Computational Geometry and Discrete Computations

Texte intégral

(1)Computational Geometry and Discrete Computations Olivier Devillers. To cite this version: Olivier Devillers. Computational Geometry and Discrete Computations. RR-3533, INRIA. 1998. �inria-00073152�. HAL Id: inria-00073152 https://hal.inria.fr/inria-00073152 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Computational Geometry and Discrete Computations Olivier Devillers. N° 3533 Octobre 1998 THÈME 2. ISSN 0249-6399. apport de recherche.

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(121) Lo8d= NJCdQS =<YDHQ H RJ=<Q Io + P:Ke8NJCNOFHI3GJY_DLo8:; DHQMK6NOFHFE=<cbvNJY_QSDH;>G3CdDHG3PdQaU R 'C0PdLo8 .pri 24 78dDHqQ -TDEC:cGJ'NJbdb:Y_GNOKe8K<NJCNJFEQ_GUV=P:QS=<cDHCN =6 N -O=<YN6R3¥ Lo8vNOLDEQ2=!C:Q_PdY DHC:I =63N -J=<Y

(122) b:YSG3bV=<YSL_DH=!QnLo8vNOCLo8:=uYS=6NJFzI3=<G3;>=LoY_DEKb:YSG3b=!YSL_DH=!Q<> G I3GTG c=¯¦dNJ;>b:FH=DEQ + P:I3DE8vNJY_N&NJC:c Y_D ¼Q

(123) NOFHI3GJY_DLo8:; .¹? p 2z-G3. Y ]=<FHNJP:CvN6RlLoYSDNOC:I3P:FHNOLoDEG3C©78:DEQNJFHIJG3Y_DLo8:; 3=<YSRD 4:=<Quc:PdY_DEC:IL_8:= DEC:Q_=!YSL_DHG3C GJ=<NJKe8 C:= bG3DECLL_8vNOLLo8:=K<GJC:QSL_Y_P:KLo=<c L_Y_DHNJC:I3P FNOL_DHGJCDEQNI3GTG cLoGJbG3FEG3I3DEK6NOFZL_Y_DHNJC:I3PdFNOL_DHGJC 4NbdFNJC:NJYIJYoNJb:8 ¥jQ_GL_8vNOL=6NJKe8 +NJK!= DHQ

(124) N L_Y_DHNJC:I3FE=NJC:cl=<NJKe8=<c:I3==¯¦ DHQ\LoQ

(125) G3C:FRG3CdK<= ¯ ZL_8:=I3=<GJ;=LoYSDHKb:YS=<c:DEK6N~Lo=<QNJYS= =¦dNJK!LaLo8d=<CLo8:=NJFEI3G3Y_DLo8d;K<G3CdQSLoYSP:K!LaLo8d = ]=<FHNJP:CvN6RLoYSDNJCdI3P:FHNOLoDEG3C©¥vGJL_8:=<Y\MDHQS=L_8:= NJFHIJG3Y_DLo8:;%I3PvNOYoNJCLo=!=<Q^Lo8:NOLZDELZcdG =!Q^C:GJL^+NJDEF:NJCdc2NOK!LoP:NJFHFRK!G3C:QSL_Y_P:KLoQ NnLoG3bVG3FEG3I3DEK6NJF LoY_DHNJC:IJP:FN~LoDEG3C©< + GuL_8vNOLMNOC:GJLo8d=<YNJFEI3G3YSDEL_8:; C:=<=!c:DHCdI2NLoY_DHNJC:IJP:FN~LoDEG3CNJQDEC:b:PdLMK<NJC P:Q_=

(126) Lo8:=Y_=!Q_P:FL¶MDLo8dG3PdLaLoY_GJP:U:FE=J aCd-G3YSL_P:CvNOL_=<FR3¥vLo8dDHQ¶NJFHIJG3Y_DLo8:; c:G =!Q]CdGJLa=<C:QSP:Y_= Lo8vNOL#Lo8:=

(127) K<GJC:QSL_Y_P:KLo=<cLoYSDNOC:I3P:FHNOLoDEG3CDEQLoG3bVG3FEG3I3DEK6NJFEFER =<W0P:DONJFH=!CLaL_GLo8:

(128) = ¶=!FNOP:CvN6R LoY_DHNJC:IJP:FN~LoDEG3CGJ#Q_G3;>= DEC:b:P L6¥zQSDHCdK<= L_8:=<YS= =¦ DEQSLLoG3bVG3FEG3I3DEK6NJF^LoYSDNOC:I3P:FHNOLoDEG3C:Q

(129) Lo8:NOL K6NJC:CdGJLMU=

(130) YS=6NJFE[D Q!=<cNJQa N ]=<FHNJP:CvN6RG3C:/= . 24 +TDH;>DEFNJYZb:8d=<C:G3;>=!CvNNJYSDHQS=MDHCGJLo8d=<YZb:Y_GJU:FH=!;>Q<¥0-G3Y^=¦dNJ;>b:FE=¶NOC NJFEI3G3YSDEL_8:; K!G3 ; b:PdLoDEC:I

(131) NJC>NOY_YoNOC:I3=<;>=!C0LjGJVFHDEC:=<Q^;N6RI3PvNJY_NJCLo=<=QSG3;>=b:YSG3b=!YSL_DH=!Q^G3CFHDEC:=DECLo=!Y_Q_=!!K LoDEG3C:Q<¥hU:P L¶C:GOL¶;>G3YS=K!G3;>b:FHDEK6N~Lo=<cG3C:=<QaQ_PdKe8lNJQ]XzNJb:bdP:Q]L_8:=<G3YS=<; .pri 24780P:Q]L_8:= K<G3;>b:PdL_=<c NJY_Y_NJC:I3=!;=!CL#;N6R>U=aC:GJL YS=6NJFE[D Q<NJU:FE=]UR2NJKLoPvNJFvFEDEC:=<Q!¥dQ_PdKe8 NJC NJY_Y_NJC:I3= ;=!CL>DHQuK<NJFEFH=!cNOCNJY_Y_NJC:I3=!;=!CL>GJab:Q_=!P:c:G ÉFHDEC:=<Q! 78dDHQ>c:D =!Y_=!C:K<=&DEQDH;>bVG3YSLoNJCL Q_DEC:K<=

(132) Lo8:=

(133) K!G3;U:DECvNOL_G3Y_DHNJFb:YSG3bV=<YSL_DH=!QaNJY_=

(134) c:D =!Y_=!C0L .p3wp 24 . - . . . . r\r ¸ sutwvtt.

(135) p. . (4. = # !$. %"! W) + G3F DEC:I Y_G3UdP:QSL_C:=<QSQb:Y_GJU:FH=!;>QDEC K!G3;>b:PdLoNOLoDEG3CvNJF

(136) I3=!G3;>=!L_YSR M8:DEKe8 YS=<FRG3CL_8:= c:DHQSK<YS=!Lo=C:NOLoP:YS=uGJ^K!G3;>b:PdL_=<Y'K<GJ;bdPdLeNOL_DHGJC:QnDEQnK<P:YSY_=!C0L_FERN U:DEI&DHQSQ_P:=DECL_8:=A4v=!FHc NJC:c;GJU:DHFERD Q<=<Q$;NJCR

(137) Y_=<QS=6NJYSKe8:=<YSQ< NOC0R

(138) cdDHQSK<P:QSQ_DEG3C:Q^NOLL_8:=ZFNJQ\L G ¤ QSRT;>bGJQ_DEP:; G3CK!G3;>b:PdLeN~LoDEG3CvNJF I3=<GJ;=LoY\RNJCdcLo8:>= 4vY_Q\L¶ G3MY -0QS8:G3blG3ClNOb:b:FEDH=!clK<G3;>b:PdLoNOLoDEG3CvNOF I3=<G3;>=LoYSR =<YS=&NJUVG3PdLLo8:DEQL_G3b:DEKJ78:= ;NJDEC DEQ_Q_Pd=<QNJYS=2L_8:= c:=!Q_DEI3C GJ#+NJQSLuNJC:c =¦dNJK!L

(139) K!G cd=LoGNJCdQS =<Y'I3=!G3;>=!LoYSDHKbdY_=<cdDHK<NOLo=!Q 4N Q_DE;>b:FE=I3=<G3;>=LoY_DEKW0P:=<Q\LoDEG3CQ_P:Ke8 NJQ H GJCM8:DEKe8Q_DEc:=GJ^Lo8dDHQnbdFNJCd=uc:GT=<Q¶L_8:DHQ¶bG3DECL'FEDH= wI NJC:cL_8:=uK<YS=6NOL_DHGJClGJZC:=! NJFHIJG3Y_DLo8:;>QM8:DEKe8c:GC:GOL2YS=<FR G3C I3=<GJ;=LoYSDHK&Lo8d=<G3YS=<;>QL_8vNOLNJYS=DH;>bVG3Q_QSDHU:FE=LoG I3PvNJY_NJCLo=<=

(140) P:QSDHCdI2K!G3;>b:PdLo=!YaNJY_DLo8:;>=LoDEKJ `M=<FHNOL_=<c>#G3SY -0Q#K<GJC:K<=!Y_C NOFHQSGC:= -0DEC:c:Q#GO©NJC:NJFERTQSDHQ!^78:=]DECLoYSDHC:QSDHK]K<G3;>b:FE=¦ DLR GJL_8:=I3=<G3;>=LoY_DEKb:Y_=!c:DHK<NOLo=!QK<NJCU=QSL_P:c:DE=<c .p ¥Mi 2³¥¶G3YY_=<Q\LoYSDHKLo=<c ;>G c:=!FHQGJ K<G3;>b:PdLoNOLoDEG3C =<Y_=G3CdFER NQ_;NJFEFC0P:;UV=<Y>GJM =<FEFc:= 4vCd=<c I3=<G3;>=LoY_DEK&b:YS=<c:DEK6NOL_=<Q NJY_=

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(142) e3 #w=?

(143) =h/= ¥:bvNJI3=!Q ¤ # .¹? 2d < K#G3DEQ_QSG3C:CvNOL NJC:c ZTDHCd=<KJ

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(154) j

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(157) oS J @ j

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(162) N C:= NJC:c QSb:8:=<YSDHK<NJF#cd=<I3=!C:=<Y_NJK<DE=<Q! C<6 #"w= ; &= 8 e #w=F J=

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(164) e3 #w=?

(165) =h/= ¥:bvNJI3=!Qai i T¥ prq3 .Ept? 2 +\ fhGJYSLoPdC:=>NJC:c ¤ d NJC R -h, P K<DE=<CLu=¯¦:NOK!LNJY_DLo8:;>=LoDEK>-GJYK!G3;>b:PdLoN LoDEG3CvNJFTI3=<GJ;=LoY\R3 C 6 $#"w= l; u= ;

(166) e3 #w=

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(168) A ¥:JG3FEP:;>A = 0 0>GO " j $ Zq

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(174) /R Z o$ 8 886 ¥6JG3FEP:;>=]p3¥tbvNOI3=<Q$?3?Ji d?Jidp3¥ prq3q .Ep#02

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(176) j &

(177) M " A q j

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(179) oS J @ $ j

(180) "M F Z "

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(182) ; 8 $ e" j

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(184) " ¥:bvNJIJ=<Q

(185) pri? hp id¥prq3qdp3 .¹? 2JG3CvNOL_8vNJCl`

(186) +T8:=! Ke80uP -h`MG3U:P:Q\L

(187) NJcvNObdLoD3= GNOLoDEC:I ÉbGJDHCLnIJ=<G3;>=!L_Y_DEKub:YS=<c:D K6NOL_=<Q< C 6 #"w= >; &= ;

(188) e #w=\

(189) =\/= ¥ObvNJI3=!Q#p p Vptk d¥ prq3oq 0d .¹? p 2 > + P:IJDH8vNOYoNnNJCdc> YSD4 GY_GJU:P:QSL$LoG3bVG3FEG3IJ$R GJY_DE=<CLo=!cuDHCdK<Y_=!;>=<CLeNJF:NJFHIJG3Y_DLo8:; -G3Y ZG3Y_GJC:G3Djc:DHNJI3Y_NJ;>Q< 8 %

(190) = 9 =

(191) =/=;?JR= ¥ 4? TEp Jq ?3?Jsd¥ prq3q .Eptk 2 >. 0. 0. '0. 0. 0. . !. . 0. 0. . 4. 0. 4. \r. .

(192) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France).

(193). . ISSN 0249-6399.

(194)