t be double po i nts of the

i n volu tion on ST,smee they are harmon i c conj ugates w ith

r egar d to every pa i r of mates of th i s i n volu t ion .

To every poi n t S of the plane there corresponds on e poi n t S ’

wh ich i s conj ugate to S with regard to every con i c of the penc i l ,or , as we shal l pu t i t for brev i ty , Wi th regard to the penci l .For draw the con ic of the penc i l through S . Let s be thetangent to i t at S . On 3 there must be a poi n t 18

" which 1s thepoi n t of contact of the second con i c of the system which touchess . S ,

S’are therefor e the doubl e poi nts of the i nvol u t ion deter

m i ned by the penci l Of con ics upon 3 and so are conjugate poi ntsWi th r egar d to the pen ci l . Also there 1s i n general only one suchpoi nt. For clearly i f any l in e t other than 3 be drawn through S ,i t cuts the con ic Of the penc i l through S . Thu s S would not bea double poi n t Of the i nvol ut ion on t and cou ld have on t no

conj ugate poi n t Wi th regar d to al l the con ics .An except ion occurs i f S be a centre of a l i ne pai r , for every

l i n e through S i s then a tangent to the l i n e - pa i r. And i ndeedi n th i s case S 1s a vertex of the common self polar tr iangleand has an i nfin i te number of poi n ts conj ugate to i tself w i thr egard to the penc i l

,namely al l the poi n ts of the oppos i te s ide of

the tr iangle .S i n ce the centres of the thr ee l i n e - pa i rs ar e the only poi nts

possess ing th is proper ty,i t fol lows that the penc i l of con ics have

on ly on e common sel f- polar tr iangle.

202 . L i n e s c onj u ga te w ith r e ga r d to a r a n ge of

c on i c s . Pr oceed i ng i n a prec i se ly Sim i lar mann er,or r ecipr o

cati ng the resu l ts of the last Arti cl e,w e can Show that :

Thr ough any poi nt S of the plane pass two l i n es 3 , s’wh ich

are conj ugate l i nes w ith r egard to every con ic of the r ange ; andthey are the double r ays of the i nvolution of tangents from S to

the con ics of the range .

To con str uct the{ 3

l i n e conj ugate to any gi ven l i ne 3 w i thregard to the r ange , fi nd the Oi n t of con ta c t S of s W i th thecon i c of the r ange determ i ne by s. The conj ugate l i ne 3

’ i s’

then the tangent a t S to the other con i c of the range throngs’ i s u n iqu ely determ i ned u nless s i s the l i ne j ommg

components of a poin t - pa i r of the range.

x1 1 ] SYSTEMS OF CON ICS 1 95

then be taken as the poi nt Of contact of s wi th the poi n t - pai r,and

to such a l i n e 3 correspond an i nfin i te number of l i nes s’, namelythe l i n es through the oppos i te vertex of the d iagonal tr iangle ofabcd.

Thi s bei ng the only case of fa i lure , thi s d iagonal tr iangle i sthe only common se l f- polar tr iangle of the r a nge.

208 . T h e e le v e n - p o in t c on i c . We proceed to find thelocu s of the poi nt S ’ conj ugate to S w i th r egard to a penc i l ofcon i cs , when S descr ibes any straight l i n e q.

The poi n t S ' may be fou nd by a con stru ct ion other than

Fig. 56.

the one gi ven in Ar t. 201 . For let c l , e , be any two con i csof the penc i l , Q1 , Q2 the poles of q w i th regard to a , , and 3 1 , 32

the polars of S w i th regard to 1 12 .

Then 3 , i s the locus of poi n ts conjugate to S w i th regard tos2 i s the locu s of poi n ts conj ugate to S w i th regard to c , .

Now S’ i s conj ugate to S with regard to both and 712 . Thu s

Bu t as S moves on q, 3 1 , 32 describe penc i ls homograph ic w i th the r ange descr ibed by S (Ar ts . 55 Hence S '

descr ibes a con i c (Ar t. 4 1 ) pass i ng through Q Q, .

Th i s con ic i s known as the eleven -

point con ic of q.

13— 2


For l et EFG (Fig . 56) be the common se lf- polar tr ia ngl eof the penc i ls . Then E i s conj ugate to the poi nt of q in whichq i s cu t by FG . Therefore E i s a po i n t on the locus of S ’

:

s im ilarly F,G are po i n ts on th i s locu s . Again the tw o double

poi nts T, U of the i nvol u tion determ i ned by the penc i l on g,

be i ng conj ugate to one another, are on the locus .Let H , I , J ,

K,L

,A] be the poi nts at wh ich q meets the s i x

s ides of the quadrangle ABCD . Then the harmon i c conj ugatesH

’

, I J’

, K'

, L’111

’

Of H ,I , J , K L

, M r espect i vely w ithr egard to the two vertices on the correspond ing sides of thequadrangle must lie on the locus . For clearly OD bei ng a chor dof al l the con ics of the penc i l , (H , H

’

) are conjugate wi th r egar dto al l such con i cs .The locus of S ’ thus passes through these eleven poi n ts .S ince the eleven - poi n t con i c of q pa sses through the tw o poles

Q1 , Q2 of q with r egard to e , and ’02 are any con ics of thepen ci l , the eleven - poi nt con ic passes through the pol es of q withr egard to al l the con ics of the penc i l .I t i s therefore also the locus of the pol es of q with r egard to

the con ics of the penc i l .

204 . T h e e l e v e n - l in e c on i c . Reciprocating the abovetheorems we Obta i n the fol lowi ng resu lts .The envelope of l in es conj ugate to the rays of a penc i l

through a poi n t Q wi th regard to a r ange of con i cs touch i nga , b, c, d i s a con i c wh ich touches : ( l ) the thr ee sides of thedi agonal tr i angle of the quadr i lateral abcd, (2) the two l i n esthrough Q conj ugate wi th regard to the r ange

,i .e. the two

tangen tSat Q to the tw o con ics of the r ange through Q, (3) thesix harmon ic conj ugates to the rays joi n i ng Q to the vert icesof the complete quadri la teral abcd, taken w ith r egard to the twos ides of the quadri lateral through each vertex .

Thi s con ic i s also the envelope of the polar s of Q wi th regardto the con ies of the range .

205 . G e om e tr i c a l c on st r u c t ion s for c omm on s e lf

p ola r t r i a n gle o f tw o c on i c s . I f two real con i cs i ntersec ti n fou r real poi nts A ,

B , 0, D , or e l se l i e enti rely ou ts ide eachother , so that they have fou r rea l common tangents a , b, c d ,thei r common se l f- pol ar tr i angle i s at once constru cted

, bemgthe di agonal tr iangle of the qu adrangl e ABC") or of the quadrilateral abcd.

I f,however, tw o of the po i nts of i n tersection , say 0 a nd D ,

ar e conj ugate imagi nary, the other two A and B bei ng real,the


1 98 PROJECTIVE GEOMETRY [CH .

of the penci l through any fifth poi n t i s readi ly constru cted byPascal’s Theorem . But i f the i ntersec t ions be not real , constructthe common selfpolar tr iangl e of the con i cs by the method of thelast Art icl e . Let P be a poi n t through wh ich i t i s requ i red todraw a con ic of the penc i l . Then by Ar t. 59 , k nowi ng P and aself- polar tr iangle

,three other poi nts Q, R ,

S of the requ ired con i car e known Now con s ider any straight l i ne a wh ich meets thetw o given con ics i n tw o pa i rs of r eal poi nts ( U, ( V,

The con ics through P , Q, R ,S and the con ics through A , B , G,

I )

determi ne two i nvol ut ions upon a ,the latter of which i s known

fr om the two pai rs ( U, ( V, The common mates of thesei nvol u tions therefore give the poi n ts i n wh i ch the r equ i red con i cof the gi ven pen ci l meets a . Hav i ng now s i x poi nts on th i s con i c,we can draw i t by Pascal’s theorem .

The student wi l l fi nd i t profitable to work out the cor r espond ingconstruct ion for the con ic of a r ange defined by tw o con i cs, wh i chtouches a gi ven l i n e p of the plan e, when fou r of the commontangents to the given con i cs are imagi nary.

207. C on i c s h a v i n g d ou b le c on ta ct . When tw o

con i cs have double contac t at A and B they define a penc i lof con ics hav i ng double contact W i th the two given ones at Aand B . The pol e E of AB i s the same for al l the con i cs ; andi f F, G be any pa i r of poi nts on the common chord of contactharmon ical ly conj ugate w ith regar d to A

,B

,EFG i s a common

se lf- polar tr iangl e of the penc i l Of con i cs . There i s thu s an

i nfin i ty of common se l f- polar tr iangles . The three l i ne - pa i rs ofthe system degenerate i n to the doubled l i n e AB , occurr i ng tw iceover , and the pai r of common tangen ts EA , EB .

Al so such a penc i l of con ics may be looked upon as forminga range, the four common tangents be i ng coi nc iden t in pa i rs .Such a set of con ics partakes of the properti es both of the penc i land of the range . Thus to any poi n t there i s a conj ugate poi n t andto any l i n e a conj ugate l i n e

,wi th regard to al l the con ics of the set.

Hen ce the locus of poles of any straight l i n e w ith regardto the con ics of the set i s a stra ight l i n e and the polars of a poi n tpass thr ough a poi n t. I t i s of i nterest to see how these occu ras degenerate cases of the eleven - poi n t and eleven - l i n e con ic

r espect ive ly.

Consi der any poi nt Q (Fig . 57) On a stra ight l i n e q. Let

meet AB at R and let R ’

be the harmon ic conj ugate of R w itgr espect to A

,B . S i n ce R , R

' are conj uga te w i th regard to all

con ics of the set, and E ,R are conj ugate w i th r egard to al l conics

x1 1] SYSTEMS OF CON ICS 1 99

Of the set, R i s the pole of ER’

q’

) w i th regard to al l the con icsof the set. And s i nce (1 passes through R , q

' i s the locus of olesOf. q w i th regard to the con ics of the set. O11 the other andcons ider the ray harmon ical ly conj ugate to E Q wi th regard toEA , E B . Let i t meet AB at Q

'

,and l et E Q meet AB at T.

Then Q'

ET i s a sel f- pol ar tr iangle for al l the con i cs of the set,

or Q' i s conj uga te to Q w i th regard to the set of con ics .The eleven - poi nt con i c correspond i ng to q therefore breaks up

i nto tw o stra igh t l i nes, of which one AB i s the locu s of poi n ts

conj uga te to poi n ts of q,and the other ER ’ i s the locu s of pol es .

Thus these tw o loc i , wh ich the same in the general case,are

Fig. 57.

now separated . In l ike man ner the el even - l i ne con ic correspond i ngto Q breaks up i nto the poi n t E which i s the enve lope Of l i nesconj uga te to l i nes through Q, and the po i n t Q

’ wh ich i s theenvelope Of polars of Q wi th regard to the set of con ics .

2 08 . C on st r u c t i on o f c on i c s th r ou gh th r e e p o in ts

a n d tou ch i n g tw o l in e s . Let i t be requ i red to con s tructa con i c to pass through three poi n ts A ,

B,0 and to tou ch tw o

l i nes p ,q (Fig. Le t the con i c requ i red touch p , g a t P , Q

respec t i vely .

Cons ider the i n vol u t ion de termi ned 011 BC by the penci l ofcon ics hav i ng con tac t wi th p and (1 a t P and Q . I f p ,

(1 mee t

200 PROJECTIVE G EOMETRY [OH .

BG at P 1 , Q1 then P , , Ql a r e mates in th i s invol ution ,for the

pai r p ,q i s a con i c of the penc i l . Al so B , G are mates i n th i s

i nvolu t ion . The double poi nts of th is i nvol u tion ar e thereforedeterm i ned . But s i nce P Q doubled 1s a con i c of the penc i l , thepoint wher e P Q meets B 0 1S on e of the double poi nts of th i si nvolut ion .

In l ike manner P Q passes through on e of the double poi n tsof the invol ut ion on A0 de term i ned by the pai rs of mates(A, (P , , P , , Q, be i ng the poi n ts where p ,

q meet AG.

There are thus fou r poss ibl e pos i tions of P Q correspond ing to

Fig. 58 .

the four l i nes j o i n i ng the double poin ts of these two in vol u tions,and so there are fou r sol u tions to the problem pr oposed .

The reader may ver i fy tha t i f P Q passes through double poi ntsof the i n vol utions 011 BC, CA ,

i t w i l l al so pass through a doubl epoi nt of the cor 1 espond1n rr i n volu t ion 011 AB .

Recip1 ocating the above construct ion w e Obta i n a cons tructionfor the con ics through tw o poi n ts and touchi ng three l i nes . Thi s,l i ke the above, has 1 11 genei al fou r sol utions .

2 09 . P r op e r t i e s o f c on foc a l c on i c s . I f tw o of theOppos i te vert ices of the quad1 ila ter al abcd a r e the c i rcu lar poi ntsat i nfin i ty Q

,Q’

,the range of con ics i nscr i bed i n th i s quadri lateral

becomes a sy stein Of con focal con ics . The i n vol u tion of tangentsthrough any poi n t P has thu s the ci rcu lar l i n es through P for

mates . I ts double r ays are the1 efor e at r ight angles (Ar t.



2 11 . P r op e r ti e s of r e c ta n gu la r h y p e r b ola . If twocon i cs Of a penc i l ar e recta ngular hy perbol as the poi nts 0 ,

Q’ areconj ugate w i th regard to two con ics of the penci l . Therefore theyare conj ugate wi th regard to al l the con ics of the penc i l . Theseare therefore a l l rectangular hyperbolas . Thu s every con i c thr oughthe i ntersections of tw o rectangular hyperbolas i s a r ecta ngularhyper bol a .

If A , B , G,B be the four i nter sect ions of two r ectangular

hyperbolas, the l i ne - pai rs are also r ectangular hyperbolas , therefore they ar e perpend icu lar. The quadrangle AB GB i s thereforesu ch that pa i rs Of Oppos i te s i des are perpend icu lar . Any on e Of

i ts four verti ces i s the orthocentre of the triangl e formed by theother three . I t fol low s that any con i c through the three verti cesof a tr iangl e and i ts or thocentre i s a rectangular hyper bola (of.Ar t. Conversely the orthocentr e Of any triangle i nscr ibedi n a r ectangular hyperbola l i es on the cu rve . For i f ABC be thetr iangle and the perpend icu lar thr ough A to B C meet the hyperbola agai n at B

,the pa i r AB ,

B C bei ng a r ectangular hyperbol a,every con i c through A

,B , G, B i s a r ectangular hyperbola .

But 0A,BB i s such a con ic, therefore 0A , BB are perpen

dicu lar, or B i s the orthocen tre of ABO.

2 12 . C e n t r e loc i . The theorems of Arts . 202 , 203 givethe fol lowi ng resu lts when q i s the l i n e at i nfin i ty.

The locu s of the cen tres of a penc i l of con ics through fourpoin ts A , B , G, B i s a con ic whose asymptotes are paral lel to theaxes of the tw o parabolas through the four poi nts and whichpasses through the vert ices of the di agonal tr i angle of the quadr angle AB GB and the m iddle poi nts of the six s ides of thi squadrangle .

I nc idental ly we have proved the theorem :

The Six middle poi nts of the s i des of a complete quadrangl el i e on a con ic wh ich c i rcum scribes i ts d iagonal triangle.The locu s Of the centres Of a range of con i cs touch i ng four

l in es a,I),c, d i s a s traight l i ne . S i nce the m id - poi n ts of the three

l i ne a irs a r e ev i dently cen tres, they lie 011 th i s locu s . Hence,i nc i ental ly : the middle poi n ts of the three d iagonals of a quadr ilater a l are coll i n ea r .

2 18 . L ocu s of foc i of a r a n ge o f c on i cs. The

i nvol ution s of tangents from two d i fferen t poi n ts P , Q of theplane to the con i cs of a range are clearly hom ly related ,s i nce e i ther tangent through

xi l ] SYSTEMS OF CON ICS 203

the range and therefore the pa i r of tangen ts from Q : and the converse i s tru e i f w e start from Q .

These homogr aph ic i nvol u t ion penc il s,however, have a sel f

correspond i ng ray,namely P Q,

for P Q i s a tangent from e i therP or Q to the con i c of the range wh ich touches P Q.

The locus of i nter sect ions of tangen ts from P and Q thereforereduces, by Ar t. 1 93 , to a cubi c thr ough P and Q.

I f now P and Q be taken at Q ,0’ th i s locus becomes the

locus of the foc i of the con ics of the range .

The foc i of the range therefore lie on a cub ic through 0 ,Q’

Such a cub ic i s known as a c i rcu lar cubi c.I f P and Q l i e on any one of the three d iagonals of the quad

r ilater al abcd which defines the range,the tangents from P to the

correspond i ng point pa i r coi n c ide along P Q ; and so do thetangents from Q to the same poi nt - pa ir . The homogr aph ici nvol u t ion s from P and Q have therefore a pa i r of double raysself - correspondi ng . Thei r product br eaks up i nto the l i n e P Qdoubled and a con i c w i th regard to wh ich P and Q are conj ugate(Exs . XI .

I f P, Q be 0 ,

Q’,the cor r espond i ng diagon al of the quad

r ilate r al i s at i nfin i ty ; the quadr i la teral i s a paral lelogram .

Hence the locu s of the foc i of al l con ics i n scr ibed i n a paral lelogram i s a rec tangu lar hyperbola (for Q ,

Q’ a r e conj ugate w i thregard to i t, by the above) , toge ther Wi th the l in e at i nfi n i tydoubled .

I f the quadri lateral , i nstead of be i ng a par allelggr a‘

m,i s

symmetr ical abou t a d iagonal , th i s d iagonal i s obv iou sly part ofthe locus . S i nce i t does not pass through 0 ,

Q’ the cub ic breaksup1in

to th i s d iagonal and a con i c through 9 ,

Q’

,that i s

,a

cu e e .

S i n ce (see Exs . VI . A,24) the components of a poi nt - pai r are

also i ts foc i the rec tangu lar hyperbola wh ich i s the locus of con icsn a paral le logr am c i rcumscribes the paral le logr am ,

andwhi ch i s the correspond i ng locu s for the quadr i lateralal about a diagonal passes through the fou r ver t i ces

not on thi s d iagonal .

2 14 . T h e h y p e r b o la of A p ol lon iu s . Let the quadrangleAB CB

,wh ich defi nes a penci l of con ics , he i n scribable i n a c ir cle.

0, Q

’ are conj ugate poi n ts i n the i n volu tion determi ned by thepenc i l on the l i ne a t i nfi n i ty . The double poi n ts of th is i nvolat ion are therefore de termi ned by two rec tangular di rec t ions .

These gi ve the axes of the two parabolas through the fou r poi n ts


and these are paral l el to the asymptotes of the centre locu s . Thecentre locu s i s then a rectangular hyperbola .

The same rectangu lar hyperbola i s the locu s of poi n ts conj ugate to poi nts at i nfin i ty with . regard to the penc i l of con i cs .I f then I be a poi n t at i nfin i ty i n the di rec t ion of on e of the

axes of a con i c s of the penci l , the conj ugate poi n t I’ i s the in ter

section Of the d iameters of s and of the c ircl e 0 abou t AB CDwh ich a r e conjugate to the d i rect ion of I But both of theseare perpend icu lar to the di rec t ion of I

”. Hence they meet at

I at infi n i ty i n the perpend icular di rect ion .

the penc i l have ther efor e axes parall el to the asymptotes of thecen tre locus .Let now I

” be any poi nt at infin i ty, I’ i ts conj ugate po i n t

wi th r egard to the penc i l ; take for s the con ic of the pen c i lthrough I ’, and l et 0, 0 be the cen tres of s, 0 r espect i vely.

From the property of the poi nt I ’, the diameters conjugate tothe d irection defined by I ” wi th regard to s and 0 pass thr oughI’

. They are therefor e 01 ’and OI ’.But OI

’ i s perpend i cula r to i ts conjugate d i rect ion by aproperty of the c ircle. Thus OI i s perpend icular to the tangentat I ’ to the con ic s, for thi s tangent i s paral lel to the d iameterconj ugate to CT and therefore passes through I Hence thenorma l at I ’ to the con i c through I ’ passes through 0, or ther ectangular hyperbola which i s the locu s of centres i s al sothe locus Of the feet of perpendicu lars from O 011 the con ics ofthe system .

Now l et 3 be any con ic, 0 any poi n t. A c i rcl e cen tre 0meets s i n fou r poi nts A B

, G, B ,and cons ider i ng the above resu lts

for the penci l of con i cs through A , B ,G, B w e Obta i n the theorem :

The feet of the normal s from any poi n t 0 to a con ic s lie on arectangular hyperbola through 0 and the centre of 3 , whoseasymptotes are paralle l to the axes of 3 . S i nce th is recta ngularhyperbola meets s in fou r poi n ts

,four normal s can i n general be

drawn from a poi nt to a con ic .Thi s hyperbola i s k nown as the hyperbola of Apol lon i u s for

the po in t 0 and the con i c s .

2 15 . J oa c h im sth a l’s T h e or em . Let L , Ill , N, K (Fig.

59) be the feet of fou r concurren t normal s to a con ic 3 . Cons iderthe i nvol u tion determi ned on the ax i s AA’by the penc i l of con i csthrough LAINK The l i n e - pair LA]

,NKdetermi nes two poi nts

P ,P

’

. The con i c 3 determi nes A ,A

’. The hy perbola of Apol

lon ius, hav i ng i ts asymptotes paral lel to the axes of s and pass ing


206 PROJECTIVE GEOMETRY [C

through the feet of thr ee Of them passes al so thr ough the poid iametr i cal ly oppos i te to the foot of the fourth .

2 16 . G e om e tr i c a l c on str u ct i on s for t r a n s

a n y tw o c on i c s i n to c on i c s of g i v e n ty p e .

already seen (Ar t. 1 72) how to transform any tw o con i csc i rcles .Any tw o con ics may be transformed i nto concentri c con ic

a real proj ect ion . For w e have seen that ther e i s always oneof the common selfpolar tr iangle wh ich i s r eal . Proj ecting

Fig. 60.

s ide to i nfin i ty the Oppos i te vertex pr oj ects into the commoncentre .

I f tw o verti ces of the common sel f- polar tr iangle of tw o con icsbe proj ected i nto 9

,Q’

,the con i cs proj ect i nto concentri c rect

angular hyperbolas .ny tw o con ics may be proj ected i nto coax ial con i cs . Thus

i f EFG be thei r common selfpolar tr iangle, proj ect FG to

i nfin i ty and the angle FE G i nto a r ight angle.Tw o coaxial con ics 31 , 32 can be transform ed i n to one another

by r ec iprocal pol ars .Let 0 (Fig. 60) be thei r common cen tre, A ,A 1

’and B ,B ,

'

the

axes of s, , A 2 14; and B 2B; the axes of 32 . Fi nd the doublepoi nts A, A

’

of the i nvol u tion determ i n ed by the pai rs of mates(A A2) (A A2

’

) and the double poin ts B , B’of the involution

x1 1] SYSTEMS OF CON ICS 207

determ i ned by the pa i 1 s of mates (B B ) (B B 2 Then fromsymmet1y abou t 0 a con i c 3 ex i sts hav i ng AA’

, BB' as axes .

Form the r ecip1 ocal polar con ic s.’

of s, w i th rega rd to s,3.

passes thr ough A A’ B B g

’

, and i ts tangen t at A .,be i ng the

polar of A w i th r ega1 d to s,i s perpend icu lar to OA and so i s

the same as the tangent to s . at A s,’

s, a 1 e thu s iden t i cal .S i n ce any tw o cou ics may be projected i nto coax i al con ics ,

proj ect i ng back w e see tha t a con ic always ex is ts w i th regard towhich tw o given con ics are reciprocal polars .

2 1 7 . S e l f - p o la r qu a d r a n gl e . Con s ider two pa i rs of conj ugate l i nes wi th regard to a con ic (p , p

’

) (q, q’

) (Fig . Let

I

Fig. 6 1 .

the poi nts p f} , pq’be called A

,B , O,

B . Let AG,BB

be 7, Let the polar a of A mee t p , p

’

, q, q’

, 7" at

P , P'

, Q, Q ,R , R

' respect i vel y. Then because p passes throughA , p i s conj uga te to both a and p

’

Thus P ’

i s the pole of p andP

,P

'

a r e conjugate poi n ts w i th regard to the con ic . S im i larlyQ, Q

'

a r e conj uga te poi nts w i th regard to the con ic . Now bya property of the quadrangle (Ar t. 1 98 ) (P , P ) (Q, Q

’

) (R ,R

'

) a r ethree pa 1rs of ma tes of a n i n vol u t ion . Hence (R , R

’

) a r e ma tesin the i n vol ut ion determi ned by (P , (Q , Q ) , tha t 1s , they a r econj uga te poi n ts on a wi th regard to the come . Al so A , R

’ are


conj ugate,s i n ce R’ l i es on a . Therefore R ' i s the pol e of AR

,

i . e. of r : thus r , 7"

ar e conjugate l i nes . Hen ce i f tw o pa ir s ofOppos i te s ides of

.

a quadr angle are formed of conj ugate l i nes w ithregard to a con ic

,the thi rd pa i r are l ikew i se conj ugate wi th

regard to the con i c .

Such a quadrangle , of which the oppos i te s ides are conj ugatew i th regard to a come , i s termed seIf-

p ola 7 wi th r egar d to thatcon i c .

2 18 . A s e lf - p o la r qu a d r a n gle exi sts h a v i n g th r e e

a r b it r a r y p o in ts a s v e r t i c e s . I f A , B ,C be any three

gi ven poi nts and w e draw thr ough A a l i n e conj ugate to B0and through B a l i n e conj ugate to AC these tw o l i n es i nter sectat the fourth vertex B of a self polar quadrangl e. B 18 u n iquelydeterm i ned u n less ABC i s i tse l f a selfpolar tr i angle, when anyl in e through A i s conj uga te to B0 and any l i n e through B ISconj ugate to AC. In th is case B may be any poi n t of theplane . Thus a self—polar tr iangle and any fourth poi nt forma self polar quadrangle.If two verti ces of a sel f polar quadrangle be conjugate, they

form wi th a thi rd ver tex of the quadra ngle a selfpolar triangle.For l et A andB be tw o conj ugate poi nts, a and 1) tw o oppos i te s idesof the self polar quadrangle through A

,B respect i vely. Then a , b

are conj ugate. Hence ei ther A 1s the pol e of I) or the pol e of b i sa poi n t P on a other than A . Now the polar ofA passes throughB and the polar of P i s b and also pa sses through B . B i s thenthe pol e ofAP ,

i .e. of a . Therefore e i ther A i s the pol e of bor B i s the pol e of a . Now suppose A i s the pol e of b and O i sthe vertex of the quadra ngle 011 I) . Then AB , GB bei ng conj ugatethe pol e of AB i s on CB . But i t l i es also on 6, s ince th i s i s thepolar of A . Therefore G i s the pole of AB and ABC i s a selfpolar tr i angle .

2 19 . T w o s e l f - p o la r qu a d r a n gle s h av i n g tw o

v e r t i c e s c omm on a r e i n sc r i b e d i n th e s am e c on i c .

Let AB CD , ABG’

B’ be tw o se lf- polar quadrangles . Then

AG, AB , AO’

, AB' are conj ugate to BB , B G, BB

'

, B G’

r e

spectiv ely . And conj ugate penc ils bei ng proj ecti veA (OB O

’

B (B OB’C’)

T B by Ar t. 2 1 .

Therefore G, B , G ,B

’are i n tersect ions of correspond i ng rays oftw o proj ect i ve penc i ls of four rays through A and B ,

that i s , theyl i e on a con ic through A and B , which proves the theoremr equ i r ed .



I n par ticular i f s be a rectangular hyperbola whosei s 0

,the tr iangle 000’ i s selfpolar w i th regard to s . I

be any other selfpolar tr iangle 0,Q

,Q’

, A , B , C l i e onthat i s

, 0 l ies 011 the c ircle c i rcumscr ib i ng AB C, or thethe centres of rectangu lar hyperbolas for whi ch a giveni s selfpolar i s the c i rcumscrib i ng c ircl e of the tri angle.

2 2 2 . C on i c s h a r m on i c a l ly i n sc r i b e d in a c on i c .

Rec iprocat i ng the theorems of Arts . 2 1 7— 22 1 we can defin ea selfpolar qu adr i la teral w i th regard to a con i c s as one whosepa i r s of oppos i te vert i ces are conj ugate w ith regard to 3 .

Also i f one selfpolar quadri lateral or tr iangle of s be ci rcumscr i bed abou t another con ic s’, an infin ity of both tr iangles andquadr i l aterals

,selfpolar wi th r egard to s

, can be c ircumscribedabout 3’ on e s i de of each tr iangle

, or three s ides of each quadr ilater al

,may be arb i trar i ly given .

3’ i s then sa i d to be ka rmonz

’

ca lly inscr ibed in 3 .

S i nce any tw o con ics may be transformed i nto one anotherby rec iprocal polars (Ar t. apply i ng such a transformat ionto transform the con i c s’ of the present Art icl e i nto 3

,the sel f

polar quadri laterals and tri angles w i th regard to s c ircumscr ibedabou t s

’ rec iprocate i n to selfpolar quadrangles and tr ianglesw i th regard to s

’

,i n scr ibed i n 3 . Hence i f s’ be harmon ically

i n scr ibed i n s, then s i s harmon i cally c i rcumscr ibed to s’ and

conversely.

We fi nd as i n the last Articl e tha t the six s ides of tw otr ia ngles selfpolar wi th regard to the same con ic toucha con 1c .

2 2 3 . T r i a n g le s i n s c r i b e d i n a c on i c a r e s e l f - p ola r

w ith r e ga r d to a c on i c . Conversely we have to show thati f two tr iangles A I B Jl 01 , A 2B 2 C2 are i nscr ibed i n a con ic s

'

,

th 1

l

er e ex i sts a con ic s wi th regard to wh ich they are both selfpo ar .We wi l l fi rst Show that a con ic 3 ex i sts w i th r egard to w h ich

a gi ven tri angle A ,B , C, i s se lf- polar and a gi ven poi nt and l i n e,A 2 and B 2O” are pole and polar . For l e t A , AQ , B 202 meetB 1 0, at L ,

M (Fig . then [ if i s the pol e of A ,A e and L ,AI

a r e conj ugate poi n ts w i th regard to the con i c requ i red . The

double poi n ts P , Q of the i nvolu tion of which (B , , (L , AI )a r e pai rs of mates g i ve the i ntersections of th is con ic w i th B 1 0] .

Joi n A QP mee t i ng B 202 at T and let R be harmon ical ly con

j ugate to P wi th regard to A QT. Descr ibe the con i c touch i ngA l P , A I Q at P and Q r especti vely and pass ing thr ough R .

xi 1] SYSTEMS or com es 21 1

Thi s i s the con i c 3 requ i red . For A , i s the pol e of B ,0, w i thregard to s and B 0, be i ng harmon ical ly conjugate w i th regardto P , Q A ,B ,C, i s a sel f- polar tr iangle for 3 . Al so (L , A")and (A , , ill ) bei ng clearly pa irs of conj ugate poi n ts w i th regardto s (s i nce 11] i s 011 the polar of A , and {PLQM

'

} l ) , A] i sthe pole of A ,L and A .

, a r e conj ugate . Al so, becau se{A , PTR } 1

,A

_T are conj ugate. Hence TM

,tha t i s B 20.“

i s the pol ar of A ., w i th regard to s .

I t now fol low s a t once that A B 202 i s sel f—polar w i th regardto s. For i f th i s were not so

,let 0;be the poi n t conj uga te to B .

on Then A ., BQCQ’ar e sel f- polar for 3 . That i s,

l ies on the con ic through A B A 2 , B 2 , i .e . 011 s’

. HenceC,

’ i s atIn l i ke manner (or by rec iproca t ion ) i t may be Shown tha t

tr ia ngles c i rcumscribed to the same con ic a r e se l f- polar for acome .

Combin i ng these resu l ts w i th those of Arts . 22 1 , 222 w e

obta i n the theorem : i f tw o t riangles a r e i n sc r i bed i n a con ic ,they are c i rcumscri bed to a con ic (cf. Exs . V . A

,Thi s may be

pu t i n to the form i f one tri angle ex i s t which is i n scri bed i n onecon ic s and c i rcumscr ibed to another s’, an i nfin i te n umber of suchtr iangles ex i st.

1 4— 2

21 2 PROJECTIVE GEOMETRY [en .

For l et A ,B ,O, be such a tr iangle ; thr ough any poin t A2 of s

w e can draw tw o tangen ts to 3’ meet i ng 3 i n B 202 . Then the

tr iangles A ,B ,O2 are i nscr ibed i n s : the ir Six Sides therefore touch a con i c . But five of these Sides touch s'. Hence thes i xth Side touches s’.

2 2 4 . C on i c s h a rm on i c a l ly i n sc r ib e d a n d c i r cum

s c r i b e d to th e sam e c on i c . Let s, , 32 be two con i cs suchthat a tr iangle AB C can be i nscribed in s, and c i r cumscr ibedto se . Let P QR be the common selfpolar tr iangle of s, , 32 . Let

A'

B’C

'be a second tr iangle inscr ibed i n s, and c i rcumscr ibed to 32 .

Such a tr iangle ex i sts by the last Ar ti cle. Al so a con ic 3 ex istsfor which ABC, A

'

B’C’ are selfpolar .

Apply a transformat ion by r ec iprocal polars w i th s as basecon i c . The tr iangles AB 0, A

'

B'

C’ be i ng selfpolar wi th regard

to s,tran sform i nto themselves . The con i c s, which i s c i rcum

scr ibed to them tr ansforms i nto the con ic 32 which i s i nscr i bed i nthem . The tr iangle P QR wh ich i s se lf- polar w i th regard to s, ands2 transforms i nto a tr iangl e self- pola r wi th regard to s, and s, ,

that i s,i t transforms i nto i tself. I t i s therefore al so selfpolar

w i th regard to 3 . Hence P QR , ABO are two selfpolar tr iangleswi th r egar d to s. Thu s P , Q, R , A ,

B, 0 ar e Six poi nts on a

come .

I t follows from the above demonstration that two con i cs s,and s, , such that a tr iangle exi sts i nscr ibed i n s, and c ircumscribed to s, , are al so such that a con i c 3 ex i sts harmon i callyinscr ibed in s, and c i rcumscr ibed to 32 .

I f in the above theorem the con i cs s, , s2 are coax ial , thei rcommon selfpolar tr iangle i s formed by the common axes andthe l i ne at i nfin ity. I f then AB C i s a tr iangle i nscr ibed i n s,

and ci rcumscr ibed to 32 a con i c can be drawn through A ,B

,0

pass i ng through the centr e of s, and hav i ng i ts asymptotes parall elto the axes . But th i s i s a hyperbola of Apol lon i u s for s, . Hencei f su ch a tr iangle ex ist the th r ee normals to s, at i ts ver t icesA , B , 0 are concu rrent .

2 2 5 . P e n c i l of c on i c s h a r m on i c a l ly c i r cum sc r i b e d

to a c on i c . I f two con i cs s, , 32 of a pen ci l are harmon ical lyc i rcumscr ibed to the same con ic 3 , l et three of thei r commonpoi n ts A ,

B , 0 be taken for the verti ces of a quadrangle self~polarfor 3 . Then by the property of the harmon ical ly c ircumscr ibedcon ics, the fou r th vertex B of th i s quadram le l i es on both 3,

and The fou r poin ts of i ntersect ion of sue con i cs then form


21 4 PROJECTIVE GEOMETRY [on

But the intersect ions of two such con i cs form a quadrangle selfpolar wi th regard to each of the fou r con ics determi n i ng the n et.The con ic through any poi nt P and the verti ces of th i s quadranglei s a con ic of the n et ; and i t i s the only con ic of the net through P .

The net therefore r educes to a penci l of con ics . S im i lar ly the w ebof the first gr ade reduces to a range .

I t fol lows from the above cons iderations that a n et of thesecond , th ird , or fourth gr ade can be made to pass through two,three

,or four gi ven poi n ts i n the plane, and that a web of the

second , thi rd , or fourth grade can be made to touch tw o, thr ee , orfour gi ven l i nes respect i vely .

In general when the terms net and w eb are used wi thoutfurther qual ification , a n et or web of the second grade i s meant .

2 2 8 . S im i la r c on i c s . Tw o coplanar con i cs are sa id to bes im i lar and s im i larly Situ ated i f they can be brought i nto a plan eper specti ve relat ion i n w h ich the axi s of col l i neation i s at i nfin i ty .

I t fol lows that the poi n ts at i nfin i ty of tw o such con i cs coi nc ide,or the i r asymptotes are para l l el .Con versely i f tw o con i cs s, , s2 have thei r asymptotes paral le l

they are s im i lar and s im ilarly s i tuated . For l et I J°° be their

tw o common poi n ts at infin i ty, t on e of thei r common tangentstouch i ng s, at P , , 32 at P 2 . Thr ough P P 2 draw any tw opara l lel l ines meet i ng s, , s2 at Q, , Q, respect i vely, and let Q2meet t at 0. With 0 as pol e of perspec t i ve , the l i ne at i nfin i tyas ax i s of col l i neation and P , , P 2 as a pai r of correspond ing po i nts,con struct the con ic s,

’ i n plane perspect ive w i th s, . Then s, , s,’

have i n common the poi nts I J Q P 2 and the tangent at P , .

They ar e therefore identi cal , that i s , s, , 82 a r e s im i lar and s im i larlys i tuated .

Any tw o con i cs may be proj ected i n to Sim i lar con ics byproj ecti ng on e of thei r common chords to i nfin i ty. Proj ecti ngback w e see that any tw o con ics may be brought i nto planeperspect i ve by tak i ng any one of the i r common chords as ax is ofcol l i neation .

EXAMPLES XI I .

1 . Show that the cross - rat io of the flat penc i l formed by the polarsof a poin t U w i th regard to four conics of a penc i l is independen t of thepos i tion of U in the plane .

x1 1] EXAMPLES 21 5

2 . Show that the cross - rat io of the range formed by the poles of a.l ine u w i th regard to four conics of a range is i ndependen t of the pos it ionof u in the plane .

3 . Prove that the harmonic conj ugates of any poi nt w i th regard tofour given pairs of po i n ts in involution have a constant cross - rat io.

4 . Show that the c i rcle s on the th ree d iagonals of a quadrilateral asd iame te rs have a common rad ical axi s .[For the se c ircle s ar e orthopti c c i rcle s of the poi n t - pairs of the range ]5 . Show that i n a range of conics tw o ar e rectangular hyperbolas

and that the i r orthopt ic c i rcle s a r e the po int - c i rcles o f the system of

coaxial orthopt ic c i rcles of the range .

6. Prove that i f tw o conics of a penc i l have the i r axes parallel , allthe conics of the penc i l have the i r axes parallel and one of these coni csi s a c i rcle .

7. Show that in the construct ion of Ar t. 2 10 i t i s poss ible to choosethe rad ius of the c i rcle w i th regard to wh ich w e rec iprocate so that bothl imi t ing poi n ts shal l rec iprocate i n to foc i .

8 . Show how to con struct the centre of a con ic touch ing fiv e givenl ines.

9 . The locus of centres of rectangular hyperbolas c i rcumscrib i ng atriangle is the n ine - po in ts c i rcle of th e triangle .

10. I f a penc i l of comics c ircumscribe a rectangle , show that theeleven - poin t con ics ar e re ctangular hyperbolas .

1 1 . The centre of the locus of centre s of conics of a penc i l i s thecentro id of the quadrangle defin ing the penc i l .[For the cen tre - locus passes th rough the m id - poi nts P , Q ,

R,S of

AB,B C

, CB ,BA . Bu t P QRS is read i ly shown to be a paral le logram .

Hence the in te rsect ion of P R,(28 , wh ich is the centroid of the quad

rangle , i s also the centre of the cen tre - locus ]1 2 . A system of conics have a g iven c ircle of curvature at A and

pas s th rough B. Show that the locus of the i r cen tres i s a conic whosecentre d ivide s AB in the rat io 1 3 .

1 3. I f a po i n t U de scribe s a norma l to a conic the fee t P , Q ,R of

the th re e othe r normals draw n from U to the eon ic form a trianglec i rcumscrib ing a parabo la touch i ng the axes .[ I f T (Fig. 59 ) de scribe L T, the range s [U ] , [ V ] , and [Q

’

]ar e s imi lar NK enve lops a parabola ]

21 6 PROJECT IVE GEOMETRY [e rr xr r

1 4. A ,B , A, B , C2 ar e two triangles i nscribed in the same con ic.Conics s, , s2 ar e described about A ,B QC2 having double contactw i th one another. Show that the i r common chord of contact touchesthe con ic for wh ich A 2B 202 ar e selfpolar.

1 5 . I f a con ic s, be harmon ically i n scribed in a con ic s, , two poi ntsof 82 , the tangents at wh ich ar e conjugate w i th regard to s , , lie on atangen t to s, . Conversely i f P be the pole w ith regard to 32 of atangen t to s, th e tangents from P to s2 a r e conjugate w i th regard to s, .

1 6. I f a con ic s, be harmon ically c ircumscr ibed to a conic s, , twotangents to 32 whose poi nts of con tact ar e conjugate w i th regard tos, , i nte rsect on s, . Conversely th e polar w i th regard to s2 of a poi nt ofs, mee ts 32 i n poi nts conjugate w i th regard to s, .

17. From the theorem that i f tw o tr iangles a r e c i rcumscribed to aconic they ar e i nscribed i n another con ic prove , by tak ing tw o vert icesof one triangle to be th e c i rcular points at infin i ty

,that the c i rcle c ir

cumscr ibing a triangle formed by th ree tangents to a parabola passesthrough the focus .

1 8. The locus of the centre of a rectangular hyperbola wh ich isi nscribed i n a given triangle i s the c ircle for wh ich the triangle is sel fpolar.[For the hyperbola i s harmonically i nscribed i n the c ircle , i .e . tri

angles selfpolar for the hyp erbola a r e in scribed i n the c i rcle,

byAr t. 22 1 the centre of hyperbola l ie s on th i s c i rcle ]

1 9 . I f a circle cut harmon ically the s ides of a triangle c ircumscribedto a con ic, i t cuts orthogonally the orthoptic c i rcle of the con ic.[For the c i rcle is harmonically c i rcumscribed to the conic ]20. I f tr iangles exi st wh ich ar e i nscribed i n a c i rcle 0 and c i rcum

scribed to a c i rcle 0’ i t is necessary and suffic ien t that the rectangle con

ta ined by segmen ts of chords of 0 th rough centre of 0’Should be

numerically equal to tw ice the product of the radi i .2 1 . Show how ,

by a real projection , to proje ct tw o con ics wh ichi ntersect i n only tw o real po ints i nto tw o s imi lar and s imilarly s i tuatede llipses.

22.F rom a given po int A a variable chord A P Q is drawn to a

given conic 3 . Through P , Q and another given point B a con ic isdrawn s imilar and s imilarly s ituated to s . Prove that th is con i c passesthrough a certain fixed poin t othe r than B.

[Proj ect the poi nts at i nfini ty 011 3 into the c i rcular points ]23. I f tw o coaxial con ics be such that a triangle exis t wh ich is

c i rcumscribed to on e con ic and inscribed i n another,prove that the

axes and the s ides of the triangle touch a parabola.


21 8 PROJECTIVE GEOMETRY [OH .

Conver sely to a con ic in the plane corr esponds a con e of thesecond order in the sheaf.Chasles’Theorem gi ves : i f a be a fixed tangent plan e to a

con e of ver tex 0, a: a fixed generator, W a var iable tangent plane ,p i ts generator of contact, the flat penc i l a (7r ) i s homogr aph i cw ith the axi al penc i l x (p ) .We deduce , as in Chap . I I I , that a var iable tangent plan e 7r

cu ts four fixed tangent planes in a flat penc i l of consta nt crossrat io or tak i ng any tangent l i n e t ly i ng i n t cuts these fourtangent plan es in a r ange of constan t cross - ratio.

Al so a var iable generator to the cone determi nes wi th fou rfixed gener ators an ax i al penc i l of con s tant cross - r atio.

Conversely the product of tw o homograph i c ax ial penc i lswhose axes a , , as, i ntersect at 0 i s a con e of the second ordervertex 0,

hav ing m,m, for a pai r of generators , and the envelopeof the pla nes de termi ned by the correspond i ng r ays of tw o homegr aph ic flat penc i ls hav i ng a common vertex bu t not ly i ng in thesame plan e i s aga in su ch a con e .

l f two homogr aph i c axia l penc i l s have a self- correspondi ngplan e

,thei r axes i ntersect i n th i s p lan e . The flat penc i ls i n

wh ich they in tersect any plan e are perspecti ve and the productof the tw o homegr aphic ax ial penc i ls i s a plan e, together withthe selfcorrespond ing plan e .

Pascal’s Theorem gi ves i f a s i x - faced sol id angle be i nscr i bedi n a cone of second order, the l in es of i n tersection of Oppos i tefaces ar e coplanar .Br ianchon

’s Theorem gi ves : i f a s ix - faced sol id angl e be

c i rcumscr ibed to a con e of the second or der, the planes joi n ingoppos i te edges pass through a l i ne. By tak i ng arb i trary poi n tson the s i x edges of thi s sol id angl e and joi n i ng them we ob ta i nthe somewha t d i fferent enunc ia t ion : i f a skew hexagon be c i rcumscr ibed to a con e of the second order, the three diagonal sjoin i ng oppos i te vert ices i ntersect a l i n e through the vertex of

the cone .

2 3 1 . P ole a n d p ola r p r op e r ti e s of th e c on e of

s e c on d or d e r . To any l i ne p through the vertex of a coneof second order corresponds a plane 7r through the vertex wh ichi s call ed the d i ametral plane of the cone conjugate to thed iameter p . These ar e obtai ned by joi n i ng to the vertex of thecone the polar of the poi n t i n whi ch p cuts any plane a w i thr egard to the sect ion of the cone by a . Al so s i nce cross - rat io i sunal tered by pr oj ection , i f P be a ny poi nt ofp and a l i ne thr ough

x1 1 1] TH E CONE AND SPHERE 21 9

P meet the cone at Q , R and vr at P ’then P,P

’

a r e harmon i cal lyconj ugate wi th regard to Q , R . 1 r i s therefore al so called thepolar plane Of P

,and conversely P 1s a pol e of W. We see that

any plane th1 ough the vertex has an i nfin i te number of poles ,which al l lie 011 i ts conjugate diameter .

If P' l ies on the polar pla ne of P , con v e1 sely the polar plane

of P passes through P .

Such planes ar e cal led conj ugate d i ametral planes . Theymeet any plane in tw o l i nes wh ich are conjugate w i th regard tothe sect ion Of the con e by tha t plan e . From the property thattwo such conj ugate l i nes a r e harmon ically conj ugate w i th regardto the two ta ngents from the i r i n tersection , w e see that t o

conjugate diametral planes for the cone a r e harmon ical ly con

juga te with regar d to the tw o tangen t plan es through the i ri n tersection .

S im i larly conj ugate d iameters of the cone are harmon ical lycc

l

mjuga te w i th regard to the tw o generators Of the cone i n thei rp ane .

The polar plane of the vertex i s i ndeterm i n ate : converselyev e

l

er y plane not pass i ng through the ver tex has the vertex for0 e .pTo a triangle sel f- pol ar wi th regard to any plane sect ion

Of a cone Of second order corresponds a tr ihedra l angle or t/n'eeedge selfpolar w i th regard to the cone, the edges of which passthrough the verti ces Of the tr i angle . These edges form a set ofthree d iame ters conj uga te pa i r and pai r

,and such tha t each i s

conj uga te to the Oppos i te face of the three edge .

he rds paral lel to any d iame ter of theo

cone ( i e . any l i n ethrough the vertex) ar e bi sected by the conj ugate d iametralplane . Thi s plan e also con ta i n s the gener ators Of con tact of thetwo ta ngen t planes to the cone through the diameter i n question .

A diame ter Of the cone passes through the cen tres Of the

sections of the cone by planes paral le l to i ts conj ugate d iametra llane .DFor l et u , v ,

10 be a sel f- pol ar three - edge,i t w i l l meet a plan e

paral le l to o w i n the vertices U,V lV

°°

Of a selfpolar tr ia nglefor the sec t ion . U i s thus the pole w i th regard to thi s sec t ion ofV

"0

IV”

,tha t is , Of the l i ne at i nfin i ty i n the plane Of the sec t ion .

U i s therefore the cen tre Of the sec t ion .

2 3 2 . C on e s h a r m on i c a l ly i n s c r ib e d i n a n d c i r c um

s c r i b e d to oth e r c on e s . If a cone 0, be ci rcumscn bed abou ta three - edge selfpolar for another cone C

,i t con ta i ns an 1nfin 1ty


Of such three - edges and is sa i d to be harmon ical ly c ir cumscr ibed toC. If a con e 0, be i nscr ibed i n a three - edge selfpolar for C i ti s i n scr ibed i n an i nfin i ty Of such thr ee - edges and i s sa i d to beharmon ically i n scr ibed i n 0.

If 0, be harmon ical ly c i rcumscr ibed to 0, then C i s harmon ical ly i n scr ibed in

2 3 3 . T h e c om p le te fou r - e dge a n d fou r - fa c e . Corr espond i ng to a complete quadrangle i n the plan e w e have acomplete four - edge i n the sheaf. Such a fou r - edge has s i x facesand the meets Of the pa irs of Oppos i te faces form its d iagonalthr ee - edge .

S im i larly to the complete quadr i lateral corresponds the complete four -f ace, with s ix edges and thr ee d i agona l planes , whichform its diagonal thr ee - edge . The harmon i c proper ti es Of thecomplete quadra ngle and quadr i lateral are transferred at once tothe fou r - edge and fou r - face . Thus two faces of the d iagonalthree - edge Of a complete fou r - edge are harmon ically conj uga tew ith regard to the two faces Of the four - edge through thei r i ntersect ions and tw o edges Of the d iagon al three—edge of a completefour - face are harmon ical ly conjuga te w ith regard to the two edgesOf the four - face i n thei r common plane .

2 34 . T h e c i r c le a t i n fi n ity . Any Sphere S cu ts theplane at i nfin i ty r

°° i n an imagi nar y c ircle . Cons ider any plan e W.

Thi s cuts S i n a c i rcl e . The poi n ts wher e th i s c i rcl e meets thel i ne at i nfin i ty of n are the c ircu lar poin ts at i nfin i ty of W. They areon the Sphere S and therefore 011 i ts c i rcle Of i ntersect ion wi th theplan e at infin i ty. Thi s c i rcle of in tersect ion i s therefor e the locusOf the c i rcu lar poi n ts at i nfin i ty in al l planes . I t i s cal led theci r cle a t i nfin i ty , and al l spheres conta i n i t.

2 3 5 . T h e sp h e r i c a l c on e . The con e formed by join i ngany poi nt 0 to the poi n ts of the c i rcle at i nfin i ty w i l l be cal ledthe sp her ica l cone through 0. Every plane 7r meets a spher i calcone in a c i rcle . For take the l i n e at i nfin i ty Of I t meets thecon e on the c ircle at i nfin i ty . The two c ircu lar poi nts at i nfin i tyOf 71 ar e therefore on the sect ion : hence the latter must be ac i rcle.Such a con e, being a su rface of the second order passi ng

through the c i rcle at i nfin i ty , i s to be also cons idered as a sphere .

I t i s,i n fact

,a p oint- spher e and i s the l im i ti ng case of a sphere of

van i sh i ngly smal l rad i us— prec i se ly as a pai r Of c i rcu lar l inesform a poi n t - c ircle . Hence the name spherical cone .



2 3 8 . Foc a l l in e s of a c on e . Cons ider the inter sect ion of any g iven con e and the spher ical con e Of same vertexby any plane W. The two con ics s , t form i ng the in tersect ionhave four common tangents

,w hi ch meet i n six poin ts. Hence

the two cones have fou r common tangent planes wh ich meet ins i x l i nes . These Six l i n es are cal ledfoca l lines Of the cone.

Any two planes through a focal l in e whi ch ar e harmon ical lyconj ugate w i th regard to the two common tangent planes throughthat l i n e are conj ugate with regard to both cones . Hence theyar e perpendicular by Ar t 236 . The focal l i nes have therefore thepr operty that conj ugate d iametr al planes thr ough them ar e perpendicular . The focal l in es are s i tuated in pa i rs 011 the threepr i nc ipal di ametral plan es . For they form the s ix edges Of acomplete four - face whose d iagonal three - edge i s the commonself- polar three - edge Of the gi ven cone and the Spher i ca l con e,that i s , the pri nc ipal three - edge Of the given cone.Any pa i r of conj ugate d iametral planes Of the g iven con e

wh ich are perpend icular are also conj ugate for the spher ical con e(Art . They meet any plane W i n two l i nes I, m which a r e

conj ugate for both con i cs s,t and therefore for al l the con ics Of the

r ange determi ned by s, t. They d i v ide ther efore harmon ical ly thethr ee poi n t - pairs Of th i s range. Bu t these poi n t - pai rs ar e thei ntersect ions w i th 7r of the thr ee pai rs of focal l i nes . A pai r Offoca l l i n es are therefore harmon ically separated by a pa i r ofperpend icu lar conj ugate d i ametra l pla nes .Tak ing for the per pend icu lar conj ugate d iametral planes the

tangent and normal planes thr ough a generator, we see that theselatter plan es b i sect the d ihedral angles between the planes thr oughth i s generator and any pai r of focal l i nes . The s tudent shouldcompare th i s property wi th that of the tangent and normal to acon i c whi ch bi sect the angles between the focal d i stances .Agai n , the planes wh ich bi sect the dihedral angles between

two tangent planes a , ,8 to the cone through a d iame ter are clea rlyconj uga te perpend icu lar planes . Thus they also b i sec t the dihedralangles be tween the tw o planes ( r , 1

' through this d iameter and apair of focal axes .The d ihedra l angle between a and a i s thus equal to the

d ihedral angle between,8 and

2 3 9 . C y c l i c p la n e s of a c on e . The comics 3 , t Of thela st Ar ti cle

,i n wh ich a gi ven cone and the spher i cal cone Of same

vertex mee t a plane have fou r common poi nts, determ i n i ngthr ee l i ne - pa i rs .

x1 1 1] THE CONE AND SPHERE 223

Any tw o poi n ts conjuga te w ith regard to both the con ics s , ta r e l ikew i se conj ugate wi th regard to these l i ne - pai rs .I t fol lows tha t the gi ven cone and the spher ical cone have fou r

common generators , de term i n i ng three pai rs of d iame tral planeswhich separa te harmon ical ly any tw o d iame ters conj uga te forboth cones , tha t i s, any two perpend icu lar conj ugate d iame ters ofthe gi ven cone .

These s i x planes a r e cal led the cycl ic p la nes of the gi ven cone .

Any plane paralle l to them conta i n s tw o Of the common poi n ts a ti nfi n i ty Of the cone and the spher ical cone , that i s, the section Ofthe cone by thi s plane passes through the c ircular poi n ts at i nfin i tyof the plane and i s a c i rcle . Thus the cycl i c planes are planespar al le l to the planes Of c i rcu lar sect ion .

A pa i r Of cycl i c planes pass thr ough an edge Of the commonself- polar three - edge Of the tw o cones , that i s , through an axi s ;and Si nce they separa te harmon ical ly the tw o other axes ( thesebe i ng perpend icu lar conj uga te d iameters) , the i r traces 011 the

plan e of those axes are equally i ncl i ned to those axes. Hencea pa i r Of cycl i c planes pass through an axi s and are equallyi ncl i ned to the pri nc ipal d iame tral planes through that ax is .Cons ider any plane a through the vertex 0 Of the cone,

meeti ng the con e i n tw o genera tors a , b and a pa i r of cycl i cplanes i n rays y. The conj ugate diameters i n a form ani n volu t ion flat pencl l. Let u ,

c be rec tangular rays of thi s penc i l .Then .z

'

, y/a nd a,b a r e harmon i cally conj uga te wi th regard to u ,

’t‘

,

therefore .r, y a r e equal ly i ncl i ned to u

, 11. S im i larly a,b a r e

equal ly i nc l i ned to u , r . Hence the angle between a and re i sequal to the angle be tween 0and y .

2 40. R e a l foc a l l i n e s a n d cy c l i c p la n e s . S i ncethe two c ircu lar poin ts a t i nfin i ty i n any plane a r e conj ugateimagi nary i t fol lows that the conj ugate imagi nary po i n t to a nypoi n t 011 the ci rcle at i nfi n i ty i tse l f l ies 011 the c i rcle a t i nfin i ty. The

c i rcle at i nfin i ty i s therefore i ts own conj ugate imagi nary locus’,and the same holds clearly for a spher ical cone w i th a real ver tex .

The studen t may ask why th is does not make i t a real c i rcle , s incei t was stated on the same grounds, in Ar t. 125 ,

that a straigh t l ine w h ichw as i ts ow n conjugate imag inary w as real . The answe r i s that the c i rcleat i nfin i ty is i ndeed de termined by tw o r ea l equat ions, name ly that ofany sphere and of the l ine at i nfin i ty . But the locus de te rmined by suchreal equat ions need not i ts elf be real , unle ss the equat ions a r e bo thli near

,wh i ch is the case for the str aigh t l ine .


The r eason i ng u sed i n Ar t. 1 25 therefore appl ies her e , to Show thatthe common genera tors and the common tangen t planes of a realcon e and a spherical cone hav i ng the same vertex a r e conj ugateimagi nary i n pa i rs . Th i s shows , bear i ng i n m ind the correspond i ngresu lts of Ar t. 205 that : (a ) the common se lf- polar three - edge i sentirely r eal , or the three pr i nc ipal planes and axes of any con eare real ; (b) of the three pa i rs of focal l i n es , one only i s real , theother tw o pairs be i ng imagi nary ; (e) of the three pai rs of cycl i cplan es

, on e only i s real .

2 4 1 . R e p r e se n ta t i on of th e sh e a f on a sph e r e .

I f w e descr ibe a Sphere of arb i trary radi us , whose centre i s thever tex of a sheaf, every plane Of the sheaf determ i nes a greatc i rcl e on the Spher e and every l i ne of the sheaf a poi n t on thesphere . A11 ax ial penc i l Of planes determ i nes a spher ical penc i lformed by great c i rcles pass i ng through a poi n t, a nd of coursepass i ng also through i ts an t i podal poi n t .A flat penc i l of l i nes of the sheaf determ i nes a Spher ica l range

Of poi nts on a great c ircl e. The r ange i s real ly a tw i n range,

s i nce each l i ne of the penc i l meets the sphere i n tw o antipodalpoi nts . The cross - ratio Of four elements i s determi ned from fou rarcs of a spher i cal r ange , or from fou r angles Of a Spher ical penc i l ,by a formula i n vol v i ng the Si n es Of these arcs or angles, i den ticalw i th that pr oved for flat penc i l s i n Ar t. 22 . Al so al l gr eatc i rcles mee t a spher i cal penci l Of four great c ircles in spher icalr anges Of the same cross—ra t io. We have thu s a whol e theory ofproj ective and perspect ive forms of the first order 011 the Spherewhich corresponds to the theory already developed for the plane.There a r e cer tai n d ifferences , for example , bear i ng i n m i nd that twopo i nts Of a Spher i cal range correspond to one l i ne of the defin i ngfiat penc i l through the centre Of the sphere

,we see that there are

two poi n ts of a Spher i cal range determ i n i ng a given cross - ratiowi th three given poi n ts Of the range .

Also the pri nciple of dual i ty wi ll hold for spheri cal figures .For s i nce the angles between two great ci rcles are equal to thearcs joi n ing thei r pol es (measu red by the angles sub tended at thecentre) , i f w e make a great c ircle correspond to i ts pa i r of polesand con versely , we have Spher ical penc i ls correspond i ng to equ ianharmon ic spheri cal ranges and conversely.

2 4 2 . S ph e r o - c on i c . The twi n cu rve i n which a cone ofthe second order meets a concentric Sphere i s cal led a spherecon i c . The properti es Of sphero - con ics are merely a resta tementin su i table language Of the pr opert ies of the con e of second order.


226 PROJECTIVE G EOMETRY [OH

The angles FTP, P TS ,

[ 1TQ, QTG are al l equal .angle FTS angle HTG .

Add i ng angl e STH w e have angle FTH = angl e STG,ar e

FT z a r e ST and ar e TH ar e TG .

The spher ical tr iangles FTH,STG are congruent and ar e

Now i f the tangent gr eat c i rcl e P T move r ou nd the curve,

TQ remai n i ng fixed , G remai ns fixed and SG remai ns fixed,arc FH a constant length .

Joi n i ng PF, P S ,PH , angle FP R = SPR by symmetry ;

and SPR TPH (equ al i ncl i nation of foca l d i stances to

ta ngent) . Hence angl e -PP R TPH or FPH i s a great c ircle.Thus SP PH FP PH FH constant .I f w e take tw o foc i S ’

, H not i n s ide the same oval , thenSP semi - c ircumference S

’P . Thus PH S

’P consta nt.

EXAMPLES X I I I .

1 . Show that propert ies of a sph er ical figure may be rec iprocatedby mak ing a great c i rcle correspond to i ts pole s and conversely . E i therfigure may then be called the polar figu re Of the other.Prove that the polar of a sphere - conic i s a sphe ro- con ic

,the cycl ic

l ines Of the on e rec iprocating in to the foc i of the other.Show that th is is a particular cas e of ordi nary polar rec iprocat ion

w i th regard to a spherical cone .

2 . Show that i f a pair of tangent planes th rough a d iame ter d of

a cone of second order touch th e cone along s,t and f be a focal l ine ,

the planes f s, f t ar e equally incl ined tof d.

Deduce that tangents to a sphe ro- con ic subtend equal or supplementary angles at a focus.

3 . F ind the envelope Of a plan e Of a sheaf wh ich moves so that i tstraces 011 tw o fixed plane s of the sheaf subtend a fixed d ihedral angleat a fixed l ine of the sheaf.

4 . I f a tangent plane to a cone of the second order mee t the tangplane s perpend icular to a princ ipal d iame tral plane in l ine s x, y , the

l ines or, y subtend a righ t d ihedral angle at a focal l ine s i tuated in the

g iven princ i pal d iame tral plane .

State the correspond ing theorem for the sphero- coni c.

x1 1 1] EXAMPLES 227

5 . I f a princ ipal d iame tral plane of a cone of the second orde rmee t the cone i n a

,a'

and p be any generator of the cone , the planesp a , p a

’mee t e ithe r cycl ic plane th rough the axis perpend icular to thegiven princ ipal plane i n tw o l ines at righ t angles.

6. Show that a plane perpend icular to a focal axis cuts the cone in

a conic,the focal axi s in que st ion pass ing through a focus o f th is conic.

7. Through any ray of a sheaf through 0 tw o cones of a con focalsystem ha v ing 0 for vertex can be described and these a r e orthogonal .

8. Show that i f one re ctangular th ree - edge can be in scr ibed in or

c ircumscr ibed to a cone of the second order, an infini te number of suchthree - edge s can be so i nscribed or c i rcumscribed.

9 . I f one s ide Of a spherical tr iangle move so that the area of the

tr iangle remains constan t, i t enve lops a sphere - con ic of wh ich the othertwo s ides (wh ich remain fixed) ar e the cycl ic l ine s.[Deduce from Ar t. 242 by the me thod of Ex. I ]

10. I f two non - coplanar con ics touch one another a con e Of the

se cond order passes through both of them.

[Use Arts. 1 2,1 1 . Two cones of the second order wh ich touch one another along

the l ine join ing the i r tw o vertices i nte rsect in a conic.1 2. Prove that any Sphere wh ich passe s th rough a sect ion of a cone

made by a plane paralle l to one cycl ic plane mee ts the cone again in aplane sect ion paralle l to the other cycl ic plane of the pai r.

CHAPTER XIV

QUADR ICS .

2 4 3 . O r d e r,C la ss , D e g r e e . The or der of a su r face i s

the n umber of po i nts i n wh ich i t i s met by any straigh t l i n e notlying i n i t.The class of a surface i s the number of tangen t plan es wh ich

may be drawn to i t thr ough any stra ight l i n e not lyi ng i n i t.The degr ee of a skew or twi sted curve ( that i s, a cur ve wh ich

does not l i e i n a plane) i s the n umber of poi n ts in which i t i s metby any plane.

Note that th i s defin it ion of degr ee coinc ides wi th the one

pr ev iously adopted for a plane cur ve , for the i ntersections of thelatter wi th any stra ight l i n e in i ts plan e may also be looked

ppon as i n tersection s w ith another plan e thr ough thi s str a ightme .

Note al so that a plane sect ion of a su r face of the nth or der i sa curve of the nth degree .

The fol lowing r esu lt wi l l be assumed : three surfaces of orderm

,n, p in tersect i n mnp poi n ts, r eal or imagi nary. Thi s i s

ev ident fr om analyti cal con s ideration s i f w e r emember that theequation to a surface of or der n i s of the n th degr ee in thecoor di nates .

2 4 4 . D ev e lop ab le s . A decelep able i s a surface envelopedby a plan e conta i n i ng only one variabl e par ameter. I t i sgenerated by the i ntersection of two consecutive plan es of thesystem . The class of a developable i s the n umber of tangentplanes whi ch can be draw n to i t from any poi nt.The locu s of i ntersection of con secuti ve generators of a

developable , or of thr ee consecutive tangent planes, i s cal led thecusp z

’

da l edge of the surface.Developables and curves ar e closely r ec iprocal .


230 PROJECTIVE G EOMETRY [ca

24 8 . Q u a d r i c a s p r odu c t of h om og r a ph i c r a n ge s

a n d axi a l p e n c i ls . I f .r,x’ be two generators of the same

system of a quadri c Q,P any poi n t on re, P

’the poi n t in wh ich

the generator y of the other system through P meets w’, thenclearly the cor r espondence between the ranges [P ], [P

'

] 1s oneone and algebr a i c . 'l hese ranges are therefore homogr aphi c .Hence e i ther regu l u s be longi ng to a quadri c determ i nes homogr aph i c ranges on the l i nes of the other regu lu s . A quadr i c canthu s be obtai n ed as the product of two non - coplanar proj ecti ver anges .

Aga i n l et .r ,.e’be two generators of one system ,

11 any planethrough x . vr determi nes a generator y of the other system ;g/mee ts a

“. Let the plan e yx

’z vr

'

. Then the cor r esponden cebetween the plan es 7

,7r' bei ng on e on e and algebra i c , we have

1[7r ]

_

A or a quadr i c can be obtai ned as the pr odiict of twoi omogr aphic axial penc i l s not belongi ng to the same sheaf.Conversely the product of any tw o such axial penc i ls i s a

quadr i c . For they determ i ne on any stra ight l in e tw o proj ect iver anges. These have two and only tw o sel f- correspond i ng poi n ts

,

whi ch are the poi n ts where the product locus meets the str aightl i n e . This product locus i s therefore of the second order .Aga in : the pr oduct of any tw o non - copla nar proj ective

r anges i s a quadr i c . For a poi nt of e i ther range determinesa plane thr ough the base of the other. In th i s way two homographic ax ial penc il s are formed , hav ing the bases of the rangesfor axes

,and the Joi ns of correspond i ng poi n ts of the or igi nal

r anges are clearly the i ntersections of cor 1 esponding planes of thetw o

b

axial penc i ls . These joi n s therefore define a quadric by thelast paragraph .

The planes of e i ther ax i al penc i l i n the above are the tangentplanes at the correspond i ng poi n ts of the r ange 011 thei r ow nax i s (s i nce each such plane conta i n s the other gen erator throughi ts correspond ing poi n t) . Hence the axial penc i l formed bytangent planes to a quadr i c through a generator i s homograph icw i th the range formed by thei r poin ts of contact on the samegenerator.I f we are given three non - i n tersecti ng l i nes w

,x’

,w a

var i able poi n t P on x determi nes tw o homograph i c axial pencil sof plan es .13

'

[P , w [P ]. The i n tersection of correspond i ngplanes of these 1s a stra ight l i ne meeti ng {0 at P and meeti ngalso a "

,x

"

,and thi s i s obv 1ously the on ly stra ight l i n e wh ich can

be drawn through P to meet a “, x Hen ceA str a ight lme meeti ng three given non - i nter sect ing stra ight

xi v ] QUADR ICS 231

l i nes (call ed di'r ectms ) descr ibes a regul us of a quadr i c , the threegi ven l i nes belongi ng to the other regu lus .And agai nA quadri c i s determ i ned by any three genera tors of one

r egulu s lyi ng in i t.

2 4 9 . C la ss of a qu a d r i c . Let a quadr i c be defined bythe proj ective ranges [P ] , [P ] determi n ed by one of i ts regu l i ontw o generators a', a

“

of the other . Let n be any straight l i ne .

The coax ia l homograph ic ax ial penc i ls a [P ], n [P'

] have twosel f- correspond i ng planes . Each one of these conta i ns a generatorP P

'

of the quadr ic . I t therefore conta i ns a second generatorand touches the quadr i c at thei r i ntersection .

Thus through any stra ight l i ne a tw o tangent planes canble dr awn to a quadri c or a quadr ic i s a surface of the secondC ass .Conversely every su r face of the second class i s a quadric .

For i ts sect ion by any plan e i s obv iously a plane curve of thesecond class, that i s , a con i c . Hence any stra igh t l i ne a will cu tthe su rface of the second class in two poi nts, for the i ntersectionsof n wi th the surface are i ts i ntersec t ions w i th any plan e sectionof the su rface thr ough a .

2 50. T w i ste d cu b i c . A tw isted cubic i s a cu r ve of thethird degree : i t may be obta i ned as the produc t of three homogr aph ic ax ial penc i ls . For take any three chords 0

,b, c of the

twisted cubi c . Let P be any poi n t on the curve and denote theplanes aP , bP ,

cP by 7m,my , Now s i nce or already mee ts the

cubic i n two poi nts (bei ng taken a chord) , a plan e vrl through i t

can meet the cubi c aga in in one poi n t P only. Thu s when w, i s

given , P , and ther efore and 7i'

s , are u n iquely determ i ned .

Sim i larly i f 7r 2 or ”3 be given , the other tw o a r e u n iqu ely de term i ned . Hence 71 [ml ] , [7 3] are three homograph ic ax ial penc i lsof planes , of wh ich the tw i s ted cub ic i s the product.The i ntersection of tw o quadr i cs hav i ng a common genera tor

( 1 i s a tw i sted cubic . For the two quadrics meet any pla ne i ntw o con ics . The i r cu rve of i n tersec t ion therefore meet s the

plane i n the fou r oi n ts of i n tersect ion of these con ics . I ngeneral therefore t i e i n tersection of tw o qu adric s i s a tw i s tedquar t ic . Bu t i f they have a common genera tor , thi s genera toraccounts for on e of these fou r i ntersec t ions . The remai n i ng par tof the i ntersection of the tw o qu adric s then mee t s any plane i nthree poi nts , that i s , i t i s a tw isted cubic .

232 PROJECTIVE GEOMETRY [OIL

I t follows that a ny thr ee homogr aph ic ax ial penc i ls of plan es[mi], [

73] determ i n e a tw i sted cubi c as thei r product. For

the pr o not of [7 5] and 1 112] i s a quadric Q2 hav ing the a xes of[7 a] as generators o the same system ; and the produ ct ofand [W3] i s a quadr i c Q3 hav ing the axes of [m] and [a s] as

generators of the same system . Q2 and Q3 have a commongen er ator

,namely the ax i s of [m]. The r ema i nder of the i r i n ter

section i s a twi sted cub ic,wh ich 13 the pr oduct of the thr ee g i ven

ax ial penc i ls.I n par t icu lar cases th i s twi sted cub i c w i l l degen erate into a

stra ight l in e and a con i c , or i nto three str a ight l i nes . Forexample if the ax ia l pen ci l s [7i

'

s] have a self- corr espond ingplan e a

, Q3 breaks up i nto th i s plan e a and another B byAr t. 230. The intersect ion of Q2 , Q3 then con s i sts of tw o str a ightl ines in a (one of which i s the ax i s of a ]?

and a con i c i n 3.

Aga i n i f the axes of the three penci s [W 2] , [7l'a] all lie inon e plane (1 and th i s plane be self- corr espondi ng for any pa i r ofpenci ls

, Q2 and Q3 break up i nto pla nes a , Band a, 7 r espectively.

The twi sted cub i c then reduces the straight l i n e By, togetherwith two indeter mina te straight l i n es i n the plan e a , for i t i s easyto see that afi,

a'

y do not i n thi s case give a defin i te locus s ince athi r d stra ight l in e in a

, of the same type, can be got by pa ir i ngthe pen c i ls d ifferently .

I f two quadr i cs have two gen er ators x, a“of one system in

common , the r ema i nder of thei r in tersection con s i sts of tw o

generator s of the other system . For l et P be a poin t on ther ema inder of the i r i n tersect ion . Thr ough P a l in e 31 can bedrawn to meet a; and a

" at Q and R . P , Q, R are three pointson each quadri c : the l i n e y whi ch contai ns them i s thu s a

generator of each quadr i c, s i nce a stra ight l in e which meets aquadri c at three poi n ts must l i e enti rely in the quadr i c .

2 5 1 . I n te r s e c t ion s of th r e e qu a d r i c s . By Ar t. 243thr ee quadri cs Q1 , Q2 , Q3 wi l l i n ter sect in gen eral in eight points .A tw i sted cubi c t meets any quadr ic Ql in s i x oi n ts . For

su ch a cub ic i s obta i nable as the product of three omogr a hieax ial penc i l s

,that i s

,as the i n tersection of the two quadiics

Q2 , Q3 determ i ned by tw o pai rs of these penc i ls and hav i ng acommon gen erator anOf the e ight i n ter sections of the three quadri cs Q1 , Q2 , Q3

tw o are accounted for by the i n tersections of Q1 wi th x . Ther emai n i ng six are the i nter sections of Q1 wi th the tw i sted cubic .A twi sted cubic t w hich i s the i nter section of tw o quadr i cs Q1


234 PROJECTIVE GEOMETRY [on

The symmetry of th i s last relat ion shows that the poles ofplan es through 7rp l i e on PR .

Tw o such l i nes P R ,7rp are sa i d to be conj ugate or polar l i nes

w i th r egard to the quadr i c .Two conj ugate planes 7r

, p divide harmon i cal ly the d ihedralangle between the tw o tangent planes through the i r i ntersec t ion 7rp . For let S , T be the poi nts of contact of r and l etS Tmeet W, p at P , B respect ively. Then i f U be any oin t of7rp the polar plane of U passes through ST. W)

, ST are t er efor e

conj ugate l ines . Thu s the pol e of p l ies on ST,bu t i t al so l i es

on 7r,hence it must be P ; M -{PSBT} 1

,or { 7rp0

'

r } 1 , whichproves the r equ i red r esu lt.The polar plane of a point P on the quadri c i s the tangen t

plan e at P . For the polar pl an e of ever y poi nt R m the tangentplan e at P passes through P .

2 5 3 S e l f -

p ol a r tetr a h e d r on . Let P be any poin t, 7 i tspolar plane , R a poi nt of p i ts polar plane , which passes throughP

,S a poi nt of or i ts pol ar plan e, which passes thr ough P , B

a1id meets 7rp at T. Then T l ies on r, p, 0

'

i ts polar plane 7 i sPBS . A tetr ahedron such as PE ST 1s sa i d to be self - conjugateor sel f- polar wi th r egard to the quadri c. Ea ch vertex i s the pol eof the oppos i te face . Any tw o of i ts vert i ces, or any tw o of i tsfaces

, or any two of i ts oppos i te edges, are conj ugate w i th r egardto the qu adr i c .The three- edge formed by any three faces of a self- pola r

tetrahedron i s sai d to be a selfpolar three - edge for the quadri c .Any tw o of the faces a r e conj ugate and the pol e of any face l i esin the oppos i te edge . The polar plane of the vertex of such a

three - edge meets i t in a tr i angle selfpolar for the con ic in whichthe same plane meets the quadr i c. Thi s three - edge i s thereforeal so (Ar t. 23 1 ) selfpolar w i th r egard to the tangen t cone to thequadr ic from i ts ver tex.

2 5 4 . A sym p tot i c c on e . The tangent planes at thepoints of intersection of a plane 71 with the quadri c pass throughthe pole of 7r . Tak i ng 11 to be the plane at i nfin i ty, the tangentplanes at i nfin i ty to the quadric envelop a cone of the secondorder which 1s call ed the asymptoti c con e of the quadr ic . I tsvertex i s the pol e of the plane at i nfin i ty. Thi s i s cal led thecen tre of the quadri c . Li n es and planes through the centre ared i ameters and d iametral planes r espect ively . As i n the case ofthe con ic al l d iameter s a r e bi sected at the centr e.

x1v ] QUADR ICS 235

A selfpolar three - edge whose vertex i s the centre forms asystem of three conjugate d iametral planes . Any one of i tsedges i s said to be a d iameter conj ugate to the oppos ite d iametralplan e , although str ictly speak i ng the conj ugate to a d iameter i sthe l i ne at i nfin i ty of i ts se - called conj uga te d iametral plane .

The pol e of a d i ame tral plane i s the,poi n t at i nfin i ty on the

conj ugate d iameter. Hence chords paral le l to a d iameter areb i sec ted by i ts conj ugate d iametral plane .

Also i f 0 be the poi n t where a d i ameter meets a plane 7'

paralle l to i ts conj uga te d iametral plane y , 0’i s conj uga te to the

poi nts of the l i n e a t i nfin i ty of y or y’ and i s the cen tre of the

sec t ion of the quadri c by y'

.

A set of three conjugate d iametr al planes form ing also byAr t. 253 a selfpolar three - edge for the asymptot i c cone, proper t iesof su ch three—edges fol low immed ia tely from the theory of thecon e of second order i n Chap . XI I I . Among other resul ts wesee that there ex i sts one set and one only of three r eal mutual lyperpendicu lar conj ugate d iametral planes . These a r e cal led thep r incip a l p la nes of the qu adri c . They are planes of symmetryboth for the quadri c and i ts asymptot i c cone. Thei r i ntersectionsar e cal led the axes of the quadri c .

2 5 5 . P la n e s of c i r cu la r s e c t i on . S i nce the quadr i cand i ts asymp toti c con e have thei r points at i nfin i ty common , i tfol lows by Ar t. 228 that the sections of the quadr i c and the coneby any plane have the same poi nts at i nfi n i ty and are therefores im ilar. I n part i cu lar, i f one of these sections i s a c i rcle , so i s theother. Hence planes paral le l to the cycl i c plan es of the asymptotic cone cu t the quadri c i n c i rcles . The tangen t planes paralle lto the planes of c i rcu lar sect ion meet the quadri c i n point ei r cles .

Thei r poi nts of contac t are cal led umbi lics of the quadr ic . Since

there ar e s i x cycl i c planes , of which tw o are real , there aretwe lve umbilics, of which four are r eal ; and they l i e i n fou rsin the three pri ncipal planes .Al so i f the asymptot i c cone i s r ight c ircu l ar, the quadri c i s a

su rfac e of revol u tion whose ax is i s the ax i s of i ts asymptot i c cone .

I n th is ca se the section of e i ther the asymp tot i c cone or the

quadri c by the plane a t i nfin i ty has double con tac t w i th thec ircle a t i nfin i ty (Ar t. Co nversely i f a quadr i c has doublecon tact w i th the c i rcle a t i nfin i ty i t i s a su rface of revolu t ion .

I f the asymp tot i c cone be a spherical cone the quadr ic is asphere and every set of conj uga te d iametra l planes arer ectangular.

236 PROJECTIVE GEOMETRY [011

2 5 6 . C la s s ifi c a ti on of h y p e r b oloi d s a n d p a r a b olo i d s .

The hyper boloi d, s i nce i t meets the plan e at i nfin ity in a r ealcon i c

,has a r eal asymptot ic con e. Ther e ar e tw o classes of

hyper boloi ds, accor d ing as the surface l i es i n s ide , or outs ide,i ts

asymptoti c cone . I n the first case there are tw o sheets to thesu rface

, one ly i ng i n s ide each half con e, and s i nce no stra ight l i n enot pass i ng thr ough the vertex can l i e enti rely in s ide a con e ofthe second order

,ther e can be no r ea l genera tor s of su ch a quadri c .

I t is called a hyper boloid of tw o slzeets. In the second case therei s on ly one sheet to the surface . Al so a tangen t plan e at i nfin i tyto the su r face, being al so a tangent plan e to the asymptoti c cone,has poi nts lyi ng i ns ide the sur face and so cuts the la tter i n r eall i nes . Th i s quadr ic has ther efor e r eal generator s . I t i s cal led ahyp er boloid of one sleeet.

S i nce the paraboloi d touches the plane, at i nfin ity, i t meetsth i s plan e i n two stra ight l i n es . The cen tre i s here the poin t ofcon tact of the plan e at i nfin i ty and i s i tse l f at i nfin i ty. Thetangen t pla nes to the asymptot ic con e become two sets of parall elplan es through the generator s at infin i ty. Al so in the r angesdeterm i ned by one r egulus upon the other , s in ce each r egu lu scon ta i n s a r ay at i nfin i ty, the r anges have the ir poin ts at i nfin i tycor r espond ing and so are s im i lar . Conversely joins of corresponding poi n ts of tw o s im i lar r anges generate a paraboloi d .

There are two classes of paraboloids accord i ng as they meetthe plan e at infin i ty i n tw o real or i n tw o imagi na ry 'l i nes . Thefirst class are cal led ll p er bolic p a r aboloids. Any tangent plan emust con ta in tw o r ea poi nts at i nfin ity on the quadri c and so

meets the latter i n r ea l gen erators .The second class are cal led ellip tic p a r aboloids. Every plane

sect ion of these has imaginary poi n ts at i nfin i ty and so i s anell ipse

,except sections by planes parallel to the d i rect ion of the

a ctua l point of contact of the quadr i c w i th the plan e at i nfi n i ty ;these meet the quadri c i n parabolas. I t i s obv ious that such aquadr i c can have no rea l gen erator s.

2 5 7 . R e c ip r oc a l p ola r s w ith r e ga r d to a qu a d r i c .

As in the case of plane figures,so i n space

,we can con struct

a rec iprocal transformation in which to each poin t corr esponds i tspolar plan e wi th r egard to a quadri c Q cal led the base quadri c ,and conversely. To po i n ts of a plane w i l l correspond planesthrough a poi n t : to a plan e figure or a sheaf wi l l corr es end asheaf or plane figure respec t i vely

,and to r anges

,flat penci s , and

ax ial penc i ls w i l l corr espond homogr aph ic ax ial penc i ls , fiat penc i ls,


238 PROJECTIVE G EOME’I‘

RY [ca

2 6 1 . S e l f - p o la r te t r a h e d r on of a p e n c i l of qu a d r i c s .

Let Q1 , Q, be tw o quadri cs , P any poi n t. The poi nts of spacewhich are conj ugate to P wi th regard to both quadr ics are on

theént

eqr section p of the polar plan es w , and of P with r egar d

to 1 ) 2 '

Al so i f two poi nts P,P

’are conj ugate wi th regard to bothQ1 and Q2 they ar e cl early conjugate wi th regard to every quadr i cthrough the i ntersection of Q1 and Q2 , for , be i ng harmon ical lyconjugate wi th r egard to tw o pa i rs of mates of the i nvol ut iondeterm i ned on PP ’by the quadr ics of the system ,

they are thedouble poi nts of th i s i nvol u t ion and so harmon ical ly conjugatew i th r egar d to every su ch pa i r of mates ; therefore they areconjugate for every quadri c of the penc i l .Thus to every poin t P of space corresponds a l i ne p every

poi nt of which i s conj ugate to P with regard to al l the quadricsof the penc i l . We may call th i s the l i n e conjugate to P wi thr egard to the penc i l .I f P describe a s tra ight l i n e a ,

the planes m ,a , sweep out

two homograph ic ax ial penc i ls abou t the polar l i n es a , , u2 of awith r egard to Q1 , Q2 . Thus p generates a quadr ic U w h ich i sthe locu s of poin ts conjugate to poi nts of n for the penc i l .Th is quadri c U con ta i n s a , and f

a g . By symmetry i t mustcon ta in the polar l ine of u wi th regard to every con i c of thepenci l . Thus the polar l i n es of a and the l i nes conjugate topoi n ts of a form the two systems of generators of U.

Let u , c , w be three l i nes thr ough a poi n t A ,a the l i n e

conj ugate to A for the penc i l . The correspond ing quadri csU, V, W have a common generator a . They have thus fou rcommon poi n ts P , R ,

S , T (Ar t. There are three poi n ts P '

,

P”, P

’"on n

,c, 10 respecti vely, which are conj ugate to P for the

penci l . The plan e P ’P P i s therefore the polar plan e of P for

all quadrics of the penc i l . Conv er sely a poi nt wh i ch has the samepol ar plane w ith regard to al l qu adrics of a penci l i s n ecessar i lyan i ntersection of U, V, W There are then only four su chpoi nts . Now the polar planes of P , B , S meet at the pol e of

the plan e PBS for al l quadrics of the penc i l . Th i s pol e mustaccord i ngly be T. S im i larly P

, B , S are the pol es of R ST,STP ,

TPH respect i vely . Hence PBST i s the common self- polartetrahedron for al l quadr ics of the penc i l .

2 6 2 . C on e s th r ou gh th e i n te r s e c ti on of tw o qu a d

r i c s . Let P be on e of the ver tices of the common self- polartetrahedron of the quadrics, A a poi n t of the twi sted quar t ic

x1v ] QUADR ICS 239

in which they i ntersect. Let P A meet the polar plane of P

at L . Then i f A '

be harmon ically conj ugate to A wi th regardto P and L ,

A’ l i es 011 both quadrics and therefore 011 the quart i c .

Hence the fou r po i nts A ,B

, A’

,B

’ i n whi ch any planethrough P meets the quarti c l i e i n tw o pai rs (AA

'

) (BB’

) on

rays through P . The l i nes joi n i ng P to the quart i c form acone which has tw o generator s in any plan e through P ,

that i s,

a con e of the second order. A s im i lar resu l t holds for the otherver t i ces B ,

S , T of the common selfpolar tetrahedron . HenceFou r of the quadr ics of a penc i l are con es , whose vertices are

the vert ices of the common selfpolar tetrahedron of the penc i l .2 63 . P r op e r ti e s of a r a n ge of qu a d r i c s . The proper

t i es of a r ange of quadr ics are immediately der i vable from thoseof a penc i l of quadrics by r ec iprocat ion . We wi l l note thefol low i ng :The ta ngent cones fr om any po i nt to the quadri cs of a r ange

form a system touch i ng fou r planes .Through any poi nt three quadr i cs of the r ange can be made

to pass.To any plane 71 cor responds a l i n e p through which pass al l

planes conj ugate to vr for the quadri cs of the r ange and wh ichi s al so the locus of pol es of W for the quadr ics of the range .

Tak i ng 71 a t i nfin i ty the locus of centres of the quadr i cs of ar ange i s a stra ight l ine .

The surface generated by the l i nes p correspond i ng to planes71 through a gi ven poi nt P i s a quadr i c touch i ng fou r fixed planesi ndependent of the pos i t ion of P . These four fixed planes arethe faces of a tetrahedron selfpolar for al l the quadrics of therange .

Bear i ng in mi nd that a cone rec iprocates i nto a con i c (wh ichi s therefore to be cons idered as a spec ial case of a surface of the

second class) we see, rec iprocat i ng the property of the lastAr t icl e , that :Fou r of the quad r i cs of a range are con i cs , w hose planes ar e

the faces of the common selfpolar te trahedron of the r ange .

2 64 . C on foc a l qu a d r i c s . Con s ider the range dete rm i nedby any quadric Q and the c i rcle a t i nfin i ty (a degenera te cof a quadri c) . There a r e three con ics of thi s range

,bes ides

the c i rcle at i nfin i ty . Thei r planes a, B, y and the plane at

i nfin i ty form the sel f- polar te trahedron of the range : a, ,B, y a r e

therefor e three conj ugate d iame tral planes of any quad r ic Q’

of the r ange . They form a sel f- polar three - edge of the asymptot i c


cone of Q’

. But al so they must meet the plan e at infin ity ina selfpolar tr iangl e for the c ircle at i nfin ity . Hence they forma selfpolar three - edge of a spher i cal con e, that i s , a r ec tangularthree - edge . Or a

, ,8 , y are the three pr in c ipal planes, of Q’

.

Hence the quadr ics Q’

of such a r ange are concentr i c andcoax ial . There are three con ics of the r ange lyi ng each in one of

the three common pr incipal planes .These con ics ar e cal led the foca l con ics of Q

' ever y point ofthem i s cal led a focus of Q

’.

The quadr ics Q’ are sa id to form a confoca l system .

Let F be any poi nt of a focal con ic . Then the tangent conesfrom F to the quadrics of the confocal system form by Ar t. 263

a system of con es touchi ng fou r fixed planes through F. Now

cons ider the ta ngent cone to a con i c from any poi nt in i ts plan e.This tangent cone (treated as an envelope) r educes to the twotangents from the poi n t to the con ic. Hence the tangen t con efr om F to the focal con i c cons i sts of two coi nc i dent tangentsto thi s con ic at F. The four fixed planes therefore cons i st ofthe two tangent planes to any cone of the system thr ough thetangent l i ne to the focal con ic at F, each such tangent plan ebei ng doubled , that i s, i ts l i ne of contact bei ng gi ven . Henceevery con e of the system touches tw o fixed plan es thr ough Falong given l i nes through F i n these plan es , or the tangent con esfr om F to the system of confocals have double contact. But one

of these tangen t cones i s the tangent con e to the c ircl e at i nfin ity,that i s

,i t i s the spher ical cone through F. The tangen t cones

fr om F to the system of confocals have therefore double con tactwi th the spher ical cone that i s

,they are right c i rcu lar con es .

Foc i of a quadr i c are thu s poi nts, the tangent cones fromwhich to the quadr ic are right c i rcu lar .

2 6 5 . N e t of qu a d r i c s . I f a quadri c passes through seveng i ven poi nts , we can show as in Ar t. 258 that i ts equation may beput i n to the form

A1 8 1 Ass -

z A35 3 2 0:

S 1 , S2 , S , bei ng gi ven express ions of the second degr ee in thecoord i nates and A, , A

, arb i trary parameters . This quadri cpasses through the i ntersections of the thr ee quadr i cs

S 1 = O, S 3 = O,

that i s,quadr i cs sa ti sfy i ng such a cond i t ion pass through eight

fixed poi n ts . Thus, i n add ition to the seven iven poin ts , therei s a n eighth fixed poi n t, which i s determ i ned y the seven fir st;~

and thr ough which the quadrics pass.


242 PROJECTIVE G EOMETRY [cm

of fou r quadr i cs of a net by a plane. The theor em i s thenobv ious .

2 6 6 . C onju ga te p oin ts w i th r e ga r d to a n e t of

qu a d r i c s . I f two poi nts P,P

’ are conj ugate wi th r egar d tothree quadr i cs Q, , Q, , Q, of a net, they are conj ugate w i th r ega r dto a ll quadr i cs of the n et. For i f Q, be any other quadr i c of then et, w e have seen by the above that a quadr ic Q ex ists belongi ngto both penc i ls (Q, , Q,) and (Q, , If P

,P

’are conj ugate w i thr egard to (Q Q, ) they are also (by Ar t. 261 ) conj ugate w i thr egard to Q . Also Q, Q, , Q, are quadr i cs of a penc i l . Hen ceP , P

’be ing conj ugate wi th r egar d to Q, Q, , they are al so conjugate w i th regard to Q, .

Thu s to every point P of space corr esponds a poin t P ’

conj ugate to P wi th regard to the net. P’ i s obta i n ed as the

i ntersect ion of the polar plan es of P with r egar d to any thr eequadrics of the n et. Hence the polar planes of P wi th r egard tothe quadri cs of the n et pass thr ough a fixed poi n t P ’

.

2 6 7 . W eb s of qu a d r i c s . A w eb of quadr ics i s the setof quadr ics tou ch i ng seven fixed planes . Rec iprocat i ng thepr operti es of a n et of quadr i cs w e obta in the fol low ingThe quadr i cs of a web touch an eighth fixed plan e. If fou r

quadr i cs Q, , Q, , Q, , Q, belong to a w eb the common tangen tplanes to Q, and Q, and the common tangent planes to Q, andQ, al l touch a fixed quadri c Q.

To ever y plan e of space there i s one plan e conj ugate wi thr egar d to a w eb of quadr ics

,i .e . the poles of a fixed plane

w

l

ith r egar d to the quadr i cs of a w eb l i e on another fixedp ane .

I n par t icular i f the given plan e be taken at i nfin i ty the locusof centres of quadr ics of a web i s a plan e .

EXAM PLES XIV

1 . The locus of the vertex of a cone of the second order inscribedin a g iven skew hexagon i s a quadric.

2 . A regulus ly ing in a quadric and an axial penc i l homograph icw ith the regulus generate a tw isted cub ic.

xi v ] EXAMPLES 21 3

3. I f a shea f of tangen t planes to a cone of the second order behomograph ic w i th an axial penc i l not pass ing through the vertex of thecone , but such that the sheaf and the axial penc i l have one sel f- cor respend ing plane , the i r produc t is a quadric.

4 . A range of poi nts on a con ic is homograph ic w i th a range on astraigh t l ine not coplanar w i th the conic but mee ting the conic at A . I f

A be a se l f- correspond ing poi nt show that the jo ins of corre spond ingpoi nts of the tw o ranges li e on a quadric .

5 . Two fixed straigh t l ine s a and b mee t a conic 3,but ar e not

coplanar w i th s or w i th each other. Show tha t a straigh t l ine wh ichmee ts s

,a,b describe s a quadric .

6. A tangen t plane to the as ymptot ic cone meets the quadric inpar allel generators be longing to oppos i te systems.

7. Through the poin ts where the planes of an axial penc i l mee t astraigh t l ine ar e drawn perpend iculars to these plane s . Show that theseperpendiculars lie in a hyperbol ic paraboloid .

8 . Show that i f a quadric contain a tw isted cub ic, the generators ofon e set mee t the cubic in one poin t

,wh ile those of the other set mee t i t

i n two po ints.9 . Show that i f tw o quadrics have a common generator the gene

rate rs of the othe r system in each quadric, wh ich intersect on the i rcommon tw isted cub ic

,form homograph ic regul i .

10. Prove that through any point P of space a quadric can be

drawn contain ing a g iven twi sted cub ic and a g iven chord of i t .Show that through P on e chord

,and one only

,of a g iven twi sted

cub ic can be drawn .

1 1 . Show that a regulus projects from any point upon any planei nto a homograph ic penc i l of the second order.

1 2 . a , B a r e tw o plane s ; a , 1) tw o non - coplanar l ines in space wh ichboth mee t aB. Show that i f P , ,

P , he poi n ts of a , B respect ive ly suchthat P , P mee ts a and b

,the correspondence be tween the plane s

[P is homograph ic.

1 3. Prove that the locus of the poles of a fixed plane for thequadrics of a penc i l is a tw is ted cub ic.[I t is the i n te rsect ion of the quadrics of conjugate po in ts for two

l ines in the fixed plane ]Deduce that the locus of centre s of quadrics of a penc i l is a twisted

cub ic w hose a symptotes a r e paral lel to the d irect ions of the po i nts ofcon tact w i th the plane at infini ty of the parabo lo ids of the s v s tem.

1 13— 2

244 PROJECTIVE GEOMETRY [ca x1v

1 4. I f tw o quadrics of a penc i l meet a plane i n conics harmonicallyc i rcumscribed to the same conic s, show that th is is tr ue of all thequadrics of the penc i l.By tak ing 3 to be the circle at i nfini ty, show that i f tw o quadr ics of a

penc i l ar e equ ilate ral (i .e . a r e such that rectangular three- edges can beinscr ibed in the ir asymptot ic cone) then all quadrics of the penc i l possessthe same property.

1 5 . Show that a s ingle focal con ic defines a family of confocalquadrics and that through any po int of Space three quadrics of the

family can be drawn, wh ich cut orthogonally.1 6. Prove that the poles of a fixed plane w i th regard to a sys tem oi

confocal quadr ics lie on a fixed l ine normal to the plane .


246

cum scr ibed to another, 209 , 210;i s determ ined by five points orby fi ve tangents, 52Construction of from givencond itions , 53, 72 , 77, 1 99Defin i tion and types of 45

E lev en ~ l in e 1 96

E leven - po int 1 95

E very curv e of second degr ee or

second class is a 1 38

Intersections of a straigh t l i neand a 1 2 1

Product of tw o projective penci l sor of tw o proj ective ranges i sa 48 , 50

Conics, confocal , 1 63, 200; harmonica lly inscr ibed in and c ircumscribed to th e same con i c , 2 1 2 ;hav ing double contact , 1 98

Focal of a quadr ic , 240S im i lar 2 1 4

Th e e ight tangents at the pointsof intersection of tw o 1 65

Conj ugated iameters , 68 , 69 ; sum an d d i fference of the ir squares , 90, 92

elements of a harm on ic form,34

imaginary e lements,1 31 ; e lem ent

determ ined by them is rea l , 1 32l ines through a po int a r e harm onica l ly conj ugate w i th regard toth e tangents from th e po int , 60;they form an involution , 1 59l ines with regard to a range of

con i cs , 194paral le logram , 89 , 9 1

points a nd l ines with regard to acircle

,59 , 62 ; w i th regard to a

con ic,64

,159

po ints wi th regard to a penc i l ofcon ics , 1 93ranges and penci l s a r e projective , 60Coordinates , 1 26 , 1 27Correspondence , 1Cross - ratio , 25 , 27, 1 28

determ ined by arm s of an angleof given m agnitude with the

circular l ines , 161determ ined by tw o correspond inge lements of coba sa l projectiveform s w ith the se l f- corresponding e lements , 1 20

of four harmonic e lements , 34

I NDEX

Cub icA symptotes of a 1 79P lane 179 , 1 87; w ith a doublepo int , 1 79 , 1 88Twisted 231

Degree , of a plan e curve, 66 ; of a

twi sted curve , 228Curve of second 1 38 ; of fourth

1 86, 1 88

Deve lopable , 228 ; cusp idal edge of

228

D iameters , con jugate of a conic ,68 , 69 ; a r e paral le l to supplemental chords , 70Imaginary of a con i c ; the ir reallength s, 86

Director c ircle,1 62

D irectors of a regulus , 231Directrix

,101 ; i ts d i stances fr om

th e centre an d foci , 103Duali ty , principle of, 67, 1 28

E ccentric ity , 102E lements , 1 ; at infin ity

,4

Co incident 78

Imaginary 1 29 ; number of realincident w ith an imaginary

one , 1 32 ; conj ugate imaginary

Se l f- corr espond ing 8 ;of tw o co

basa l proj ective form s,30, 1 20;

h ow constructed , 1 20; they determ ine with any tw o cor r espond ing a con stant cr oss ~ r atio,1 20; se l f- correspond ing of

tw o coplanar projective figures,1 47 ; of tw o homograph ic formsn ot on th e same ba se , 1 75 , 1 76

E l l i pse , 46, 72 , 88E l l ipsoid , 229E nve lope

,46

Faure and G ask in ’s Theorem , 213

F igures , in plane perspective , 7; inspace perspective , 2

COr i ‘eSpondin g 1

Homograph ic 1 39

Projective 1 7R eciprocal 1 43 , 1 48

Forms , e lementary geometric, 27; ofthe second order

,1 1 8

,174, 1 76

Coba sal 29 ; identical i f th r

INDEX

e lements a r e selfcorrespond ing,30

HarmonicHomograph ic 1 33— 136

Inciden t 173 , 1 77, 1 8 1

Projective an d perspectiveFocalaxi s of a con ic , 102chord of curvature

,1 1 3

con ics of a quadri c , 240d i stances of a po int on a con i c ;the ir sum and d ifference , 104 ;angle s wh ich they make withtangent and normal , 106l ines of a cone , 222perpendiculars on tangent, 108spheres , 100Foci , 101 ; four in number , 1 63 ; n ot

more than tw o rea l , 101 ; the ird istan ces from th e centre , 103

Focus , conjugate l ines through aar e perpend icular, 101

Tw o tangents to a con i c subtendequal or supplementary anglesat a 104

Four - edge , complete , 220Four - face , complete , 220Fr é gi er point, 1 68

G ask in and Faure’s Theorem , 21 3

Harmon icproperty of th e complete quadr ilateral and quadrangle , 36ranges and penc il s , 33

Homography , determ ined by tw o

correspond ing tetrads , 1 40; i sa proj ective transform ation andconverse ly , 1 46

G eometrica l evidence of

Hyperbola.Construction of from givencond i tions , 80

Rectangular 93 , 94 , 1 23

Tw o con j ugate d iame ters o f aa r e harm on ical ly conjugate

w i th regard to th e asympto te s ,69

Hyperbo lo id , 229 ; o f one or tw o

sheets , 236

Inc ident e lem ents , 1forms , 1 73 , 1 77, 1 8 1

Joach imsth al’s Theorem ,

204

L ine - pa ir and po int- pa ir, 52 , 65

Mates in an involution,1 5 1 ; har

m on ica lly conjugate with regardto double e lements , 1 52

Mene laus’ Theorem , 40

Ne t,of comics , 2 13 ; of quadrics , 240

N ewton’s theorem on the product ofsegments of chords o f a con ic ,8 4

Normal to a con ic ; bisects angle

be tween foca l d ista nce s , 106 ;i ts inte rcept on the foca l axis ,107 ; i ts length inverse ly p r oportional to the central pe r pend icu la r on ta ngent , 109

points and l ines of two rec iprocalfigures , 1 48

Infi n i tyC ircular points at 1 60; con

j ugate with regard to a rectangular hyperbo la, 1 6 1

E lements at 4

Tangents at 45

Involution,

1 51 ; determ ined bypenc il of con ics on any stra ightl in e , 1 92 ; by tangents to arange of con ies from any point,1 93 ; determ i ned by two pa irs ofmates , 1 51 ; of con jugate ele

m ents with regard to a con ic ,1 59 ; of points on a con ic and

of tangents to a con ic , 1 66Centre and axis of 1 52 , 1 66 , 167Double e lements of a n 1 52 ;h ow constructed , 1 53

E l l i ptic or hyperbol ic1 55

Mates in an 1 5 1

R ectangula r 1 57; every e ll i pticon a straight li ne may be

regarded as a section of a rectangular 1 57

Re lation between s ix po ints in1 52 ; between s ix rays , 1 56

In volutions on th e same base ; the ircommon pai r of mates , 1 57

Homograph ic 1 8 1 ; product ofhomograph ic 1 86

248

Notation for homogr aph y , 1 38 ; forpo ints , l ine s and planes , 1 for

proj ective ranges and penc i l s ,46 ; for segments, 2

Order of a surface , 228Ord inate and abscissa referred to

con j ugate d iameters,86

ParabolaConstruction of from given cond ition s , 48 , 80, 8 1 , 82

Diameters of a 69Parameter of paral le l chords of a

9

Special focal properties of th e09

Tangent to a makes i nverse lyproportiona l intercepts on tw o

fixed tangents , 48Parabolas through four po i nts , 1 22Parabolo id , 229 ; e l l i pti c or hyper

bolic , 236Pascal’s Theorem , 16Penci ls , axial , 1 28 ; flat, 27 ; of

con ics , 1 92 ; of quadrics , 237Concentric , perspective , directlyand oppos itely equal 28

,29

Homograph ic 1 34—1 36

Product of projective or homo

graph ic 48 , 1 79 , 1 83 , 1 88

Projective of first and secondorders , 28 , 1 1 8 ; how con

structed , 31 , 1 75 ; cross - centreof tw o such pencil s

,38 , 1 1 9

Perspective figures, in a plane , 7i n space

,3

Particular cases of figures in

plane 1 3

Pole or centre of 9Tw o figures in plane ar e p r o

jection s of a th ird figure in

another plane , 8Ways of bringing tw o con ics i ntoplane 5 1

P lanes of circular sect ion of aquadri c , 235

Polar 57, 64 ; as chord of contact,58 ; of centre , 61 ; plane of apoint w i th regard to a quadric ,233

Con structions for th e of a pointwith regard to a con ic , 64

INDEX

durin g r a

Quadrangle , sel f- polar for a con ic ,207

Complete its harmon ic p r operty , 36

Quadric , 229 ; as product of

homograph ic ranges or axial penc i ls , 230; generators of a229

QuadricsConfocal 239

In tersections of two or three232 , 238

Quadri lateral sel f- polar for a con ic ,2 10

Complete i ts harmonic pr operty

,36

Quartic, twi sted 231 , 237, 238

Rabatment, 7 ; of vertex of pr ojec

Polars of points of a range form apenc i l equianharmon ic with th er ange , 60

R eciprocal 66Pole of a l ine with regard to a con ic ,

64 of a plane w i th regard to aquadric, 233

Product of homogr aph ic axial penci l s whose axes intersect

,218

w hose axes do n ot intersect, 230of homograph ic involutions, 1 86of i nvolution an d homograph ics imple form ,

1 88

of n on - coplanar homographi cr anges , 230

of penci l s and r anges of first order,48, 50

of penci ls and ranges of secondorder

,1 83

,1 88

of three homograph ic axial penc i l s ,232

ProjectionCentral or con i calCyl indr ical 10

Drawing of 1 5

Locus of vertex of

batmen t, 10

Orthogonal 10; every e l l ipsecan be derived from a c ircleby orthogonal 88

Particular cases ofProblems in 1 8

Success ive 17


QlismhrmgcPR INTED BY JOHN CLAY, M .A.

AT THE UN IVERS ITY PRESS .


14 DAY USETO DESK FROM

LOAN DEPT.

Th is book is due on th e last da te stamp ed below ,

or on th e date to w hi ch r enew ed . R en ew als on ly :Tel . N o . 64 2 - 3405

Ren ew als may be made 4 days p r i or to date due .

R enew ed books ar e subject to immedi ate r eca ll .

U C . BERKELEY LIBRARIES

( 05 1 3 5 1 03 3

BRARYUNIV ERSITY OF CALIFORNIA LI

An Introduction to .Projective Geometry - Forgotten Books · PDF filePREFACE MY object in writing the following pages has bee n to supply the growing n eed of t mathematical students

Documents