Aa Seminar - Seminar

8/10/2019 Aa Seminar - Seminar

1/25

BASIC PROPERTIES OF

LINE SEGMENT,POLYGON

Presented by REENA BHATI


2/25

Applicti!n "!#ins

Computer Graphics and Virtual Reality 2-D & 3-D: intersections, hidden surface elimination, ray t

racing

Virtual Reality: collision detection (intersection

Ro!otics "otion planning, assem!ly orderings, collision detection,

shortest path finding

Glo!al #nformation $ystems (G#$

%arge Data $etsdata structure design

'erlaysind points in multiple layers

#nterpolationind additional points !ased on 'alues of)no*n points

Voronoi Diagrams of points


3/25

Applicti!n "!#ins c!ntin$ed

Computer +ided Design and "anufacturing (C+D C+"

Design 3-D o!ects and manipulate them

.ossi!le manipulations: merge (union, separate, mo'e Design for +ssem!ly

C+DC+" pro'ides a test on o!ects for ease of assem!ly, mainte

nance, etc/

Computational 0iology

Determine ho* proteins com!ine together !ased on folds i

n structure $urface modeling, path finding, intersection


4/25

C!#p$tti!nl Ge!#etry

#ntersection of line segment

0asic geometric o!ects

.oint : ( 1, y

%ine : (x1,y1 , (x2,y2

%ine segment : si4e of line is gi'en

(x2,y2)

(x5,y5)

(x2,y2)

(x5,y5)


5/25

Line Se%#ent Pr!perties

6*o distinct points: p57(15,y5, p27(12,y2

Define a ne* point p37 ap58(5-ap2, *here 9 a 5

p3is any point on the line !et*een p5and p2

p3is a convex combinationof p5and p2

line segmentp5p2is the set of con'e1 com!inations of p5and p2/

p5and p2are the endpoints of segment p5p2

p1

p2

p3

x

y


6/25

Cr!ss Pr!d$cts Can sol'e many geometric pro!lems using the cross product/

6*o points: p57(15,y5, p27(12,y2 Cross product of the t*o points is defined !y

p51 p27 the determinant of a matri1 15 12

y5 y2

7 15y2 12y5

7 the signed area of the parallelogram of four points: (9,9, p5

, p2

, p5

8p2

#f p51 p2is positi'e then p5is cloc)*ise from p2

#f p51 p2is negati'e then p5is countercloc)*ise from p2

p1(x1,y1)

p2(x2,y2)

p1+p2

x

y


7/25

Cl!c&'ise !r C!$ntercl!c&'ise .ro!lem definition

Determine *hether a directed segment p9p5is cloc)*ise from a directed segment p9p2*/r/t/ p9

$olution

5/ "ap p9to (9,9, p5to p5;, p2to p2;

p5; 7 p5 p9, p2; 7 p2 p9

2/ #f p5; 1 p2; < 9 then the segment p9p5is cloc)*ise from p9p2

3/ else countercloc)*ise

p0

p1

p2

a) p0p2is counterclockwise

from p0p1: (p2-p0)x(p1-p0) < 0

p0

p2

p1

(b) p0p2is clockwise from p0p1:

(p2-p0)x(p1-p0) 0


8/25


9/25

Intersecti!n !( * Line Se%#ents


10/25

T'! Se%#ents Intersect +*

$i%e cases

p1

p&

p2

(a) p1p2, p3p&intersect

p3

p1

p2

p&

(b) p1p2, p3p&'o not intersect

p3

p1

p2

p&

p1p2, p3p&intersect

p3

p1

p2

p&

(') p1p2, p3p&intersect

p3p1

p3

p&

(e) p1p2, p3p&'o not intersect

p2


11/25

T'! Se%#ents Intersect +-

onsi'er two cross pro'ucts:

1 (p3* p1) x (p2* p1)

2 (p&* p1) x (p2* p1)

p1

p&

p2

(a) p1p2, p3p&intersect:

(p3* p1) x (p2* p1) < 0

(p&* p1) x (p2* p1) 0

p3

p1

p2

p&

(b) p1p2, p3p&'o not intersect:

(p3* p1) x (p2* p1) < 0

(p&* p1) x (p2* p1) < 0

p3


12/25

T'! Se%#ents Intersect +.

p1

p2

p&

(c) p1p2, p3p&intersect

(p3* p1) x (p2* p1) < 0

(p&* p1) x (p2* p1) 0

p3

p1

p2

p&

(') p1p2, p3p&intersect

(p3* p1) x (p2* p1) 0

(p&* p1) x (p2* p1) 0

p3


13/25

T'! Se%#ents Intersect +/

Case (e

=hat are the cross products > (p3 p5 1 (p2 p5 7 9

(p? p5 1 (p2 p5 7 9

6he cross products are 4ero;s, !ut they do not intersect

$ame result *ith Case (d

p1

p3

p4

(e) p1p2, p3p&'o not intersect

p2


14/25

T'! Se%#ents Intersect 0 Al%!rit)#

6*o@line@segments@intersect (p5p2, p3p?

5/ (15;,y5;7(min(15,12, min(y5,y2

2/ (12;,y2;7(ma1(15,12, ma1(y5,y2

3/ (13;,y3;7(min(13,1?, min(y3,y?

?/ (1?;,y?;7(ma1(13,1?, ma1(y3,y?

A/ if not ((13; 12; and (15; 1?; and (y3; y2; and (y5; y?; then

B/ return false case (e and more

/ else

/ if (p3p5 1 (p2p5 or (p?p5 1 (p2p5 are 4ero then

E/ return true case (c and (d59/ else if (p3p5 1 (p2p5 and (p?p5 1 (p2p5 ha'e different sign then

55/ return true case (a

52/ else

53/ return false case (!


15/25

Orderin% Se%#ents

6*o nonintersecting segments s5and s2are compara!le at

1 if the 'ertical s*eep line *ith 1 coordinate 1 intersects!oth of them/

s5is a!o'e s2at 1, *ritten s5


16/25

E1#ple


17/25

M!2in% t)e S'eep Line

$*eeping algorithms typically manage t*o sets of data:

5/ 6he s*eep line status gi'es the relationships among the o!ects intersected !y the s*eep line/

2/ 6he e'ent point schedule is a seFuence of 1 coordina

tes, ordered from left to right, that defines the haltingpositions (e'ent points of the s*eep line/ Changes to

the s*eep line status occur only at e'ent points/

6he e'ent point schedule can !e determined dynamicall

y as the algorithm progresses or determined statically, !a

sed solely on simple properties of the input data/


18/25


$ort the segment endpoints !y increasing 1 coordi

nate and proceed from left to right/

#nsert a segment into the s*eep line status *hen its left e

ndpoint is encountered, and delete it from the s*eep line

status *hen its right endpoint is encountered/ =hene'er t*o segments first !ecome consecuti'e in the t

otal order, *e chec) *hether they intersect/


19/25


6he s*eep line status is a total order 6/ for *hich *e reF

uire the follo*ing operations: #$HR6(6, s: insert segment s into 6/

DH%H6H(6, s: delete segment s from 6/

+0VH(6, s: return the segment immediately a!o'e segment s

in 6/ 0H%=(6, s: return the segment immediately !elo* segment

s in 6/

#f there are n segments in the input, *e can perform each

of the a!o'e operations in (log n time Replace the )ey comparisons !y cross product compariso

ns that determine the relati'e ordering of t*o segments/


20/25


21/25

3)et)er P!int is Inside

.ro!lem definition

Gi'en a simple polygon . and a point F, determine *hether the point is insi

de or outside the polygon/

6he polygon can !e a non-con'e1 polygon

Dra* a line from the point to a point outside of the polygon, count the num

!er of intersections *ith the edges of the polygon/

6he point is inside the polygon if the num!er of intersections is odd


22/25

3)et)er P!int is Inside +*

.oint@in@polygon(., F

.: polygon *ith 'ertices p5, p2, //, pn, edges e5, e2, //, en

F: point

5/ pic) an ar!itrary point s outside the polygon

2/ count 7 9

3/ for all edges eiof the polygon do

?/ if eiintersects the line segment Fs then

A/ count88

B/ if count is odd then return true

/ else return false

Comple1ity

. (n *hen n 7 si4e of the polygon


23/25

3)et)er P!int is Inside +-

+ssume special cases don;t occur

Case (a and (! !ser'ation

#s it indeed important to find an optimal line (i/e/, the line segment Fs tha

t minimi4es the num!er of intersections>

o/ inding the num!er of intersections is more important

#dea: Can use the 'ertical line that pass the point F

s

(a) s#oul' not count (b) s#oul' count

s


24/25

3)et)er P!int is Inside +.

.oint@in@polygon@2(., F

.: polygon *ith 'ertices p5, p2, //, pn, edges e5, e2, //, en

F: point (19, y

9

5/ count 7 9

2/ for all edges eiof the polygon do3/ if eiintersects the line 1719then

?/ yi7 y-coordinate of the intersection

A/ if yiI y9then count88 this can !e yi< y9

B/ if count is odd then return true

/ else return false

Comple1ity

.

(n *hen n 7 si4e of the polygon


25/25

THAN4 YO5

Aa Seminar - Seminar

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