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p1 = x2 + y2 − R2

p2 = z + H/2p3 = z − H/2

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Ax + By + Cz + D = 0Y7K8 [

Q]f?BRZTE?BU#S@RZ<.>]n>@I:A9<TG]v?]v_aUZ>@?BI:.DERNC%RMcw]fCE?@AKI>BR x = a1.t + a2

y = b1.t + b2

z = c1.t + c2

Y7K8 e

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t =Aa1 + Bb1 + Cc1

Aa2 + Bb2 + Cc2 + D

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t1 = c1

c2+H/2

t2 = c1

c2−H/2

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d−1∑

k=0

((−1)knk(P )) = 1 − (−1)d Y798

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n0(P ) = scJRa<EAK_ WE?BRC%RSmAK_a_fRZ>BS

n1(P ) = acJRa<EAK_ WE?BR

C3| ]v?V>@R&S n2(P ) = f

c:RM<EA9_WE?@RMC%RNo]KFVR&SRV>n3(P ) = v

c:RM<EA9_WE?@RMC%RM KA9c:DE_aR#S#8EQ]?@R#c:]v>@I:A9<C3| DEcJR#?jT(A9DE?

d = 2, d = 3RV>

d = 4C%A9<E<GRO?BR#S@T(R#FH>BIJ KRZ_aRZ<.>

d = 2 : s − a = 0d = 3 : s − a + f = 2

d = 4 : s − a + f − v = 0

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d = X +18 ]9?RZ^%RZ_aTEc:R#nT(A9DE? DE<GRxTG]v?@>@IJ>@I:A9<

C%DTGc:]9<(X = 2)

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d = 3s − a + f = 2

8 A9DE?DE<ER T,]v?@>@IJ>@I:A9<LC%Rc!| R#S@TG]9FZR(X = 3)cJRN<EAK_ WE?BRNC%RT(A9c9b&C%?BR#SR#Sm>jC%A9<E<GUTG]9?cJRN<EAK_ WE?BROCERO KA9c:DE_aR#SC%ROc:]f?BRZcw]n>BIJAK<`T(A9DE?

d = 4s − a + f − v = 0

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Y7K8 d A9D

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o]9FZR#S cccJRN<EAK_ WE?BRMC%RFVAK_aT,A.S@]9<K>BR#SFVAK<E<ERZ^R&STG]v?j]9?BFORV>

tcJRM<GA9_ WG?@RMC%RO>@?BA9D,SZ8

Q]S@DE?mo]KFVRMC3| DE< SmAKcJIwC%RM]0 9R&FODE<>@?BA9D<| R&SY>jT,]9SxS@IJ_aTEc:RZ_aRZ<.>¥FZA9<E<ERZ^%R GIJcRZ^%I:Sm>@R[aFERZ_aI:<GSoqR#?@_aU#SjIJ<GCEUZT(RZ<GCE]9<.>BSj<GA9<k?@U&C%DGFV>@I:WEcJR&SxRZ< DG< T,AKIJ<.> qRZ^%RZ_aTEc:R C%Dk>@AK?@R H8Q]f<EA9>@I:A9<kC%RN>@?BA9DRV>¥C%RNyKRZ<E?BRMC3| DG<ERSmDG?mo]KFVRMSmAK<.>>B?@b&STE?BAFGR#Sq KA9I:?§Gy9DE?BRf798\[#H8+-,,C, 1 365E;2S1 #< <E: 7 :($& 795 <>:(' 1Q]f?@R#TE?BU#S@RZ<.>B]v>@I:A9<poq?BA9<.>BIJb#?@RM]fc:]aSm>@?BDGFH>BDE?BRNC| DE<kyK?B]9TEER98


7K8J7 & uxR#c:]v>@I:A9<kC3| DEcJR#?¥S@DE?j.DERZcw.DER#ST(A9c.>@A9T(R#SQ5]x?@R#c:]v>@I:A9<C3| ~DEc:RZ? S@R KUZ?BI§GR]vIwSmU#_fR#<.>5SmDG?C%R&ST(A9c9b&C%?BR#SF#]v<EAK<EIwKDGR#S 4−6+4 = 2T(A9DE?cJR¥>@UZ>@?]vb#CE?@R

8− 12 + 6 = 2T,AKDE?c:RFZDEW(R#_]vIwS]vD,S@S@I,T(A9DG?C%R&S~TG]9?m>BI>BIJAK<GSC%R

c¢| R&SmT,]9FVRK8E¦>m>BRZ<.>@I:A9<LCE]9<GSjFZR FZ]KSxIJco]vD%>]9DGSBSmIFZA9_aT%>BRZ?jc:R 9AKcJDG_fRMI:<%§G<GI5FVAK<.>@RZ<,]v<.>c:RNT,R#<K>]n>BA9T(R

5 − 10 + 10− 5 = 0AKDc¢| T(RZ?FVDEW(R

16− 32 + 24− 8 = 0H8

7K8 [ & ORZ<G?@RMC%R&SxSmDE?@o]9FZR#SRZ>x>@?BA9DQ5]SmDE?@o]9FZR C3| DE< >BA9?BR<| R#Sm>¥TG]9SSmI:_aTEc:RZ_aRZ<.>OFZA9<E<GRV^%R ,I:cRV^%I:Sm>@Rf[F %FVc:R#S¥IJ<,C%UZT(RZ<%¡CE]9<.>BSj<GA9<L?BU#C%D,FH>@I:WEc:R#SRZ< DE<LT(A9I:<K>&83QR>BA9?BRR#Sm>NDE<ERS@DE?@o]9FZR C%A9<.>OcJRfy9R#<E?@R R#Sm>N[%8QR`>BA9?BR`TE?BU#S@RZ<.>@R 7S@RZDGc>B?@AKD8 QR#SaS@DE?@o]9FVR&SFZA9_a_aRc:R?@DEW,]v<*C%R AR#WEIJD,SAKD-cw]W(A9D%>BRZI:cJc:RNC%R Nc:RZI:<p<ERO?BRZTE?BU#S@RZ<.>@R#<.>TG]KScw]oq?BA9<.>BIJb#?@R¥C| DE< 9AKcJDG_fR_]vIwSc:RyKRZ<E?BRCERc:RZDE?jS@DE?@o]9FZRR&SY>¥]9DGS@S@IC%R [E8


' W DE<ERoq?BA9<.>BIJb#?@RMR&SY>NFVA9<,SY>BI>BDEU TG]9?[R#<GS@RZ_ WGcJR&S ,DG<LRZ<GS@RZ_WEc:R C%RS@A9_f¡_aRV>BSpRZ>jDE<kRZ<GS@RZ_WEcJRMC3| ]v?V>BR#Sx¦ G]vI:<GS@IKDGRMcJRNy9?]vTGEROCER#Sj]9?$Z>@R#S ¦

ai = (sj , sk)Ec¢| ]9?$Z>@ROI%XYA9I:<.>jcJRMS@A9_a_aRV>

sj]9D SmAK_f_aRZ>

sk

A9D SmAK<yK?B]9TEERNC%DG]9c qc:RMy9?]vTEERMC3| ]9CnXm]9FVR#<GFVRMC%R&Sx]v?V>BR#S ¦

si = (aj , ak)Ec¢| ]9?$Z>@R

aj]9?@?BIJ KRO]9DS@A9_a_aRV>xI Ec¢| ]9?$Z>@R

akR#<?BRZT,]v?@>

]9?fFZA9< 9R#<.>@I:A9< c:R#Sa]v?V>BR#S S@A9<.>fA9?BIJR#<K>BUZR&ST(A9DE?a.DERcw]L_]v>@I:bZ?BRS@A9IJ> >@AKDnXYA9DG?BS¨kyK]9DGFERA9D ¨ C%?BA9IJ>@RK8Q]C%R#SBFV?BIJTE>@I:A9< C3| DE<-C%A9_]9IJ<ERpFZA9_aT(A9?@>B]v<.>C%R&S FZ]0 IJ>@U&SSmRo]vIJ>¨c!| ]vIwC%R`CERTEcJD,SmI:RZDE?SF FZcJR&SC3| ]v?V>BR#S DG<©F %FVc:RpFVA9?B?BR#S@T,AK<GCE]v<.>M]vD M9('$#(25/4Uf3$#./n42/ GCER#SxF %FZcJR&SxFVA9?B?BR#S@T,AK<GCE]v<.>]vD%^ MK,'$&,25/n p'$&./0!42/n 8QRM_aA%C%bZc:RMC%A9IJ>j?@R&SmT(R#FV>@R#?jC%R#S MK,'$#/0"G'$#4 p U !'$#),/0$# & c:R#Sj]9?$Z>@R&S<GRMT,R#DE 9R#<K>jS#| I:<.>@R#?BS@R#FV>@RZ?j.D| RZ<c:RZDE?SxSmAK_a_fRZ>BSxFZA9_a_ DE<,S & DE< S@A9_a_aRV>¥]9TETG]v?@>@I:RZ<.>x¨p[f]v?V>@R&S & I:c,RZ^%I:Sm>@RjDG<ER¥AK?@I:RZ<.>B]v>@I:A9<C%R&S]9?$Z>@R#S >@RZc:c:R.DERx>BA9D%>SmAK_f_aRZ>R#Sm>AK?@I:y9I:<ERjC3| DE<ER]9?$Z>@RNRV>jRZ^.>B?@R#_aI>BUNC| DE<GR]vD%>B?@RK8 ' W DE<ER oq?@AK<.>@I:bZ?BRT,R#D%>V>@?BR ?BRZTG?@U&SmR#<K>BUZRMTG]9?OepRZ<GS@RZ_WEcJR&S (DE< RZ<GS@RZ_WEcJR

C%RS@A9_a_aRV>BSj EDE<RZ<GS@RZ_WEcJRC| ]9?$Z>@R#S¦ RZ>jDE<kRZ<GS@RZ_WEcJRMC%RNo]KFVR&S 8AKDE?jFG]KKDGRo]KFVRNA9< C%AK<E<ERNc:RMy9?]vTEERNC%R&Sx]v?V>@R&S

¦ ai = (sj , sk)

Ec¢| ]9?$Z>@ROI%XYA9I:<.>jcJRMS@A9_a_aRV>sj]9D SmAK_f_aRZ>

sk

A9D SmAK<yK?B]9TEERNC%DG]9c qc:RMy9?]vTEERMC3| ]9CnXm]9FVR#<GFVRMC%R&So]9FZR#S ¦

ai = (fj , fk)5c:]o]9FZR

fjR#Sm> ¨ mC%?BA9IJ>@R aC%Rc¢| ]v?V>@RaI cw]o]9FVR

fkR&SY> ¨

Yy.]vDGFGROC%RNc¢| ]9?$Z>@RNI

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SUIV[1]

DEBUT[2]

(6,-3)

(0,0)(10,0)

(7,0)

(6,4)(4.5,4)

(4.5,0)

(COORD[1][1],COORD[1][2])

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a.cdir

a.cind

a

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a.apcd

a.asci

a.apci

a.orig

a.extr c

c.trou

c.trou->trou

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f4./*MKJMK4)" M+ 7N* +&. C +.-4-L!<"K(:+.-D2Z C (:"%'A4+.-D("e "!#%$(* !# $6+ C -+42.$/!<"^(* M780"0"R"+2 '+)( C "!#$6+@[ f4./*MKJMK4 )A2 C !#R"+W78+.-GA$O041+.-W(:+G7n* 0"7 9U C !#$ ?EG+j;.!5(:A$6+ C E)!#'+G-4!d "!#%$+.-4$!#%$OA C !<+. C "K+n&3$6A C !<+ C cdm "7 ( C +a+.$/!#7 !V-0" $T7N* !# $6+ C -+42$ !xCG]v<GST(A9cy9A9<GRQ~| ]vc:y9AK?@IJ>@E_aRFVcw]9SBSmRFG]KKDGRS@RZyK_aRZ<.>~C%DaT,AKc y9AK<ERTG]v? ?]vTET(A9?@> ¨cw]NC%RZ_aIJ¡ CE?@AKI>BR98n;=cp]aSmIJ^`FVcw]9SBSmR&S %T,]9SC3| IJ<.>@R#?BS@R#FV>@I:A9< Fv7 %I:<.>@RZ?S@R#FH>BIJAK<S@IJ_aTEc:R F&[#%S@IJ<Gy9DEcw]v?BI>BU98:8J8 QRTEcw]v<R#Sm>CEU#FVAKDET(U¥RZ<pS@RZTE>?BUZy9I:A9<,S q?tK?#7K8J8:89? V89;=cSmD%a>C%R¥c:A%FZ]vc:IwSmR#?~cJR&SRZ^>@?BRZ_aI>BU#SC%R#SNS@RZyK_fR#<.>BSOT,AKDE?NI:CERZ<.>@IJ§GRZ?Mc:R#SNF#]9S>@?BIJ Iw]vD%^ 9AKIJ?>B]9WEcJR&]vD V8 ]v?ORV^%RZ_aTEc:RSmI c:R#SC%RZDE^-RV^>@?BUZ_aIJ>@U#SaS@A9<.>fCG]v<GSfcw]?BUZyKIJAK< ?Bt IJc<ERT(RZD%> -]0 9AKIJ?fC3| IJ<.>@R#?BS@R#FV>@I:A9< !Fv7 H8¤¥R _ Z_aR S@IOcJR&S[ T(A9I:<.>BSSmAK<.>CE]v<,S`cw]-?BUZyKIJAK<£?&7#?H[©A9D ?eE8:8:8QR&SS@I:<Ey9DEcw]v?BIJ>@U#SF#e FVh j]vTETG]9?B]9I:SBS@RZ<.>jKD,]v<GC c:R FZA9DETEc:R ?@I ?X$<| R&SY>OTEcJD,SS@D%pS@]9<.>¥T,AKDE?CEU#FVIwC%R#?C%Rc!| I:<K>BRZ?SmR&FH>BIJAK<AKDaCERxc:]O<EAK<%¡¢I:<.>@R#?BS@R#FV>@I:A9<8 ]v? RZ^%RZ_aTEc:RxcJR&S~SmI:<EyKDEc:]9?@IJ>@U&S SmI:_aTEcJR&S F#e ]vTET,]v?]vIwS@S@RZ<.>j.DG]v<GCLA9<L]c:R#SFVA9DGTEcJR&S ?Bt ?e ?[E?#7 A9D c:RZDG?BSS _aUV>[email protected]#S#8G~c:c:R#ST,R#DE 9R#<.> Z>@?BR`>B?B]9I>BUZR&S¨c¢| ]9I:CERkCER>BR#Sm>BSayKUZA9_aUZ>@?BI:.DER&S c:A%FZ]vDE^*]vc:A9?Sf.DERc¢| AK< R&SY>A9WEc:I:y9UMC%RNTG]v?FVAKDE?BIJ?c:ROT(A9cy9AK<ERNT,AKDE?>@?]vIJ>@R#?xcJR&SxSmI:<Ey9DGc:]9?@IJ>@U&SFZA9_aTEc:I:.DEU#R#S!FVh V8

?Bt ?&7 ?H[ ?Be ?Bh ? ??Bt F97 F#[ Fv7 F#e Fv7 FZh F#?#7 F&[ Fv7 FZe F97 Fv7 FZh F#?[ F97 FZe Fv7 F97 Fv7 FZh F#?Be F#e Fv7 Fv7 F97 Fv7 FZh F#?@h F97 Fv7 Fv7 F97 Fv7 F# F#? FZh FVh FVh FZh FZ FZh F#?B F& F# F# F& F# F& F#

Fv7 ]KSxC3| I:<.>@R#?BS@R#FV>@I:A9<F#[ ;=<.>BRZ?SmR&FH>@I:A9<SmI:_aTEc:RFZe IJ<Gy9DEcw]v?BI>BUMSmI:_aTEcJRFVh %IJ<EyKDEcw]v?BI>BUNFZA9_aTEc:I:.DEU#RF# %DE?xDE<ERMRZ^.>B?@R#_aI>BUNCEDkS@RZyK_aRZ<.>FZ DE?xc:RS@RZy9_aR#<K>(!¨c¢| IJ<.>@R#?@I:RZDG?


QR&S¥]9cJyKA9?BI>BE_aR#SxFVIJ¡=C%R#SBSmD,SjSmAK<.>¥C%A9<E<GU#SxT(A9DE?cJRFZ]9S¥e9¤ G_]vIwSxI:c:SjoqA9<GFV>@I:A9<E<ER#<.>]9DGS@S@IT,AKDE?jc:RMFZ]9Sj[9¤ CG]v<GSc:RMTEcw]v< V8F 4VM9"GJM9254 U "Gq/n4 ,2 2 ,(2+-4TK¤O]v<GScJR&SFZ]KSS@IJ_aTEc:R#S3c¢| ]vI:?@RC| DE<MT(A9cy9AK<ER~C%R

n]v?V>BR#SR&SY>5CEA9<E<EU#R~TG]9?c:]xoqA9?B_ DEc:R

c:RT,AKc yKA9<ERa<| R#Sm>TG]9SOoqAK?BFZUZ_aRZ<.> S@IJ_aTEc:RZ_aR#<K> FZA9<E<GRV^%Ra_a]9I:ScJR&S]v?V>BR#SMC%AKIJ KRZ<.>Z>@?BROAK?@I:RZ<.>BUZR#S

A =n

∑

i=0

∫ xi+1

xi

Yi(x)dxY798

Ic:R#Sx]9?$Z>@R&SxSmAK<K>jC%R&SxSmR#y9_aRZ<.>SxC%RC%?BA9IJ>@RNA9<A9W%>BIJR#<.>

A =

n∑

i=0

1

2(yi+1 + yi)(xi+1 − xi)

Y798:7#t ¤O]v<GSc:RFZ]KSyKUZ<EU#?B]9cIJco]9D%>jD%>@I:c:I:S@RZ?xcw]f?@R#TE?@U&SmR#<.>B]n>BIJAK<pTG]9?B]9_fUZ>@?BI:.DERNC%RNcw]yKUZA9¡

_aUV>B?@I:ROoq?@AK<.>@I:bZ?BRORZ>o]vI:?@RNDE< FG]9<Ey9R#_fR#<.>xCERO n]9?@Iw]vWEc:RNT,AKDE?xc¢| I:<.>@UZyK?B]v>@I:A9< ∫ xi+1

xi

Yi(x)dx =

∫ 1

0

Yi(t(u))t′(u)dum798:797

~< e9¤ c:RNTE?BIJ<GFZIJT(RNR#Sm>xcJRN_ Z_aR#%_]9I:SI:c3R#Sm>xD%>@I:cJR E]9D`TE?BU#]9c:]9WEc:R#%C%RO>@?BI:]9<Ey9DGcJR#?c:R#S o]KFVR&S C%Rjcw]Ooq?@AK<K>BIJb#?@RT(A9DE?~o]9FVI:c:I>BRZ? c¢| RZ^TG?@R&S@S@IJAK<fCERjc!| I:<K>BRZ?B n]vc:RC| I:<.>@UZyK?B]v>@I:A9<8n<?BRZ n]v<,FERcw]>@?BI:]9<Ey9DGc:]v>@I:A9<`T(RZDE> V>@?BRM.DERZcwFVAK<G.DERNRV>x_ Z_aRMTE?BU#S@RZ<.>@R#?CER#S>@?BIw]v<EyKcJR&SI:<.>@RZ?S@R#FH>]v<.>BS#8

∫ ∫

Zi(x, y)dxdy =

∫ 1

u=0

∫ 1−u

v=0

Zi(x(u, v), y(u, v))Det(J)dudvY798:7&[

ADet(J)

R#Sm>jc:RC%UV>BRZ?B_fI:<G]9<K>jC%RNcw] K]KFVAKWEIJR#<E<ER

J =

(

dxdu

dydu

dxdv

dydv

) Y798:7#e Ic:R#S>B?@Iw]v<Gy9c:R#SxS@A9<.>xcJI:<EU#]9IJ?BR#SAK<T(RZD%>_fAK<.>@?BRZ?x.DER

V =

n∑

i=0

zi0 + zi1 + zi2

2(xi1 − xi0)(yi2 − yi0) − (xi2 − xi0)(yi1 − yi0)

Y798:7Zh

+-,C, 0 6:/3 (<>:/7 @ 1 .Q5E1 3<kFVA9<,SmIwC%bZ?BR[<EIJ KR#]9D%^C%RN_]v<EI:TEDEcw]n>BIJAK< & QRM<EI: 9R&]vDUZc:UZ_aRZ<.>B]9IJ?BRM.DEIR#Sm>jy9U#<EUZ?]vc:RZ_aR#<K>>B?B]9<GSmT,]v?BRZ<.>T(A9DG?jc!| D%>BIJc:I:SB]n>BRZDE?_a]9I:ST,]9ST(A9DE?xc:RMTE?BA9y9?]v_a_aRZDG? !FVRS@A9<.>xcJR&S_fUZ>@EA%C%R&SV8& QRM<EI: 9R&]vDCERMc!| IJ<.>BRZ?@o]9FVRND%>BIJc:IwS@]v>@RZDG?#8


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?@R#c:]v>@I:A9<©C%D©_ Z_aR<EA9_k8;=cwSMT(RZ?B_aRV>m>BRZ<.>C%RpFZA9<GSm>@?BDEI:?@Rf<| IJ_aT(A9?@>@Rp.DER#c~T,AKc .>BA9T(R98;=c:SS@A9<.>TG?@I:<GFVI:TG]9cJR#_fR#<.>D%>BIJc:I:S@U#SR#<e9¤ T(A9DE?c:R#ST,AKc Kb#C%?BR#S qR#<k[9¤ cw] Sm>@?BDGFV>@DE?BR¥R&SY>SmIxS@IJ_aTEc:R .D| R#cJc:Rk<ER<EU#FZR#SBSmIJ>@RTG]9SFZR#SfA9T(UZ?]n>BRZDE?S H8 ! A9D,S<GA9DGSfc:IJ_aIJ>@RZ?BA9<,SC%A9<GFCE]v<,SxFVRV>@>@RS@R#FV>@I:A9<k]vDkFZ]9SxCER#SxT(A9c9b&C%?@R&SZ8;=cjRV^%IwSY>BR DE<ERkIJ<E§G<EIJ>@UC3| A9T(UZ?]n>BRZDE?SfRZ< e9¤_]9I:SI:cwSaT,R#DE 9R#<K> Z>@?BRk?@U&C%DEIJ>BSp]9D

<EA9_WE?BRCER%89¤| ]vD%>B?@R&S~SmAK<.>~<EU#]9<E_aA9I:<GS D%>BIJc:I:S@U#S~T,AKDE?C%R&S ?]vIwSmAK<GS C3| A9?C%?@RTE?]n>BI:.DERA9Dk]vc:y9AK?@IJ>@E_aIw.DER98

!O L]"#9R ! RZ?@>@RV^ ER#cJc]9<GC G]9FZR(s + 1) − (a + an) + (f + 1) = 2((cc + 1) − t) ! ] #9RNCEy9RM]v<GC ! R#?m>BRV^

(s + 1) − (a + 1 + an) + f = 2(cc − t) L]"#9ROC%y9R]9<GC G]9FZRs − (a + 1 + an) + (f + 1) = 2(cc − t) Mu L]"#9RNC%yKR NI:c:cuxI:<Eys − (a + 1 + an − 1) + f = 2(cc − t) u" NIJc:c G]KFVRML] #KRMuxIJ<Gya]9<GC "jAKcJR

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%RX+ C -+.EG+.%$ P+G 0 C $60"%$(* G "74( C +G;. +7824"M;. +P"W1+$0:F4"$/! C cG7n* :F +.$%710 C 03'7 !<240"$/!<" C A6 A.$6A4+D(3+L-u!# R"+ C -.+.- (:+.-D A C 0"$6+. C - (:+)F40X-+Z[QRp>BEUZAK?@b#_aRC3| IJ< KRZ?SmI:A9<©TE?BU#S@RZ<.>@R`DE<*I:<.>@UZ?V>>@EU#A9?BI:.DER`_a]9I:Sf]vD,S@S@ITE?]n>BI:.DER

[email protected]| I:c T,R#?@_aRV>MR#<K>B?@R]9D%>@?BRfCERaSm>@A%F #KRZ?NcJR&SNT,AKc Kb#C%?BR#SOS@A9DGSNDE<GRoqAK?@_aRaSY>]v<GCG]v?C_ Z_aRSmI3R#cJc:RM<| R#Sm>xTG]9SDE<GI:.DERK8~cJD,SmI:RZDE?S]vc:y9A9?BIJ>@E_aR#SC3| IJ< 9R#?BS@I:A9<aA9<.>UZ>@UTE?@AKT,A.SmU&SZ8.QRTEcJD,SFZA9<E<DpR&SY>TE?@AKWG]n¡

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$#" A%C%U#cJIwS@]v>@I:A9<y9U#A9_aUV>[email protected] 3

MVSF MEV

MEV, MEV MEF

MEV,MEV, MEV, MEV MEF, MEF, MEF, MEF

s1

f1s2

a1

f2

s3s4

a2

a3

a4

a5 a6

a7a8

s5 s6

s7s8

a9

a11a12 f5

f6 f4f3

a10

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! 7 t t 7 7 tL ! 7 7 t [ 7 tL ! 7 [ t e 7 tL ! 7 e t h 7 t [ h t h 7 tL ! [ t 7 tL ! [ t 7 tL ! [ P t P 7 tL ! [ d t d 7 t e t d 7 t h 7&t t d 7 t 7K7 t d 7 t 70[ t d 7 t


C%UV>B?@DGI>cJRTE?@R#_aIJR#?SmAK_a_fRZ>5RZ>5c¢| ]9?$Z>@R]9IJ<GS@IKFZ?@U#UZR~T,]v?DE< N ! 8vG]9.DERFVA9_aT(AKSB]v<.>@RFVAK<E<ERV^%RN<| R#Sm>jTEc:DGSoqA9?B_aUZRM]vc:A9?SKDGRC3| DG<ERNo]9FZR#EC3| DE< FVAK<K>BA9DE?RZ>¥C3| DG<kS@A9_a_aRV>&8F 4. aXN./n"%$#!,' (4'5'4 QRLTE?BA9WGcJb#_fR R&SY>pc:RS@DEI: 0]9<.> UV>]v<.>`C%AK<E<EU[©C%AK_a]9IJ<GR#S

ARZ>

BFZA9<GSm>@IJ>@DEU&S

C3| DE<ERAKDTEc:DGS@IJR#DE?BSFVAK_aT,A.S@]9<K>BR#S5FVA9<G<ERV^%R#S RZ>5?BRZTE?BU#S@RZ<.>@U&STG]9?RA

RV>RB

FVA9_a_aRZ<.>A9W%>BRZ<EI:?jDE<GR?@R#TE?BU#S@RZ<.>B]v>@I:A9<

RCC%D C%AK_a]9IJ<GRM?@U&SmDEcJ>B]9<.>¥C%RMc!| DE<EI:A9<

C = A ∪ B(C%R

c!| I:<K>BRZ?SmR&FH>BIJAK<C = A ∩ B

A9Dc:]aC%I UZ?BRZ<GFZRC = A − B

8W DO '5$#!,' gj04: +.78+j8N./n"%$&('5 (4''54 /0)(25!"%/n &4. D78+.- A C 0"$/!<" -F4@b78A4+.%'+.-a2.780"-4-4!<; +.- ∪,∩,− EeZ("! tIA+.- " C M+i C Z("! C +k; +k(:+.-:F +.$ - C AO9:7 !<+ C -[=I7#78+.-u-" $93A'A C 0b78+.EG+ $I'"$6A4+L- ∪∗,∩∗,−∗

[

A

B

AnB

A

B

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?BIJR#DE?A ¨fc!| RV^>BUZ?BIJR#DE?xC3| DG< T,AKc y9AK<ER98


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nF9>@U#SnA9<fT(RZD%>~>B?@AKDE 9R#? C%R#S ]9cJyKA9?BI>BE_aR#S C%Rxc:AF#]vc:I:SB]n>BIJAK<

C%A9<.>NcJR&SMFVAK_fTGcJRZÎJ>@U&S qRZ< >BRZ_aTGS3RZ< _fU#_aA9I:?@RfRV>MRZ< >@R#_aTGSMC%RaTE?BUZTG]9?B]v>@I:A9< ¥SmAK<K>cJR&SxSmDGIJ n]v<.>BR#S AKc yKA9<ER R#_aTGS U#_fAKIJ?BR ~?BUV¡¢>@?]vIJ>@R#_fR#<.>I:_aTEcJR

O(n) O(n)¡

>BA9I:cJUO(log2n) O(n) O(n)A9< KRV^%RO(log2n) O(n) O(n)

convexe

etoile

simple

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cJR&SLe FZ]KS qT(A9cy9A9<GR-SmI:_aTEcJR jUV>BA9I:cJU FZA9< 9RZ^R O(n)

8O~< ?BRZ n]v<,FER KD,]v<GC DE<y9?]v<GC-<EA9_WE?@RC%RkT,AKIJ<.>S

m >> nC%A9IJ> V>@?BRkFVcw]9SBSmUTG]v?f?B]9TET(A9?@>f]vD T,AKc yKA9<ER

>B?B]9I>BRZ_aRZ<.>BSN?BUZT(UV>BI>BIoS Ncw]kFVA9_aTEc:RV^%IJ>@UC%R&S]vc:y9AK?@IJ>@E_aR&S<| R&SY>MTEc:DGSMU&KDGIJ n]vc:RZ<.>BR#S O(m ∗ log2(n)) + O(n) << O(m ∗ n)

8Q~| ]vc:y9A9?BIJ>@E_aRC%Rc:A%FZ]9cJIwS@]v>@I:A9< C3| DE<LT(A9I:<.>CE]v<,SjDE<LFZA9< 9RZ^RMR&SY>TE?@U&SmR#<.>@UN§GyKDE?@R

[%8 eE8

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A9DLS@I>BR#S N = p ∈ Ed, Card(N) = n

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NC%ROFZAA9?C%A9<E<GUZR#S §E^%UZR#S

x ∈ Ed q?BR#.DZ>@R¥T(A9I:<.> A9DpR#<GFVAK?@R¥cJR&ST(A9I:<.>BSC%R

NFVA9<.>BRZ<DGSxCE]9<GSDE<kIJ<.>BRZ?B 0]9c

[x, y] ∈ Ed ^ Ed q?BR#.DZ>@RMW(A9IJ>@R H8


P1

P2P3

Pi

Pi+1

Pn

a1

a2a3

ai

P1

P2Pn/2

Pi

Pi+1

Pn

an/2ai

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c:RS@R#FV>@R#DE? FVA9<.>BRZ<G]9<K>cJRT(A9I:<K> RZ<O(log2n)

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A9DE?MFZR#SO?@R&FERZ?FER&Sj?BUZT(UV>@U#R#Sc:]_aI:S@RR#<TEcw]9FZRfCERfSm>@?BDGFH>BDE?BR#SN]9CE]9T%>@U#R#SOSZ| I:_¡T(AKS@R98RLS@A9<.>fTG?@I:<GFVI:TG]9cJR#_fR#<.>CER#S]v?BWE?BR#S#8 ij<£TE?@UZ>@?]vIJ>@R#_fR#<.>FZA9<GSm>@?BDEIJ>aCEA9<GFcw]Sm>@?BDGFH>BDE?BRxRV> >@?BI:RjcJR&S T(A9I:<K>S~C%R

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x ∈ EdR&SY>RZ<GS@DEIJ>@R?B]9TEI:CERZ_aRZ<.>FZc:]KS@S@UCG]v<GS5DE<ERFVR#cJc:DEc:RcCERFZRV>m>BRSm>@?BDGFV>@DE?BR RZ<

O(log2(n))TG]9?xRV^%RZ_aTEc:RMT(A9DE?jDG< M¤ 5?@R#R V8GQR&SjFVAA9?C%AK<E<EUZR&SC%R#SjS@IJ>@R#S¥CE]v<GSjc:]FZRZc:cJDEc:RcS@A9<.>

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?BA9<.>T(A9DE?C%R#S?]vIwSmAK<GSTE?]n>@Iw.DER#S%TE?BU#S@RZ<.>@U&SI:FZIRZ< [v¤8E;=cS#| ]9y9IJ> & C%R#SxyK?@I:c:cJR&S 9AKIJ?§GyKDE?BRN[E8 h & C%R#Sj.DG]KC>@?BRZR&Sq 9AKIJ?§,y9DE?BRN[E8 & C%R#S M¤ 5?@R#R#Sq KA9I:?§Gy9DE?BR[%8\# A9DE?cJR&SpyK?@I:cJc:R#S`RV>cJR&S`.DG]9C%>@?BRZR#c!| R#S@TG]KFVR R#Sm>C| ]9W,AK?BC c:IJ_aIJ>@U ¨ DE<ERoqRZ<Z>@?BR

(xmin, ymin, xmax, ymax)

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nRV>OCERfFZA9c:A9<E<ER&S

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oqR#?m>OT,R#?@_aRV>@>B]9<K>NC%R#SOFVcw]9SBS@RZ?jc:R#SOSmIJ>@R&SCG]v<GSc:]`SY>B?@DGFV>@DE?BR _]9I:S]vDGSBSmI C%R?@U&]vc:I:S@RZ?jc:R#S?BR#.DZ>@R#ST(A9I:<K>RZ<òqA9<GFV>@I:A9<C%Rc:RZDG?BSFVAAK?BC%AK<E<EU#R#S

(x, y)8 AKDE?c:]y9?BIJc:c:R¥cw]oqAK<GFH>BIJAK<

R&SY>x 9R#<.>@I:c:]v>@I:A9<kC%A9<E<GUZRNTG]v?

i = n (x−xmin)(xmax−xmin

j = m (y−ymin)(ymax−ymin

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pWEI>SZ8

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0

1

2

3

4

5

6

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y

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[E8 h & uxR&KD V>@RNW(A9IJ>@RS@DE?xDE<ERMyK?@I:c:cJRQR&SMIJ<.>BRZ?B 0]9c:S FVA9?B?BR#S@T,AK<GCE]v<.>SO¨kcw]kW(A9IJ>@R[xmin, xmax], [ymin, ymax]

S@A9<.> F#]vcwFVDEc:U#SCE]v<,S c!| R#S@TG]9FZRxC%Rxc:]OyK?@I:cJc:R

[imin, imax], [jmin, jmax]89QR&S FVR#cJc:DEc:R#S~FZA9?B?@R&SmT(A9<GCG]v<.>@R&S

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nH8

QR M¤ ?BRZR ?BUZT(A9<,C£¨ FZRLTE?BA9WEc:bZ_aRLF#]v?c:R#S`CE?@AKI>BR#SpS@UZTG]9?B]9<.>S@R#S`FVRZc:c:DEcJR&S<ERSmDGIJ KRZ<.> TG]9S C%R&S~FVAAK?BC%AK<E<EU#R#SRZ<.>@I:bZ?BR#S AKDfCER#S TEDGI:SBS@]9<GFVR&S C%Rj[%8.R&S C%?BA9IJ>@R

DSmR#?@AK<K>

XYDGSm>@RZ_aR#<K>aFEA9IwS@IJR&ST(A9DE?fSmU#TG]v?BRZ?U&.DEI>]vWEc:RZ_aR#<K>fc!| RZ<,SmR#_ WEc:R`C%R&SS@I>BR#S RZ< [ S@A9D,SY¡RZ<GS@RZ_WEc:R#S

SD+

RV>SD )

]n§G<kC3| ]9SBSmDE?BRZ?x.DERCard(SD+ ) = Card(SD ) ± 1

8QR M¤ 5?BRZRaR#Sm>MDE< ]v?BWE?@RfWEI:<G]vI:?BR ¨kFG]KKDGR<EI: 9R&]vD c!| R#S@TG]KFVRfR#Sm>MS@UZT,]v?BUR#< [

qR#< h T,AKDE?DE<-.DG]KC>@?BRZR NSmDEI: n]v<.>DE<ERSmR#DEcJRFVAA9?C%A9<G<EUZR&SZ83¦jD <GIJ KR#]vD S@DEI: 0]9<.> c:]SmU#TG]v?]n>BIJAK<pSmR¥oqRZ?] S@DEI: 0]9<.>c¢| ]9D%>@?BRFZA.AK?BCEA9<E<EU#R98.Q~| ]v?BWE?@R¥A9W%>BRZ<D`R&SY>U#.DEI:cJI:WE?BU¥RV>xC%RTE?BAvoqA9<,C%RZDE?

log2nEDE<GRM?@R&KD V>@RNT(A9I:<.>jR#Sm>¥FVRV>@>@RNoqAKI:Sm¡ FZI3?@U&]vc:I:S@UNRZ<>@R#_aTGSxA9TE>@I:_ DE_

Fv| R#Sm>j¨C%IJ?BRNRZ<O(log2n)

AKT,U#?B]v>@I:A9<GS#8Q]FVA9<,SY>B?@DGFV>@I:A9<C| DE< M¤ ?BRZRT,R#D%>V>@?BRf?@U&]vc:I:S@U R#<

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3.58 ' 4RZ>¥KDGROc:RM?BR#.DV>BR#SxS@RZ?BA9<.>xC%A9<GFO?BU#]9cJIwSmU#R#SRZ<>BRZ_aTGSxAKT%>@I:_ DE_k8


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Rd 0o'+d2bE5M78+&:!#$6Ah+ $6+.ET%-,(3+O(nlog2(n) + n(d/2))

+$Y(:+O(n(d/2))

+.'780324+@[

! A9DGS>B?@AKDE 9R#?@RQ$¥DE<ERNC%U#_fAK<GSm>@?]n>@I:A9<`y9UZ<GUZ?]vc:R¥RV>x?BIJyKA9DE?BRZD,SmR¥CED`>BEUZAK?@b#_fRNCE]9<GS6( ) 8

0

20

40

60

80

100

120

140

160

-10 -5 0 5 10

y

x

Tri de x par calcul de l’enveloppe convexe

donnees triees

pointsy=x^2

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Pb(tri) ⊂ Pb(EC)C3| A

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C3| A#O(EC) ≤ O(Tri)

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80Q]¥C%RZD%^%I:bZ_aR]9TETE?BAFGR~FVA9<,SmIwSY>BR¨xI:< 9RZ?S@RZ?c:R#S5]v?BbV>@R&SC%R¦|(]n§G< C%RC%I:_aIJ<DER#?jTE?BA9yK?@R&S@S@IJ KRZ_aRZ<.>cJR#DE?j<EAK_ WE?BR Fv8 ]E8 C38(KD| ]9TE?@b&SIJ< KRZ?SmI:A9<RZc:cJR<| IJ<.>@R#?BS@R#FV>@RTEcJD,S cJRSmR#y9_aRZ<.>H8 <T(RZD%> _aA9<.>@?BRZ? .DERcJR>@?]vIJ>@R#_fR#<.>R&SY> FVAK<. KRZ?By9R&]v<.>_ Z_aRpSmIFVRp<| R#Sm>TG]KS]vD S@RZ<GSMSm>@?BI:FV>#8ij<ERa>BR#FE<EIw.DERpC3| IJ< KRZ?SmI:A9<GS m?]v<GC%AK_aI:S@UZR&S T,R#D%> V>B?@RND%>BIJc:IwSmU#R 9A9I:?xFG]vT8%hpC%R ( ) V8

(a) (b)

(c) (d)

[%8\[nh & ij<ERO>@?BIw]v<EyKDEc:]v>@I:A9<kC3| DE<kT,AKc y9AK<ER] ¤ | RV>BR#FH>BIJAK<LC%R#Sx>@?BI:]9<Ey9c:R#SI:<.>@RZ?S@R#FH>]v<.>BSc¢| ]v?BbV>@R T,AKIJ<.>@I:c:cJU&SV8 qW EDGS@I:A9< C%R#Sx>@?BI¡]v<EyKcJR&SZ8 !FjU#TG]v?]n>BIJAK<C%R&Sj[T,AKc y9AK<ER#S#8 C ?BI:]9<Ey9DGc:]v>@I:A9<CER#ST(A9c.yKA9<ER&SZ8


(a) (b)

(c) (d)

1 2

1

2

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$ . d ? $. . $ Y Q~| A9WEXYR#FH>BIo5R#Sm>IwFVIC3| A9WE>@RZ<GIJ?DE<ERO>@?BIw]v<EyKDEc:]v>@I:A9<pC%AK<.>c:R#S>@?BIw]v<EyKcJR&SAK<K>DE<ER>B]9IJc:c:R

RZ>DG<ERNoqA9?B_aROI:_aT,A.SmU#R#S#8¤RM<EA9_WE?@R#DGS@R#S_aUV>@GACER#SxT(RZ?B_aRV>m>BRZ<.>¥C%RMSB]n>@IwSmo]vI:?@RMC%R&SxFV?BI>BbZ?BR#SxC%RO>B]9IJc:c:RN>@A9DE>

R#<<ERCEUZy9?]9CG]v<.>T,]9Scw] oqAK?@_aRMC%R&SxUZc:UZ_aRZ<.>S ¡C%R&S>@R&FE<EIwKDGR#SjC%RNWEIwSBSmR&FH>@I:A9<T(\[v*)RZ>¥C%RC%U&FVA9DGTG]vyKRR#<_aAv>BIoS( h )¡C%R&S>@R&FE<EIwKDGR#SjC%RNyKUZ<EU#?B]v>@I:A9<CERM<EARZDGCESxS@DE?xc:R#Sj]9?$Z>@R&S ¡8J8:8L]9I:S¥c!| ]vTGTE?@A%FERMRZ^%T,A.SmU#R FVIJ¡ CER#SBSmAKDGSxT(RZD%> V>@?BR ]9TETEc:I:.DEU#R]vDE^ [f>T(R#SC%RFZ?@IJ>@b#?@R&SC%U#_aA9<GSm>@?]n>BIJAK< ¨jc¢| ]9TETEDEI=(:7Zh )(\[v )(\[9[ )¢80~c:c:R]C%RTEc:DGSc!| ]0 n]v<.>B]9y9RC3| D%>BIJc:I:S@RZ? c:R_ #_aR]9cJyKA9?BI>BE_aRfC3| I:<GSmR#?m>BIJAK< C%RpT(A9I:<K>S.DERpFVR#cJDGI~TE?BU#S@RZ<.>@UTE?BU#FZU#C%R#_a_fR#<.> q 9AKIJ?N§,y9DE?BR[E8J7#h H8C,C,/+ 9795X: < 1 @61 5 1 1 #<

N/n'M9JN4QROTG?@I:<GFVI:T(RC%Dk?B]&%a<GRZ_aRZ<.>xR#Sm>xcJRS@DEI: 0]9<.> S@I

Tr(R) > Ts(R) ⇒ CN(R)

]0 KR#FOu Gi¥<ERM?BUZy9I:A9<C%RMc¢| R&SmTG]KFVRTrEQ]>B]9IJc:c:RMA9Dc:]aC%UZoqA9?B_a]v>@I:A9<?@U#RZc:cJRMC%R&SUZc:UZ_aRZ<.>SjCE]v<,Sc:]f?BUZy9I:A9B]9IJc:cJRMAKDcw]aC%UVoqAK?@_]v>@I:A9<kSmAKDEG]vIJ>@U#ROCER#SxU#cJU#_fR#<.>BSxCE]9<GSc:]a?BUZyKIJAK¥C%RM<GA.R#DGCESxCE]9<GSju


~cJD,SmI:RZDE?SFGA9IJ^SmAK<.>T,A.S@S@IJWGcJR&S3T,AKDE?cw]xC%UV§,<EI>BIJAK<MC%RR, Ts(R), Tr(R)

RV>CN(R)

8! AKDGSx]0 9AK<GSFEA9IwSmICEROTG?@R#<GC%?BR

& RGQ]fW(A9DEc:RMFVI:?BFZA9<GSBFV?BI>BRO¨fDE<GRM_a]9IJc:c:R

& Tr(R)EQRM?B] KA9<`CERMc:]fW(A9DEc:R

& Ts(R)Gi¥<ERNoqA9<GFV>@I:A9<k.DEIR#Sm>jUZ n]9cJDEU#RM]vDkFVRZ<.>B?@RMC%RMcw]fW,AKDEcJR

& CN(R),ij<ERMTE?BA%FVU&C%DE?BRN.DEI]vXYA9D%>BRNDE<k<EARZDGCk]vDkFVR#<.>@?BRNCERMc:]fW(A9DEc:R

RV>m>BR`C%UZ§G<EIJ>@I:A9<©R#Sm> n]vcw]vWEc:RRZ< e9¤ 5[9¤ RV> R#< 7#¤8RpFEA9IJ^T(A9DE?uRZ> T(A9DE? ! !u <| R&SY>TG]KS<GA9DE KR#]vD ( [9*) ( e9e )¢8EQ5]y9U#<EUZ?]n>BIJAK<`C%R&ST(A9I:<.>BST(RZD%>xT(AKS@RZ?C%R#STE?BAv¡WEc:bZ_aR#SSmIcJRFVRZ<.>B?@R CERc:]pW,AKDEcJRR#Sm>¥EAK?BSC%R N8¤¥I (U#?@R#<.>@R#SSmAKcJDE>@I:A9<GS¥A9<.>¥UV>BUTE?BAv¡T,A.SmU#R#SxCE]9<GScw]c:IJ>m>@U#?B]v>@DE?BRMFVAK_f_aRNc:RM?@RmXYRV>¥C%DkT(A9I:<K>xAKDkS@A9<kC%U#TEc:]KFVR#_fR#<.>#8

`)(,/0$+-4Q~| ]vc:y9AK?@IJ>@E_aR~C%R?])%<ERZ_aR#<K>5R#Sm>5c:RS@DEIJ n]9<K> 0%A9IJ> ©DE<ER>@?BI:]9<Ey9DGc:]v>@I:A9<C%R¤¥R#c:]9DE<G] C3| DE<R#<GS@RZ_ WGcJRMC%RNT(A9I:<K>SRZ< [v¤8GA9IJ>

Rcmc:RN?B] KA9<p_]n^%I:_ DE_ C%RO>@A9D,ScJR&SFZRZ?FVc:R#S

FVI:?BFZA9<GSBFV?BIJ>BS C%R#S >@?BI:]9<Ey9c:R#S C%Rj¤RZcw]vDE<,] C%R N8KAKI>Tsc:]O>]vI:cJc:RjS@A9DEG]9I>BUZRK89I

Rcm ≤1.5 ∗ Ts

c!| ]vc:y9AK?@IJ>@E_aR<ER¥o]9I>x?BIJR#< SmIRcm > 1.5 ∗ Ts

DE<<EA9DG 9R#]9DpT(A9I:<K>xR&SY>xFZ]9c:FZDEcJU]vD FVR#<.>@?BRpC%D-FVR#?BFZcJRRZ>SZ|IJcR#Sm> ¨ c!| I:<K>BUZ?BIJR#DE?C3| DE<©>@?BIw]v<EyKcJRpC%R IJcR#Sm>]nXYAKD%>@Up¨ ]0 9R&FfDE<ERp n]v?BIw]v<.>@RpC%R`c!| ]vc:y9AK?@IJ>@E_aRC%Rpcw]k§GyKDE?BR`[%8:7Zh RZ>C%A9<G<ERpDE<ER`<EA9DE KRZc:cJR>@?BI:]9<Ey9DGc:]v>@I:A9<CER¤¥R#c:]9DE<G] p<EA9>@U#RN|:8Q~| ]vc:y9A9?BIJ>@E_aRpS#| ]9TETEDEIJ>fS@DE? cw] TE?BA9TG?@I:UV>BUpC%R#S >@?BI:]9<Ey9DGc:]v>@I:A9<GSMC%R¤¥R#c:]9DE<G] ©SmDGI¡

n]v<.>@R .,/ I+-4 ,$Mb!#$ ] +1-+.EGF78+,(3+k "!#%$<-T7 +kM7 - 9 C 0b'( 24+ C 278+mR@!<(3+,!#'27 -(:0b1-D7n* +.%RX+7831+,24" R"+&U+G(3+ +L-L$K24+ C 278+W2! C 24" -2 C !#$ac,$ C !<0"U9:78+W(3+e780,$ C !#B0"9b780b$ !:/7 62 < :(/1 3 5 @ ; &9<

Q~| ]vc:y9A9?BIJ>@E_aRxTE?BU#FZU#C%R#<K> oqAK<GFH>BIJAK<E<ERjT(A9DG?~<| IJ_aT(A9?@>@R¥.DER#cJc:RxoqA9<,FH>@I:A9<TsC%AK<E<EUZR

CE]v<,SR2 8 DG]v<GC Ts

<| R#Sm>T,]9SC%A9<G<EUZRNRV^%TEc:IwFVIJ>@RZ_aR#<K>_]vIwSx.D| I:c(o]vD%>xcw]aFVA9<,SY>B?@DEI:?BRÖTG]v?@>@I:?~C3| DE<a_a]9IJc:cw]vy9RI:<EI>BI:]9cEA9DaC%Da_]vI:cJcw]vyKRCERxc:]oq?BA9<.>@I:bZ?BR#nTEc:DGS@IJR#DE?S T,A.S@S@IJWGIJc:I>BU#SSZ| A ?BRZ<.>j¨<GA9DGS#8<kFZA9<GS@IwC%bZ?BRO.DER & c:R#Sj]9?$Z>@R&SCED_]9IJc:c:]9y9RNI:<EI>BI:]9cC%A9<G<ERZ<.>jc:AF#]vc:RZ_aRZ<.>xcw] >]vI:cJc:RS@A9DEG]9I>BUZR & c¢| AK<k 9RZDE>xCER#SxU#cJU#_fR#<.>BSxC%RMW(A9<E<GRN.DG]9cJIJ>@U ?@U#y9DEc:I:RZ?S& c¢| AK<k 9RZDE>c:RM_aA9I:<GSxC3| UZc:UZ_aRZ<.>SxT,A.S@S@I:WEcJRK8ij<ERòqA9<GFV>@I:A9<©cJA%F#]vc:R#TE?BRZ<,]v<.>RZ<-]9?@yKDE_aRZ<.>DE<©>@?BIw]v<EyKcJRaT(RZ?B_fRZ>C%Rp?BUZT(A9<,C%?@R

SmI:_ DGc>]v<EU#_fR#<.>xäFZR#SjefRV^%I:y9R#<GFVR&S

Ts(t) = α ∗ l(t)¢[%8 d

Ac q> R&SY>xcw]acJAK<Ey9DGRZDE?xC%RMcw]fTEcJD,SxT,RZ>@IJ>@R]9?$Z>@RMC%D>@?BI:]9<Ey9c:R>RV>

αDG<kFZAR %pFVI:RZ<.>x?@U#RZc 8RV>m>BRC%UV§,<EI>BIJAK<kR#Sm>j 0]9c:]9WEc:ROR#<k[9¤ RV>jR#< e9¤8

$#" ¦jc:y9A9?BIJ>@E_aIwKDGROyKUZAK_fUZ>@?BI:.DER 6,6

G/ I+-4 Y* 0"7 9U C !#$ ?EG+Y24"%RX+ C 9U+-L! α = 2/3[ u* +L-L$ce("! C +k-4!7N* "jM+93A C +

0X- (* 0 C v$O+5'7 - +.$/!#$6+L- ; +D24+7#78+D(3"%'A4+L- 0br(:A610 C $O[

G/ I+-4 '!u7N* 0"7 93 C !#$#?Ee+ M+ C + +.$/$6+o0b 2. "!#%$ D780K$ C !<0"9b780b$ !@?BI:]9<Ey9c:RNR#Sm>U# n]vc:DEUZR¨fc¢| ]9I:C%RMC%D FZ?@IJ>@b#?@RMS@DEIJ n]9<K>

Q(t) = β ∗ Ri(t)/L(t)¢[%8:7#t

A#Ri(t)

R#Sm>c:RM?] 9A9<C%D FVRZ?FVc:RNIJ<,S@FZ?@IJ>j¨f>Q > cw]fcJAK<Ey9DGRZDE?jC%RNc¢| ]9?$Z>@RNcw]TGcJDGSjy9?]v<GC%RNC%RN>RZ>

βDE< FZA.R %pFVI:RZ<.>j<EA9?B_]vc:I:SB]n>BRZDE?

(β =√

12)8

G/ I+-4 '!P7n* 0"7 9U C !#$ ?EG+,'+ C + +$ $6+W0" 2d "!# $!#7(:b '+W'+e$ C !<0"9b7 0"B$/!<" p(3" $I78+.-u$ C !<0bU9:78+L-Db $P'+ ; 0"7 !#$OA)F4 C 'A4+

∀t ∈ T, Ri(t)L(t) > (1 −

√3/2)

"r+'24 C + ∀t ∈ T, Q(t) > 2 ∗√

3 − 3 ≈ 0, 464

W +-,' #$Z/n"%$#!,' b C 780m(3A.EG" -L$ C 0b$ !<brR""! C .[

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_]vIwSx.DEIoqA9<,FH>@I:A9<G<ERSmDG?xcJR&Sx]v<EyKcJR&S & R

GQRN>B?@Iw]v<Gy9c:R& Tr(R)

GQ~| ]v<EyKcJRN_aI:<EIJ_DE_ CED>B?@Iw]v<Gy9c:R& Ts(R)

,ij< ]v<EyKcJRNI:_aT,A.SmU& CN(R)

0Q]oqA9<,FH>@I:A9<.DEI.FZ]9c:FZDEcJR DG<<EARZDGCM]9DMFVRZ<.>B?@RCER~c:]W(A9DGcJRFVI:?BFZA9<GSBFV?BI>BR]vD>@?BI:]9<Ey9c:R98

RZ>k]9cJyKA9?BI>BE_aRL?B]&%<ER ]9DGS@S@INc:]-oq?@AK<.>@I:bZ?BR .DG]v<,C Fv| R#Sm><GU#FVR&S@SB]vI:?BRLRV>k>@?]vIJ>@R C%R#SyKUZA9_aUZ>@?BIJR&ST(A9cy9AK<G]vc:R#STE?BR#SBKDGR#Sx.DERZcwFVAK<G.DER#S#8%;=c3TG]9?m>jC%DkTE?BIJ<,FVI:T,R G/ I+-4 '!$T+.-L$ $ C !<0"9b7 +W(* 0bU9:78+GE)!# !#E)E

θ78+e$ C !<0"U9:78+)-1* 04:' :0b $

-L C 78+ 24+.%$ C + (" 24+ C 2.78+r2.! C 24b1-2 C !#$G0 0"9b7 +hE)!# !#E)E (:+2 ∗ θ

%"! C t9b C +[ :[


[E8 [9 & i¥<RZ^%RZ_aTEc:RNCERM?B]&%a<GRZ_aRZ<.>(¤RZc:AKS#8J7K8 tG8 t <?BRZ_]v?.DERZ?]S@DE?MFVRV>NRV^%R#_fTGcJRaC%Ra_]vI:cJcw]vyKR.DERfc!| A9< ?@RZ>@?BA9DE KR ¨`T(RZD C%RFEAKS@R#STE?Bb#Sxcw]S _aUV>B?@I:RC%D _]vI:c:c:]9y9RMc:IJ<EU#I:.DERK8,QRM_a]9IJc:cw]vy9RMc:I:<EUZIw.DERR#Sm>¥FVAK<GSm>@IJ>@DEU CER e9eKd]v?V>BR#S o]9I:SB]v<.>DG<p?B]9TET(A9?@>~_a]v^%IJ_DE_ C%R 7K8J7K8KQ]M>@?BI:]9<Ey9DEcw]n>BIJAK<a?@U&SmDEcJ>B]9<.>@RjR&SY>FVA9_f¡T,A.SmU#RC%Rj7&tvh.>@?BI:]9<Ey9c:R#S3RZ>5C%RxP7#d<GA.R#DGCES#8

MIN(Q(t)) = 0.34, MIN(l(t)/L(t)) =0.34, MIN(θ(t)) = 19.5

8

B(t)

n

a

2a

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FVR#?BFZcJRMFVI:?FVA9<,S@FZ?@IJ>]fDE< ]9<Ey9c:RN_fI:<EI:_ DE_C%R2 ∗ θ

8

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