HAL Id: inria-00072833
https://hal.inria.fr/inria-00072833
Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.
Elementary Algorithms for Reporting Intersections of Curve Segments
Jean-Daniel Boissonnat, Antoine Vigneron
To cite this version:
Jean-Daniel Boissonnat, Antoine Vigneron. Elementary Algorithms for Reporting Intersections of
Curve Segments. RR-3825, INRIA. 1999. �inria-00072833�
a p p o r t
d e r e c h e r c h e
THÈME 2
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Elementary Algorithms for Reporting Intersections of Curve Segments
Jean-Daniel Boissonnat — Antoine Vigneron
N° 3825
Novembre 1999
-/.
%0213#
4658769:;<769>=?5A@BDC*=FE&EGC*99H7JI8KHLM9NIOC*=?9>5QP<=?R*9>5AS&C*9
TVUXW&Y[Z\]_^a`&bXcdZfehgjikchlcdZ&e
Zmnlopeqlrseutwv8Yx6gjehcqykrsZ
zH{|gp}~ZmnzH{|chtYZ
opsJgk{wmZ {|Z&lUXZ&{|lUXZb6jA\j$]gkZYDxX{Z[&kj]&DsojijZ&t
M&Xjj Zas{gkJgAtwZ twZ&jZ&{|ojeJopehijgk{cdmUsYtm|gD{|Z6gj{mm|UXZ3chbAmZ{tZ&lm|cdbsiDsojcd{tVopYgjbsiDo[twZmVgj
lrX{|jZ#tZikY[Z&bAm|t&Jop{mH {gkY¡mUXZ#chbAmZ&{|tZ&lmchgjbs{ZGchl&om|Zj¢pgkrX{£opehijgk{cdmUXY3tgjbsedvDrstwZ¤m~¥¤gftwchYXedZ
s{ZGchl&om|Z&t3¦mUXZ§s{ZGchl&om|ZmUsopmDlgkY[Jop{|Z&tamUsZ'lg8gj{chbsom|Z&t gj¤m~¥¤g¨6gjchbkmtopbJ©mUXZ§s{ZGchl&om|Z
m|UsomtoOvtcdfoªJgkcdbAm'cqt3x6Zehg¥f¢#gjbu¢Vgj{Mopx6gjZ¨o«twZ&ijYZbAmG¬®b¯soj{wm|chlrXeqop{G¢,m|UXZ$X{|Z&Xchl&omZGt¥Z
rJtwZ$XgªbXgpm<opehedg¥°mg±lgkrXbAm§mUXZ²b8rXYx6Z{MgpfcdbAm|Z{twZGlmchgjb¯6gjchbAm|t'bXgj{'mg³tgj{m3mUsZY<¢nopbJ´mUXZ
m|cdYZlgjYXehZµcdm~v'gpHgkrX{nopehijgk{cdmUsYt#XZ6ZbsXtngjbmUsZb8rXYx6Z{¶gpHcdbAm|Z{twZGlmchbXisojcd{t¢sbXgpmngkb<mUXZ
b8rXYDxJZ&{Hgp*cdbAm|Z{twZGlmchgjbDJgkcdbAmt·?cd¸*Z{|ZbAmehv {|gjY¡m|UXZ#gpm|UXZ{Hbsgjb[m{|cd8cqopeXojedikgj{|c¹m|UXY3t|º Z#X{ZGtwZ&bAm
ojbQopehijgk{cdmUXY gj{#m|UXZ{|Z&»¼xXehrXZfop{|cqopbAmngp>m|UXZX{|gjxXehZY!¥#UXZ&{Z¥¤ZfUsoOkZo§twZmngpHxXehrXZ[twZ&ijYZbAm|t
ojbs½o¾tZmgpa{ZG½tZijYZ&bkmttwg²mUsopm3bXg©m~¥g¾twZ&ijYZbAmtgjnmUsZ<t|opYZ<twZm3cdbAmZ&{|tZ&lm&±TVUXZ'm|cdYZ
lgjYXehZµcdm~v«cht§¿[ÀÀ
©Á
6ÂkÃdÄkÅ
´TVUXcqt§opehijgk{cdmUXYÆcqtmUsZb±rstwZGªm|g«twgkedkZ'm|UXZ¨ijZ&bXZ{opeVloktwZ<chb
¿[À
Ç
ÃdÄkÅ
£mchYZj¤¬®bmUXZl&ojtZ gpHstZrsgj»ÈtZikYZbAm|t · c¼ZjVtwZ&ijYZbAm|t¤m|Usom¶cdbAmZ&{|tZ&lm#cdbQopmY[gAt~m
gkbXZaJgkcdbAmº,¥¤Z¶X{|gj6gktZ ox6Zmwm|Z{ojedikgj{|c¹m|UXY¥#UXgktZnm|cdYZflgkYXedZµc¹m~vcqt¿[ÀwÀ¼
Á²
ÂjÃhÄjÅ
Á© Ç
6Â
nede%gjrX{Vm|cdYZlgkYXedZµc¹m~v§{|Z&trXedm|t#oj{Z o[edgkiD?oklm|gj{V {|gjYÉgkmchY3ope¼
Ê&Ë¾Ì£Í OGsÎ6QÎ Í &Ï kЯ[Ñ cdbslZ[mUXZ¥#{|c¹m|cdbXi<gj,mUscht {|Z6gj{m&¢uo'ÒJoO¥UJojtaxJZ&Zb² gjrXbJ$chb$m|UXZ3X{|g8gp
gjTVUXZ&gj{|ZY sHntolgkbstZ&yArXZbJlZj¢Am|UXZopbsojedvtcht¤gj%mUsZopehijgj{|cdmUXY gk{VmUXZfikZbXZ&{|oje*l&ojtZaojbsMmUXZ
ojbsopehvtwcqtagpmUXZ§ojedikgj{|c¹m|UXYÓ gj{amUsZ3stwZ&rsgj»ÈtZikY[Z&bAm l&ojtZ[oj{Z[bXgpmlgj{|{ZGlmGDbXZ&¥{|Z6gj{mam|Usom
gkbXehv'lgkbkmopchbstmUXZf{|Z&»¼xXehrXZfloktwZ cqtoOopchehojxXehZaoktV¬~n#¬~¡#Z&Jgk{wm#ZGtwZGop{lUM#»Èkjjs
Ô
ÐAÕ*Ö|×
Í
Øj Ù
gkYXrm|opmchgjbsoje¤^aZ&gjYZm|{vk¢>gjxXrstwmbsZ&t|t¢HÚ>µXojlmDop{|cdmUXYZm|chlj¢
Ñ
¥¤ZZX»¼ehcdbsZ'opehijgj»
{|cdmUXY3t
ÛuÜOÝßÞà*áâFã¶ÜäÞ>åpæ~æwçèäâ éFÝêäëßëì Þ¼íGèOèpáâ?éFæ~îDåìaéFÜOæVï6ðGñ6òuóôÛ$óqõ«öGÛ%ò©ñsâFá|÷Fæ~ø®é£ùáúûGüwýýþôÿö8óäçOî é¼ÜGæ
%ø®éFÝßáçaîOæ£òuæ~øÈÜOæ~â?øÈÜOæNááè~â¼ä|éFÝ æVÿ~á®éFâFÝßøwä
+¤ Æ+¤ . ²#
Ë b«X{|gj6gktZ3XehrstwchZrs{|tDopehijgj{|cdmUXYZ&tJgkrX{D`mZGlmZ&{&¢HXojbstrsb«ZbstZYDxXedZMZ
op{lt
XZQlgjrs{x6Z¨YgjbXgjmgjbsZ&t§r¯sehojbu¢VedZGt'±sopch{|Z&t3yArXc¶tZQlgkrX6ZbAm& gAt§opehijgk{cdmUXYZGtbrXmchedcqtwZ&bAm&¢
Z&b½Xehrst[r½s{`Gchl&omykrscVX`&lcqZ<twcVZ&rµ½op{lt[twZlgkrXJZ&bAm&¢,yArXZZrµªs{`Gchl&omtDtcdYXehZ&tD6gjrX{
lgjYsoj{Z&{nehZ&tflg8gk{|gkbXbX`&Z&tnXZXZrµQ6gjchbAm|t Zmf`GlcqZ{ftcHrXb²6gjchbAmaZGt~mojr²ZGttrst&¢uopr²XZ&t|twgkrst¢
gkrªtwrs{rXb½oj{|ljÚHb½sop{mcqlrXehchZ{G¢bXgAtX{|`&Xchl&om|tbXZ§6Z{|Y[Zmwm|ZbAmDsoktZMlgjYm|Z{ehZ'bsgjYxs{Z§Z
6gjchbAm|tacdbAmZ&{|tZ&lmchgjbQbsc>XZm|{chZ{aedZGtaJgkcdbAm|tacdbAm|Z{twZGlmchgjb%¢6ZmaehZDmZYst ZlojehlrXeZ&twm¶ gkbslm|cdgkb
Xr²bXgkYxX{|ZDXZ[Jopch{ZGt¶op{ltayArXc£tZ[lgjrX6ZbAm ZmfbXgjb©r©bXgjYDxX{ZZ[JgkcdbAmtfchbAmZ&{|tZ&lmchgjbMeho
Xc¹¸*`{|ZbslZ[XZ&tfoprXm{|Z&t opehijgk{cdmUXYZGtnbXgkbQm|{ch8chojrµ$lgkbXb8rstuZ3lokt¶xsedZ&r»¼{|gjrsijZM·?g²edZGtftwZ&ijYZbAm|t
tgjbAm#lgkedgk{`Gt£ZbxXehZrMgjr'Z&bM{|gjrsijZ¶XZnm|ZehedZa?ogjb'yArXZ ehZ&tVtZijYZ&bkmt,{|gjrsijZ&tbXZ twZflgjrX6ZbAmVsojt¤Zm
XZYYZ6gjrX{¶edZGtxXehZrstºVZGt~mm|{|ojc¹m|`Zb<m|ZYsta¿[ÀwÀ
<Á
*ÂjÃdÄkÅ
 ٠Zm¶opehijgk{cdmUXYZfZGt~m§eho3xJojtZ
rXbªojedikgj{|c¹m|UXYZ§yArXc£m|{|ojc¹m|Z§edZMlojtij`bs`{opeHZb¾m|ZYst¿[À
Ç
ÃhÄjÅ
Â|¶ojbstfehZ'loktZ§JtwZ&rsgp»
tZikYZbAm|ta·?lZGt~m»p»Èch{|ZaZGtVop{lt¤ykrscutwZ lgkrXJZ&bAmVZ&b'rXb6gjchbkmojr'Xehrstº¢gjbX{|`&tZbAmZ¶rXbY[Z&cdehehZrX{
ojedikgj{|c¹m|UXYZ§ZMlgjYXehZµcdm`M¿[ÀÀ¼
Á ÂjÃdÄkÅ
QÁ¯Ç
J©T%gkrstbsgkt{|`&trXedm|opm|tXZ'lgkY[sedZµ8cdm`twgkbAm
gkmchY3oprµDrsbM?oklm|ZrX{#ehgjiAop{|c¹m|UXYchyArXZaX{|W&t&
¨Í &СÏË¾Ì£Í OGsÎ6&Ð "! bXZ$Z&{{|Zrs{XojbstMeho½X{Z&rXjZ©r m|UX`gk{W&YZ$³o½`m|`$m|{gkrXj`&Z$opX{|W&t'eho
{|`&sojlm|cdgkbZVlZ¤{|ojX6gj{m&%¬®et#Z&bstrXc¹mHyArXZ¤edZGtojbsopehvtwZGtXZe$ojedikgj{|c¹m|UXYZ,ik`bX`&{|ojekZm>Z¤e$ojedikgj{|c¹m|UXYZ
6gjrs{¶ehZ&t¶stZrsgj»ÈtZikYZbAm|t¶twgkbkmacdbJlgj{|{|Z&lm|Z&t&%ZDloktn{|gjrsijZ»ÈxXehZr¨ZGt~ma{Z&X{cqtaXopbst¶ehZ{opX6gj{maZ
{|Z&lUsZ{lUXZ¶¬~¶#¬~¡V»®jkj
%Í
GÖ& Q
^a`&gjY`m|{chZDojedikgj{|c¹m|UXYchyArXZk¢J#gkxXrstwmZGttZj¢*{|c¹m|UXY`m|chyArXZDZµXojlmZj¢*ehijgk{cdmUsY[ZGtnZ
xJopeqoOvkopikZ
*
(* <. +, D
TVUsZ'rstrsopeVYgZemg¾opbJopehv+&Z§ijZ&gjYZm|{cql'ojedikgj{|c¹m|UXY3tcqtDmUXZ<#Z&oje¤n,¥#UschlU³cht[ojt|twrsY[ZG¾mg
lgjYXrm|ZDZµXojlm|edv¨¥#cdmU©{|Z&ope>b8rXYDxJZ&{|t.-h&0/ÈDTVUXcqtaYg8XZeHUXcqZ&t¶m|UXZ[?ojlm m|Usom mUsZoj{cdmUsY[Zmcqlgj
{|Z&oje£lgkY[srmZ&{|t UsoktfoedchYc¹m|Z&¾X{|Z&lchtcdgkb²ojbs©cdikbXgj{|Z&tab8rXYZ{|chl&ope£ojbs${|gjxsrst~m|bXZ&t|t cqttrXZ&t&§ntfo
lgjbstZ&yArXZ&bslZ[o'ch{|Z&lm chY[sedZ&Y[Z&bAm|om|cdgkb$gjopb©opehijgj{|cdmUXY!mUJom cht lgk{{|Z&lm¶rXbJZ{¶mUXZ3#ZGope>,
YgZ&e8XgAZGtbXgpm>bXZ&lZ&t|toj{chehv¶m|{|ojbstehopmZ,chbAmg on{|gjxsrst~m>opbs21Ogj{Z%3§lcdZ&bkm>X{|gjik{|ojY¢OopbJlopm|okt~m|{gkXUXcql
x6ZUJoOAchgjrs{|tVop{|Z lgjYYgjbsedv§gkxstZ{|jZ&u
54J{|twmopXs{gAojlUQm|g¨{Z&YZ&v¨m|UXcqtX{|gjxXehZY cht m|gQrstZZµXojlmop{|c¹m|UXYZmcqlp'¬®b¾mUXZ§lgjbAmZµAmgj
ikZgkY[Zm{|chl ojedikgj{|c¹m|UXY3t¢YDrslU<X{gkij{|Z&t|t¤Usojt#x6ZZ&bQgjbsZfcdbmUsZ{ZGlZ&bkmJojtwm$-d76¢u088¢%9s¢uG\7/ÈVb»
gjmUXZ&{HopXX{|gkoklUu¢GmgaxJZ¤ gkedehg¥ZGfUXZ&{Zk¢pUsojtZ&Y[Z&{ikZ&f{|Z&lZbAmehvj nZGlcqtwchgjbJtchb[ijZgkYZm{|cql,opehijgk{cdmUsYt
XZ6Zbs©gjb²ijZ&gjYZm|{cqls{ZGchl&om|Z&ta¥#UXcqlU©oj{ZrstrsopehehvQopehijZxs{|ojchlZµX{|Z&t|twchgjbst&:sgj{fZµojYXedZk¢* gj{
oMm|{chXehZ3gpVJgkcdbAm|tijchjZb©x8vQm|UXZch{loj{wm|Z&tchojb©lgAgk{|XcdbsopmZGt¢%XZ&lcqchbXi¨¥#Usopm cqt m|UXZ3gj{|cdZ&bAm|om|cdgkb²gj
m|UXZfm|{cqopbXikedZf{|Z&XrslZGtVmg§Z&opehrsom|cdbXiQ· mUXZ[twchijb¨gj|º#o3YDrXedmchOoj{cqom|ZfJgkedv8bXgkYchojeugp£Z&ij{|ZZ m~¥¤gsV¬È
ojbªopehijgk{cdmUXY gjbXehv¾rstZ&ts{ZGchl&om|Z&tfgjnZik{Z&Z'\¨ojtDo< rXbslmchgjb½gjVmUXZMcdbXsrm[Xopm|os¢>ojbs«cd#mUXZ
chbXXrXm¶Xomo§op{|ZflgZG¨ojttcdYXehZ;4XµZG<X{ZGlcqtwchgjb<b8rXYDxJZ&{|t&¢slgkYXrm|opmchgjbst¶lojbx6ZgkbXZZµoklm|edv
rJtwchbXi<mUsZ§bsom|cdkZ'gkrXxXehZ3X{ZGlcqtwchgjb¾Usoj{|X¥¤oj{Zgp¤mUsZ'lgkY[srmZ&{&<TVUXZ'Z&ij{|ZZ3gj#ojb«opehijgj{|cdmUXY
cqtmUXZ&{Z gj{|Z§{Z&ehopmZG¾mg$mUXZX{|Z&lcqtcdgkb«{ZGykrscd{|Z&¾m|g${|rXb½ojb½opehijgk{cdmUXY rstcdbsi$ZµXoklm[oj{cdmUsY[Zmcqlp
TVUscht3YgjmchOopmZGtmUXZ²Z&tcdikb´gpfZ%3§lchZbAm'ojedikgj{|c¹m|UXY3tgp edg¥ÓZik{Z&Zj #Z&rJlchbXi¾mUXZ²Zik{Z&Zgj
m|UXZ opehijgj{|cdmUXY3t£¥#chehe6{|Z&XrslZnmUXZab8rXYx6Z{#opbJmUXZflgjYXehZµc¹m~vgjNmUXZ ZikZbXZ&{|opmZalgjb4sijrX{om|cdgkbst&¢
Y3o'<jZ[mUXZMopehijgj{|cdmUXY3t Ygj{|ZZehZYZbAm|oj{v²opbs¾Ygj{|ZijZ&bXZ{ope¼¢u{|Z&rslZmUXZMopYgjrXbAmgj¤b8rXYZ{|cqlope
lgjYXrmom|cdgkbst¢k¥#UXcqlU3chtrstrsopehedv[yArXc¹m|Z¶ofeqop{|ijZV {ojlmchgjbgp*m|UXZm|gpm|ojesZµ8ZGlrm|cdgkbmchYZnZ&t6Z&lcqopehehvcd
YDrXedmcd»¼X{|Z&lchtchgjb<lgjYXrXmchbXi3chtcdb8jg=<jZGN¢Xopbs6gkt|twchxXehv'opeqtg[{|Z {opchbMmUXZrJtwZfgj>lgjYXehchl&omZGMsom|o
twm{|rslmrX{|Z&t{ZGtwrse¹m|cdbXi¶cdbDedg¥ªtsoklZ,{|Z&yArXch{|ZYZbAm|t&?>ng¥Z&jZ&{&¢G{ZGrslcdbXinmUXZVZ&ij{|ZZ,gjJopbDopehijgj{|cdmUXY
Y3oOvcdbJl{|Z&ojtZcdm|tnmchYZDlgkYXedZµc¹m~v<chb¨mUXZ[#Z&ojen,ÆYgZe¼:sgjehedg¥#chbXi %cdgjmwmo§Zm ope¼?-hj%/¼¢6¥Z
lgjbstcqZ{amUsZ§Zik{Z&Z[gjm|UXZ3X{|Z&cqlopmZGtaoktopb¾ojsc¹m|cdgkbsopeHYZGojtrX{Zgj,mUsZ§lgjYXehZµcdm~vQgpVX{|gjx»
ehZY3t#ojbs'ojedikgj{|c¹m|UXY3t¢opbJ§chbkm|ZbsMm|gDZ&edrslchsomZam|UXZ {Z&ehopmchgjbstUXch'x6Zm~¥¤ZZbMmchYZ»®lgkY[sedZµ8cdm~v§opbJ
XZij{|ZZDgp£m|UXZ[X{ZGcqlom|Z&t&f#Zeqom|Z&Q{|Z&tZ&oj{|lUQl&opbQx6Z[ gjrXbJQchbA@ b8rmU t tZYcdbJope¥gk{B<C-hED0/£opbJ
chb¨twgkYZa{|Z&lZbAm#sop6Z{tF-s¢9X¢s/¼
¬®bMmUXcqtVsojJZ&{&¢¥¤ZalgjbstcqZ{¤mUsZas{gkxXedZ&Y gp{Z&Jgk{wm|cdbXiDmUsZf§cdbAm|Z{twZGlmchbXi[sopch{t¤ojYgjbXioDtZm
gj
YgjbsgpmgkbXZlrs{kZ tZikYZbAm|t& m|cdY3oje¿[À
ÃhÄjÅ
'Á
6ÂF»Fm|cdYZojedikgj{|c¹m|UXY3t#op{|Z;<Absg¥#bM gk{#mUscht
s{gkxXedZ&YG-H8A¢¤%/ÈI>g¥¤ZkZ{fm|UXZ&tZ'ojedikgj{|c¹m|UXY3tfrstZ'X{|Z&cqlopmZGtgp#UXchijU³Zik{Z&Zj¢Zjis¨m|g$lgkY[Jop{|Z
m|UXZojxstlcht|t|opZfgp>m~¥¤g§chbkm|Z{twZGlm|cdgkb6gjchbAm|t¶gj{#m|g§ehg8l&om|Zopb¨chbAmZ{tZ&lm|cdgkb¨JgkcdbAmn¥#cdmU${ZGtw6Z&lmmg'o
kZ{mcqloje%tweqopx%nTVUXZ&tZs{ZGchl&om|Z&t#UsoOkZo§Z&ij{|ZZojbs¨opb²opehijZxs{|ojchlflgjYXehZµc¹m~vMmUJomaoj{ZfrJtwrsojedehv
UscdikUXZ{nmUsojb¨mUXZDcdbAm|Z{twZGlmchgjb¨s{ZGchl&om|Z§¦¤mUXcqtachtncdbQJop{mcqlrXeqop{nmUXZ[lojtZf gk{nehcdbXZ[twZ&ijYZbAmtnopbJ
lcd{lehZ#twZ&ijYZbAmtJ-/È rs{,opehijgk{cdmUXY3tgjbsedvDrstwZVm|UXZ#chbkm|Z{twZGlm|cdgkbDX{|Z&Xchl&omZ#ojbsDm~¥g gpm|UXZ{twchYXedZ
s{ZGchl&om|Z&tV¦mUXZs{ZGchl&om|ZVmUsopmtwgk{wmtm~¥gf6gjchbAm|tHx8vopxst|lcqt|tojZj¢popbJ[mUXZs{ZGchl&om|Zm|Usom¤toOvt>cdNo
6gjchbAm%cqtux6Zehg¥f¢&gkbu¢Ogj{%ojxJgkZ£o#twZ&ijYZbAm&%¬®bsop{mcqlrsehoj{&¢&¥Z,Xg#bXgpmlgjYXrm|Z£m|UXZ,op{|{opbXikZYZbAmNbXgj{
m|UXZ m{op6Z+&gjcqXope%Yoj<gpmUXZtZikYZbAm|t&K,<gk{Z&gjZ{G¢kmUXZs{ZGchl&om|Z&tV¥¤ZfrstwZXg3bXgpm¶t|oOv'ojbAvAm|UXcdbsi
ojxJgkrmum|UXZ,b8rXYDxJZ&{%gj{%mUXZJgAtwcdmchgjbJtugp8m|UXZ,chbAmZ{tZ&lm|cdgkb JgkcdbAm|t&¢popbJamUXZ,mchY[Z¤lgkY[sedZµ8cdm~vagp8mUXZGtwZ
ojedikgj{|c¹m|UXY3t¶XZ6ZbsXt gkbXehv¨gjbQm|UXZb8rXYx6Z{ gp¤cdbAm|Z{twZGlmchbXiMsopch{|tagjtwZ&ijYZbAmt¢NbXgjm¶m|UXZb8rXYx6Z{
gjchbAmZ{tZ&lm|cdgkbM6gjchbAm|tf·Fc¹¸*Z{|ZbAm|edv' {gkY mUXZfgjmUXZ&{#bXgjbm|{ch8chojeuopehijgj{|cdmUXY3tF-\¢288¢6s¢s¢u/ º
Z 4s{t~m£s{ZGtwZ&bkm,ojb3opehijgj{|cdmUXY¡ gk{HmUsZ{ZG8»ÈxXedrsZ#Ooj{cqopbAmHgp*m|UXZ#X{|gjxXehZY/¥#UXZ{|ZVmUXZ¶twZ&ijYZbAm|t
oj{ZMlgjehgj{|Z&¾xsedrXZgj{[{|Z&ªtg¨mUJom[mUXZ&{ZMcht[bXg²cdbAmZ&{|tZ&lmchgjbªojY[gkbXiQm|UXZxXedrsZMtZijYZ&bkmtDojbs«bXg
chbAmZ&{|tZ&lmchgjb«opYgjbsiMm|UXZ§{|Z&¾gkbXZ&t&MTVUXZ§mchY[ZMlgjYXehZµcdm~v$cqtD¿[ÀÀ
$Á
6ÂkÃdÄkÅ
¢¥#UXcqlU«chtoQedgki
?oklm|gj{H {|gjY gjm|cdY3ojeFHTVUXcqt£{|Z&trXedm,ijZ&bXZ{opehc+&Z&t£otwchYcdeqop{£{|Z&trXedm£ gj{,mUXZaloktwZ#gjustwZ&rsgj»ÈtZikY[Z&bAm|t&¢
c¼ZjtZijYZ&bkmtm|UsomfchbAmZ{tZ&lmacdb¾om YgktwmagkbXZ6gjchbAm -0/ÈTVUXcqt opehijgk{cdmUXYÓcqt¶mUXZ&b$rstZ&Qmg<twgkedkZ
m|UXZ ijZ&bXZ{ope*loktwZacdbMmchYZ¿[À
Ç
ÃdÄkÅ
Â|¬®bMmUsZ l&ojtZ¶gjstZrJgp»®twZ&ijYZbAmt¢A¥ZaX{|gj6gktZ¶o[x6Zmwm|Z{
ojedikgj{|c¹m|UXY¥#UXgAtwZ[mchYZ§lgkY[sedZµ8cdm~v¨cqt¿[ÀwÀÈ
Á¯
ÂjÃdÄkÅ
籂
6Â|§TVUXZ&tZxJgkrXbsXtfojedYgAt~mfY3omlU
m|UXZ¿[À
Ç
Áª
ÃdÄkÅ
ÂHedg¥¤Z{Vx6gjrXbs -/È
M "
%Zm x6Z¨o$tZm3gp
Y[gkbXgpm|gjbXZtwZ&ijYZbAmt«ÓtwZ&ijYZbAm[cht & 2 2[cdncdmcqtDmUXZij{opsUªgp¶o
Jop{mcqopehedv©Z%4sbsZ&«rXbscdop{|cqomZ§ rXbslm|cdgkb· c¼Zj$ojbAv©jZ&{wm|chl&ope£ehchbXZ'chbAmZ{tZ&lm[twrslUªo$twZ&ijYZbAmchb½om
YgAt~m#gjbsZa6gjchbAmº,TVUXZ X{|gjxXehZYÉcqtVm|g[{|Z6gj{mVmUXZ'sojcd{tVgp>twZ&ijYZbAm|tVgj mUJomchbkm|Z{twZGlmG
%Zm¶ojbsJxJZ¶m~¥g[twZ&ijYZbAmt£gp opbs'edZm'ojbsfx6Zm~¥¤gZ&bs6gjchbAm|tgp%tgjYZ¶tZikY[Z&bAm|t£gj
§¢jbXgjm,bXZ&lZ&t|toj{chehvfgp#gk{ %ÀNÂojbs*ÀNÂ%XZbXgjmZ#m|UXZlg8gj{chbsom|Z&t>gj%¢kojbsjÀuÂ>ZbXgjmZGt>mUXZ
6gjchbAmgjn¥#UsgktZ¶opxJtlcht|to cqt©·?c¹trslU'oJgkcdbAmVZµchtwm|tº Z¶lgjbstchXZ{£m|UXZ¶ gjehedg¥#chbXiX{|Z&Xchl&omZGt¦
ÐAØÏ k&Ð! J "#%$
&('
ÐAØÏ k&Ð*)N %ÀNÂ,+!%À-AÂ
ÐAØÏ k&Ð/.u *ÀuÂ0+1*À-kÀ2ÀuÂwÂwÂ
ÐAØÏ k&Ð43% *À-kÀ2ÀuÂwÂwÂ+!*À-FÀ2ÀNÂÂwÂ
ÐAØÏ k&Ð*5N6 87:9%ÀuÂ<;<%ÀAÂ=trslUmUsopmjÀuÂ
&
?ÀuÂ
zH{|Z&Xchl&omZ$§chtDY3opbssomgk{vk¨zH{ZGcqlom|ZM\¨ojedehg¥t mg²twgk{wmm|UXZ§Z&bs6gjchbkmt²zH{ZGcqlom|Z'<m|Zeheht
¥#UsZmUsZ{¶ojbQZbJJgkcdbAmaedchZ&taopx6gjZfgk{nx6Zehg¥ o§twZ&ijYZbAmGnzH{|Z&cqlopmZ9'X{g8cqZ&t#mUXZ[gj{Z{ngpHm~¥¤g
tZikYZbAm|topehgjbXiojZ{mcqlojesehcdbsZnsokttcdbsifmUX{|gjrsijU'ojb3Zbs6gjchbAm&£zH{|Z&Xchl&omZalUXZGl<8t,¥#UXZm|UXZ{m~¥¤g
tZikYZbAm|t¤chbkm|Z{twZGlm#¥#cdmUXchbQo[jZ{mcqlojeNtweqopx<Z%4JbXZ&x8v3m~¥¤g[Z&bs6gjchbAm|t&
Z[Xg§bXgpm tw6Z&lc¹ vQo§X{ZGlcqtwZ{Z&X{ZGtwZ&bAm|om|cdgkb< gj{nmUXZ[twZ&ijYZbAm|t&¢6¥#UXchlU$YoOv<Z&JZ&bsQgkb¨mUXZ
ojXXehchl&om|cdgkbu :Xgk{§ZµXojY[sedZk¢,m|UXZ¨ rXbslmchgjb okttglcqomZGªm|gªo«twZ&ijYZbAm3Y3oOv½x6Z¨ijchjZ&b±ZµXedcqlcdmehv
m|gjikZmUsZ{,¥#cdmU'mUXZachbkm|Z{|opes¥#UsZ{|Zncdm¤cqtXZ%4sbXZGN¢8gj{mUXZn rXbslmchgjb'Y3oOvx6ZnijchjZ&b§chY[sedcqlcdmehv§ojt,mUXZ
ojedikZxX{opcqlH rsbslm|cdgkb3gp6XZij{|ZZ?> m|Usom£chbAmZ{|6gjeqomZGt%>
ÁA@
ikcdkZb[JgkcdbAmt· chbslehrschbXi m|UXZVZbs6gjchbAm|tHgj
m|UXZtZikY[Z&bAmº mUXZ&{{|ZX{|Z&tZbAm|opmchgjbst#oj{ZojehtgJgAttchxXedZk¤TVUsZZik{Z&ZfgpmUXZs{ZGchl&om|Z&tledZGop{|edv
XZ6ZbsXtgjb©mUsZ'lUXgktZb©{|ZX{|Z&tZbAm|opmchgjbu'¬®b²m|UXZ3mopxXehZ3xJZ&edg¥Ó·?tZZMopeqtwgC-X¢% /ôº¢%m|UXZ'Z&ij{|ZZ&t gj
z£{ZGchl&om|Z&t\»®op{|ZijchjZ&b gj{,ehchbXZ¶tZikY[Z&bAm|t£{|Zs{ZGtwZ&bkm|Z&x8v[mUXZalg8gj{chbsom|Z&tHgjNmUXZ&cd{¤ZbJJgkcdbAmt¢
gk{Usoje¹ô»®lch{|ledZGtZ4sbXZ&ªxAv¾lZ&bAmZ{t&¢{ojchc£XehrstDo¨xJg8gkedZGopb ·ôm|g$cqt~m|cdbsijrXcqtwU«x6Zm~¥¤ZZ&b¾mUXZMrXX6Z{
ojbs3m|UXZ edg¥¤Z{#oj{|lGº¢A gj{#lch{lehZatZikY[Z&bAm|tVZ%4JbXZ&Mx8v3mUX{|ZZa6gjchbkmtf· chbslehrschbXi[mUsZ¶m~¥¤g[Zbs6gjchbAm|t
gj£mUsZ[tZikY[Z&bAmº¢uojbs¨ gk{alrX{|jZ&taZ4sbXZ&²x8vQo§6gjehv8bXgjYcqope>Z&yArsopmchgjbA
&CB
ÀuÂ| xstZ{|jZm|Usom
m|UXZaX{ZGcqlom|Z&t¤op{|Zngk{|Z&{ZG3x8v3cdbJl{|Z&ojtchbXiZ&ij{|ZZ&tchbMopehe6mUXgAtwZaloktwZGt >g¥¤ZkZ{G¢jm|UXcht¤gk{|Z&{Y3oOv
x6Zfcd¸6Z&{Z&bkmV gk{twgkY[Zagpm|UXZ{Vm~v86Z&tVgjtZijYZ&bkmt :Xgk{#cdbJt~mopbslZj¢8 gj{VUsoje¹ô»®lch{lehZ&tVZ%4JbXZ&Mx8v3mUX{|ZZ
bsgjbs»¼ZµAm|{Z&YZn6gjchbkmt¢Am|UXZ¶Z&ij{|ZZ¶gp%zH{|Z&cqlopmZ \fcqt¤\'Df¥#UXchedZ¶mUXZ Z&ij{|ZZgj%zH{ZGcqlom|Z¶chtgjbXehv
9J
zH{ZGcqlom|Z Z&ij{|ZZ
ehchbXZ Usoje¹ lcd{lehZftwZ&ijYZbAm 6gjehv8bXgjYcqope
tZijYZ&bkm lch{|ledZ 6gjchbAm|t gpXZij{|ZZf
\ \ G\
\
\ \ 9
9 9 G\
9 G\
TVUXZ[mopxXehZDx6Zehg¥ ikcdkZ&tnmUXZ<8bXg¥#bQehg¥Z&{aojbsQrsXJZ&{fxJgkrXbsXt¶ gk{a{|Z6gj{mchbXitwZ&ijYZbAm cdbAmZ&{w»
tZ&lmchgjbst#XZ6ZbschbXi3gkb'm|UXZfX{|Z&cqlopmZ&tm|Usomnoj{Z rstZ&N
zH{|Z&cqlopmZGt mchYZlgkY[sedZµ8cdm~v {Z Z{|ZbslZ
edg¥¤Z{Vx6gjrXbs rXX6Z{x6gjrsbs edg¥¤Z{Vx6gjrXbs rXX6Z{#x6gjrXbJ
% Í Î Í Í ÎШ&Ð Ë Ð8Î6G
@
m|UXcqt#sop6Z{ m|{ch8choje
am|g
Ç Á«
ÃhÄjÅ
Ç
¤ÃhÄjÅ
gkcht|t» Ñ bXg8Zvk2-0/ mUXcqt#sojJZ&{
am|g.9
Ç
Á«
ÃhÄjÅ
Ç
¤ÃhÄjÅ
m|UXcqt#sop6Z{ mUXcqt#sojJZ&{
am|g3
ÃdÄkÅ
'Á
ÃhÄjÅ
§Á
Ù Uso +k»®Ú£Z&ehtxu -6/
opeqopxsojbI-h/
%Í Î Í Í ÎÐ<Ð8ØuÖ&ÐQGÐ Ë ÐÎ6&
am|g
ÃdÄkÅ
'Á
À
§Á
6ÂjÃhÄjÅ
mUXcqt#sojJZ&{
am|g3
ÃdÄkÅ
'Á
ÃhÄjÅ
§Á
ZkiJ Ù Usopb -/
% Í Î Í Í ÎÐ Ì GÐ %Ø Í Ö&Ð Ë ÐÎ6&
am|g
Ç
Á«
ÃhÄjÅ
Ç
Á À
'Á
6ÂkÃdÄkÅ
mUXcqt#sojJZ&{
am|g.9
ÃdÄkÅ
'Á
ÃhÄjÅ
§Á
opeqopxsojbI-h/
%Í Î Í Í ÎÐ<Ð8ØuÖ&Ð Ì &Ð Ø Í Ö|GÐ Ë ÐÎ6G
am|g
ÃdÄkÅ
'Á
ÃhÄjÅ
§Á
gkcht|t&» Ñ bXg8Zvk -0/
/# $ .
"!$#&%'(*),+-!$.
¬È¤¥ZgkbXedv$rstwZ[m|UXZX{ZGcqlom|Zm|UsomlUsZ&l<ta¥#UXZm|UXZ{fgk{absgpm m~¥g<tZijYZ&bkmtacdbAmZ&{|tZ&lm&¢Nm|UXZ3ehg¥Z&{
x6gjrsbs§gkb§mUsZnm|cdYZflgkY[sedZµ8cdm~vgpojb8vojedikgj{|c¹m|UXY mUsopm¤{|Z6gj{m|t,mUsZ¶chbAmZ{tZ&lm|cdbsisojcd{t,cqt/aÀ
Â
¬®bJZZGN¢AlgjbstcqZ{ 4JijrX{|Zf#¥#UXZ{|Z¥¤Z#UsoOjZVm~¥¤gtZm|t 0Vojbs
gj21
twZ&ijYZbAm|t£Z&ojlU3pmUsZb8rXYx6Z{
gj£cdbAm|Z{twZGlmchbXi§Jopch{|taopYgjbsi3mUXZ[twZ&ijYZbAmtngj, 0fcqt 40p¢Nopbs<m|UXZb8rXYDxJZ&{¶gj,chbkm|Z{twZGlm|cdbXi'sopch{|t
ojYgjbXi'mUXZ§tZikY[Z&bAm|tagp
chtf
Z3l&opb$ikcdkZm|gQ 0 opbJ©
opb8v¨opehrXZedg¥¤Z{¶mUsojb 165
7
uZm
& 0 Á
¶ttrXYZm|Usom¶mUXZ&{Zcqtnopm¶YgktwmngkbXZchbkm|Z{twZGlm|cdgkb<ojYgjbXi3m|UXZJopch{|t#gj, 098
#¨bQgk{|Z&{#mg
XZ&lcqZ[¥#UXZm|UXZ{¶mUsZ{|Z[oj{Z<gj{f
Á(@
cdbAmZ&{|tZ&lmchgjbst&¢*¥ZDbXZZG¨m|g'm|Z&twm;: 0<8
: &
1=5
>
sojcd{tngj
tZikYZbAm|tHchb3m|UXZn¥¤gj{t~m»Èl&ojtZj¢ptcdbJlZncdmcqt,ojed¥VoOvt>JgAttchxXedZnmgoj{{opbsijZ¤mUXZatwZ&ijYZbAmtHcdbMtwrslU§o¥¤oOv
m|Usomno[bXgkblUsZ&l<jZG'Jopch{¤chbAmZ&{|tZ&lm|t&
n/2 segments k1 intersections
? E1
E2 k2 intersections
n/2 segments
:chijrs{Zj¦ %g¥Z&{x6gjrsbs¥#c¹m|U<z£{ZGchl&om|Z3agkbXehvj
"!$#&%'(*),+-!$ !$
gkcht|twgkbXbsopm¤opbJ Ñ bXg8Zv8cdb(< -/%UsoOjZ tUXg¥#b§m|Usom/aÀ
Ç
Á«
ÃhÄjÅ
Â>cqtoDedg¥¤Z{Vx6gjrXbJMcd¥¤Z gjbXehv
rJtwZVzH{|Z&cqlopmZ&t¤k¢k\nojbsszH{|Z&Xchl&omZ 9aX{|gAcqZGtum|UXZVgj{Z{gpXm|UXZ#twZ&ijYZbAmtojedgkbXi¶o¶jZ&{wm|chl&ope8ehcdbXZ
Jojt|twchbXiDmUX{|gjrXikUMojb'ZbJJgkcdbAm#gp>twgkYZ tZikYZbAm&>¬ÈmtwZ&ZY3t¤bsom|rX{ope6mg3tZZa¥#Usom#cqt¤mUXZ 6g¥Z&{¤gj
m|UXcqts{ZGchl&om|Zaojbs§cdumUsZaehg¥¤Z{¤xJgkrXbs§gkb§m|UXZnm|cdYZflgkYXedZµc¹m~v§l&opb§x6Z cdYX{|gjZG3c¹c¹m#cqt¤okXZG
m|gm|UXZamUX{|ZZ4s{t~mVX{|Z&cqlopmZGt ! b gk{wm|rXbsopmZehvj¢Xo[tedchijUAmVY[gc4Jl&omchgjbMgpum|UXZ X{|gAgjugp
gkcht|tgjbXbsopm
ojbs Ñ bXg8Zv8chb<'tUXg¥tmUsopmnzH{ZGcqlom|Z 9XgAZGtVbXgpmojedehg¥¯mg3chYX{|gjZm|UXZfehg¥Z&{¤x6gjrsbsN Ù gjbJtwcqZ{
4JijrX{|Za\¥#UXZ{|Z¶¥¤ZnUJoOjZ¶o$4s{t~mVij{|gjrX'gptZijYZ&bkmt,gkb3mgk'gjumUXZ4sikrX{Zk¢opbs§o[tZ&lgkbs3ik{gkrX
gj
twZ&ijYZbAmtgkb¨o[t|opYZ ehcdbXZfx6Zehg¥ m|UXZtZijYZ&bkmt¤gpmUXZ 4s{twmVik{gkrXu
m segments
n segments
:cdikrX{|Zf\X¦ ug¥¤Z{VxJgkrXbs¥#cdmU<X{|Z&cqlopmZ&t
@
Ú£oklU'gjumUXZ twZ&ijYZbAmt,gkb§mgk'gjNmUXZF4sikrX{Z¶Usojt¤oDjZ{|v[bsop{|{g¥´6gj{mchgjb3m|Usomg8ZGt,bXgjm¤chb»
m|Z{twZGlm>opb8v jZ{mcqlojekehchbXZJojt|twchbXi#m|UX{gkrXijUDopbZ&bs6gjchbAm&¬®bgj{Z&{umg ZGlcqZ¥#UsZmUsZ{gj{bXgpmmUXZ&{Z
oj{Zagjbsedv
0
sopch{|t¤gjtZijYZ&bkmtmUsopmVchbAmZ&{|tZ&lmf·?cFZjHgkbXehvmUXZtZikY[Z&bAm|t¤gpmUXZF4s{t~m#ik{gkrXJº¢
¥¤Z[UsoOjZmg<lUXZ&l< gj{ chbAmZ{tZ&lm|cdgkbu¢*cdb²mUsZD¥¤gj{t~m»Èl&ojtZj¢JmUXZ &
/aÀ
Ç
6¤Jopch{|t lgkbstchtwmchbXi'gj
o<tZikY[Z&bAmgp,m|UXZ 4s{twm ik{gkrX¾opbs¾o<twZ&ijYZbAmgjm|UXZ§twZGlgkbs$ik{gkrXu3¬®bsXZZ&u¢%cdmchtJgAttcdxsedZ[mg
XZ gj{|YÉm|UXZtZikY[Z&bAm|t#gjm|UXZ;4s{t~m¶ij{|gjrX¥#c¹m|UXgjrXmnYg8Xc¹ v8chbXi3mUsZch{¶lgkYxXchbsopmgj{|cqopeNop{|{|ojbXijZ&YZbAm
tg3m|Usom o3ijchjZ&b<tZikYZbAm 0 gjHmUXcqtnik{gkrX<Usojtnc¹mtnYchbXchYrXY A»Èlg8gj{cdbJomZfchYYZ&cqom|ZehvojxJgkZ
ojb8v'twZ&ijYZbAm
gpmUXZDtwZGlgjbJ'ik{gkrXu UXZmUXZ&{gj{#bXgjm 0 opbs8
cdbAm|Z{twZGlmbXg¥ Z6Zbsst#gjbXehv
gkbQm|UXgktZm~¥¤g<twZ&ijYZbAm|t&¢N¥#UXcqlU²Y3o'<jZ&t¶bXZGlZ&t|t|op{|vMm|g¨lUXZ&l<<m|UXZsopch{&DTVUXZ[mchY[Z3lgjYXehZµc¹m~v¨gj
m|UXZfX{|gjxXehZY cht¤m|UXZ{|Z gk{Z</aÀ
Ç Áª
ÃhÄjÅ
Â
Ð Í OÐ Ë % . 2B % C "B%/aÀ
Ç
6Â
% A %
° < . #
¬®b<m|UXcqt¶tZ&lmchgjbu¢J¥ZokX{|Z&t|t#m|UXZtJZGlcqopel&ojtZ¥#UXZ&{Z¥¤ZUsoOkZ m~¥gMtwZm|tgj£twZ&ijYZbAm|t ojbs8
· mUXZ¶{ZG3opbs[m|UXZxXehrXZatwZ&ijYZbAm|tº>trslUmUsopmbXgm~¥gtZikY[Z&bAm|tHgjNmUXZnt|opYZntZmcdbAmZ&{|tZ&lm&¬®b3mUscht
tZ&lmchgjbu¢"!#n·?{ZGtw%$!%ºZ&bXgpm|Z&t¤oxXehrXZD·?{ZGtw%{|Z&Jº£tZijYZ&bkmG Za¥#cdeheJs{gkZ#mUXZn gjehedg¥#chbXi{|Z&trXedm&¦
Ð Í OÐ Ë )'& )(*+ ( "# & ( ( % ) ,*-%& .*J#H# ¡¿[ÀwÀ
±Á
*ÂjÃdÄkÅ
 & /D¿[À
 ' 2B
@
;0A 132
ngpmcqlZ[m|Usom mUsZ[ehg¥¤Z{axJgkrXbs²gp¤tZ&lmchgjb©Mg8ZGt bXgpmfUXgkeh$cdb²mUXZ{|Z&8»ÈxXehrXZlojtZ¦nm|UXZ3xJZGt~m
<8bXg¥#b<ehg¥Z&{#xJgkrXbs<chb<mUXcqtl&ojtZfcht¶t~m|cdehe /aÀ
ÃhÄjÅ
§Á
6Â|Vnb¨rXX6Z{x6gjrsbsmUsopmncqtx6Zmwm|Z{nmUsojb
m|UXZfgjbsZ ojxJgkZ¶cqt <8bXg¥#b' gj{#{|Z&8»ÈxXehrXZ stwZ&rsgj»ÈtZikY[Z&bAm|tF- /¼
>nZ{|Z¥ZnX{ZGtwZ&bAm,ofbXZ¥´XeqopbsZntw¥¤ZZ&§opehijgk{cdmUXY gj{£m|UXZn{|Z&»¼xXehrXZ#X{|gjxsedZ&Y> kZ{mcqlojestw¥¤ZZ&
ehchbXZQYgjZGt3ojl{|gkt|t[mUsZ$XeqopbXZ< {|gjY2ehZôm'm|g«{chijUAmG UXZ&b cdm'{ZGojlUXZGt§opb´Z&bXJgkcdbAmG¢twgkY[ZQsom|o
twm{|rslmrX{|Z&toj{ZrX*XopmZ&½opbs©chbkm|Z{twZGlm|cdgkbstlojb¾xJZ§{|Z6gj{mZGN<¤mDopb8vQm|cdYZk¢¥¤Z§gjbsedv©lgjbJtwcqZ{
=% #tZikY[Z&bAm|t¤¥#UXcqlU<oj{Z¶mUsZtwZ&ijYZbAm|t¤m|Usomnl{gAttm|UXZtw¥¤ZZ&MehchbXZj
4 Ð65>ÎÏôGÏ Í Î 87 79 ( %% ;:<! "B=#> ( % ) ?*B A@ !*-H ( %'&(
!# 1 %'&( B!% (+ C C"D) 1" ( E!# ,*# !% 1 I )= ;
%/ !%F
¬®bmUXZ gjehedg¥#chbXintrXxstZ&lmchgjbst&¢O¥Z¤Z&t|l{|chxJZVopbojedikgj{|c¹m|UXYmUsopm{|Z6gj{m|tumUXZ =7Gfchbkm|Z{twZGlm|cdgkbst
TVUsZgpm|UXZ{cdbAmZ&{|tZ&lmchgjbst,op{|Z#{Z&Jgk{wm|Z&x8vm|UXZnt|opYZnojedikgj{|c¹m|UXY/opômZ{ZµXlUsojbXijchbXi mUsZgj{|cdZ&bAm|om|cdgkb
gjXmUsZvA»®oµchtgj{mUsZ¤lgjehgj{t%gjXmUXZVtZijYZ&bkmtTVUXZVopehijgj{|cdmUXY mgax6Z¤ZGtl{chx6Z&cqtmUXZ&{Z gj{|ZojXXehcdZG
gkrX{VmchYZ&t&
4
Ð65>ÎÏôGÏ
Í
Î()
'& % C H*B ' )#@$ I %0; $@ (IH 9
) "+ ),+ 6
:sgj{mUXZVt|o <jZ,gpsleqop{|cdm~vj¢OedZmrJt 4J{|twmlgjbstchXZ{>ontcdYXehZ{X{|gjxXehZY ¥#UXZ&{Z¥ZUsoOjZgjbXZxXedrsZ¤tZikY[Z&bAm
gkbXehvj¶TVUsZDchbAmZ{tZ&lm|cdgkbstnoj{Z{Z&Jgk{wm|Z&<x8v¨lopeqlrsehopmchbXi3mUsZ 7§tZijYZ&bkm¶gp !# TVUXcqt twZ&ijYZbAm&¢
XZbXgjmZ&
B
À!#¢chtgkxm|ojcdbXZG½x8v«XrJtwUXchbXi !#g¥#b8¥¤oj{|stDoktDYDrslU´ojt[JgAttchxXedZk¢,X{|g8chXZ&«m|Usom
B
ÀA! Â <jZ&ZstnmUXZ3t|opYZehZômfZ&bs6gjchbkmokt! opbs²g8Z&tabXgjmachbAmZ{tZ&lm twZ&ijYZbAmtnm|UsomD! g8Z&tabXgpm
chbAmZ&{|tZ&lmf·FtwZ&ZF4sijrX{|Zfkº
e
f(eb)
b
:cdikrX{Zfs¦£TVUXZaÒsg8gj{tZijYZ&bkm· mUXZlcd{lehZ&t#lgj{|{ZGtw6gjbs§mgijg8gMcdbAmZ&{|tZ&lmchgjbstº
TVUXZ²Òsg8gj{twZ&ijYZbAm<lopbxJZ¾lojehlrXehopmZG¯rstcdbXi±gjbsedv±mUsZI4s{twm'mUs{Z&Z$X{|Z&cqlopmZGt5:cd{t~m¥Z
XZmZ&{YchbXZ#m|UXZn{|Z&3tZikY[Z&bAm£mUsopmcqt£chYY[ZGcqomZ&edv[x6Zehg¥³m|UXZnehZôm¤ZbsXJgkcdbAmgj ! >TVUXZb'¥ZYgkZ
Xg¥#bA¥Vop{XtrXbAmcheX¥¤Z#{ZGojlUDoa{|Z&tZijYZ&bkm>m|Usom,g8ZGt>bXgpm£chbAmZ{tZ&lm ! ZVmUXZ&b3YgjZVopehgjbXiamUscht
tZikYZbAmfrXbAmche,cdm|tf{|cdikUAm Z&bs6gjchbAm&MTVUschtfs{glZGrX{|Zchtf{|Z6Z&opmZG²rsbkm|cde,¥Z3Z&bslgkrXbAmZ{ m|UXZ§{|cdikUAm
YgAt~mV{ZG§Z&bs6gjchbkmG£gjmZamUsopm
B
À!#ÂHUsokt,m|UXZftojYZatZm#gpijg8g§chbkm|Z{twZGlm|cdgkbstVojt !# >Z&bslZk¢¥Z
}~rJt~mbXZ&Z&Mmg{|Z6gj{mVmUXZGtwZ chbkm|Z{twZGlm|cdgkbstV¥#UXchedZlgjbstwm{|rslm|cdbsim|UXZ Òsg8gj{tZikY[Z&bAm&
¬®b²mUXZ[ijZ&bXZ{ope>loktwZD¥#c¹m|U©tZkZ{opexXehrXZ[tZikYZbAm|t&¢*c¹mtZZY3t c3§lrXedm mgZ%3§lchZbAmehvQlgkYXrmZ
m|UXZ Òsg8gj{tZikYZbAm|t#oktVmUXZ&vMoj{ZfZ4sbXZ&<ojxJgkZj Z mUXZ&{Z gj{|Za{|ZeqoµMgjrX{Z4sbXcdmchgjbQojbs<opehedg¥o
ÒJgAgk{#twZ&ijYZbAm®ÀA!%ÂHmg3chbAmZ&{|tZ&lmE!#¢XX{|g8chZG3m|Usom#m|UXZfcdbAm|Z{twZGlmchgjbUsokt#oDeqop{|ijZ&{#opxst|lcqtt|omUsojb
m|UXZfbXZµAm#Z&bs6gjchbkm#m|g3xJZfZ&bslgkrXbAmZ&{ZGN
+-!$+=!$() !
TVUsZ3chZGo<chtfmgQ{|Z6gj{m mUXZ§ikg8g8©chbAmZ{tZ&lm|cdgkbst ¥#UXchedZ3srstwUscdbXi<m|UXZ§xXehrXZ'tZikYZbAm|tfg¥#b8¥Vop{Xt&
Ú,ojlU±xXedrsZ$twZ&ijYZbAm3cqt§t~m|gj{|Z&³cdb¯o¾tZm[ÀA!GpÂf gj{§tgjYZQojlmchjZ<{|Z&±twZ&ijYZbAm !%TVUXZQtZm§gj
tZikYZbAm|tfchb[ÀA! ¶¥#chedexJZMY3opchbkmopchbXZ&«XrX{chbXi¨m|UXZ't¥Z&Zu<¬ÈmlgjbJtwcqt~mtfgp#m|UXZ§xXehrXZtwZ&ijYZbAm|t
m|UsomVlojbXbXgpmxJZ¶XrstUXZ&§Xg¥#b3xJZGloprJtwZ#m|UXZv3oj{Z#xXehgl<jZGZcdmUXZ&{x8v!
gj{,x8vDmUXZ¶edg¥¤Z&twmtZikY[Z&bAm
gj[ÀA!
Â
TVUXZ§ojxJgkZ01xJZ&edg¥ {|ZeqomchgjbJtwUXch©¥#c¹m|UXcdb/
·?{ZGtw%
º¶chtfosoj{wm|choje>gj{Z{G¢N¥#UXcqlU²cqtfoMmgpmope
gk{|XZ{£¥#UXZ&b3{ZGt~m|{cqlm|Z&[m|gojlmchjZntZikY[Z&bAm|t& Zn¥#chede6Y3opchbkmopchbmUXZngj{Z{|Z&[ehchtwmgj6m|UXZ¶oklm|cdkZ#{ZG
tZikYZbAm|t¤chb<o[xsojehojbslZGMtZ&oj{|lU'm{|ZZV}~rstwmehc<kZachb Ù Usopb t#ojedikgj{|c¹m|UXY -/N gj{#{|Z&8»ÈxXehrXZtZijYZ&bkmt
:sgj{aZ&ojlU¨xsedrXZtZijYZ&bkm !%¢*¥ZDbXZZG¨m|gMlgkbstchZ&{¶o'{ZG¨twZ&ijYZbAm ~À!#Â,mUJom XehoOvtmUXZtojYZ
{|gjehZojtHmUXZnÒsg8gj{,tZikY[Z&bAm£cdbm|UXZnX{|Z8chgjrst,twZGlmchgjb%!>ng¥Z&jZ{G¢F!#HY3oOvDcdbAmZ&{|tZ&lm ®ÀA!#Âojbs[cht,gjbXehv
{|Z&yArXch{|Z&3m|g§topmcqt~ v§mUXZ gkedehg¥#cdbsiD¥¤Z&o'<jZ{#lgjbscdmchgjbJt¦
Í6Ì ÐAO|Õ( & 79 ( %% (; ! @ C B. & % H 6*B #0®ÀA! Â# *%
%#7 + : <! "
>F
Í6Ì ÐAO|Õ ) 7 $ 1 $+@ 2 ~À! Â=B 0%3@ B% .# ) # '
'&( )
:sgj{,ZGojlU§oklmchjZn{ZG3twZ&ijYZbAm !%p¢A¥¤ZnZbsgpmZ¶x8v [ÀA!%Â%m|UXZ¶tZmgp%opeheJoklm|cdkZxXehrXZ¶tZikY[Z&bAm|t !#
trslU3m|Usom~À!#Â
&
!%p>¬Èm¤cqtVt~m|gj{|Z&3chbMo[Xopm|otwm{|rslmrX{|Znm|Usom#opehehg¥t£rXbXchgjb%¢AYchbXchYrXYÉZµ8m{ojlm|cdgkb
ojbstrXXs{ZGttcdgkbchb¿[ÀFÃdÄkÅ
Â6mchYZ· ZjisonxscdbXgkYchojeUXZ&ojJº>TVUXZ¤rXbsZ&{ehv8cdbXiagj{Z&{cqtmUXZVkZ{mcqlope
gk{|XZ{gj#oklm|cdkZ3{ZG«twZ&ijYZbAm|topehgjbXi<m|UXZMt¥Z&Z¾ehcdbXZkQgjmZ3m|Usom¥¤Z'tUsopehe,bsgpmY3opchbAm|ojcdb ~À!
Â
ZµXedcqlcdmZ&edvk¢Nojt¶c¹mfcqtnbXgjm ledZGop{a¥#UXZm|UXZ{a¥¤ZDlojb²gMc¹m ¥#cdmUXchb$m|UXZtojY[ZmchYZDx6gjrXbJXt ¬Èmt¶ojedrXZ
l&opbxJZ {|Zm|{chZkZ&§chb<edgkikoj{cdmUXYcqlm|cdYZa {|gjY m|UXZfUXZ&ojst [À! Â :Xgk{opb8v3{|Z&twZ&ijYZbAmE! ¢8mUXZtZm
À! Â,t|omcqt4sZ&t¤m|UXZ gjehedg¥#chbXi[X{|gj6Z{m~v6¦
Í6Ì ÐAO|Õ . & &.(&. "& [ÀA!%OÂ =E% ( % !%F & C"% %'&( ) [À!%Â
C ( % $!% B %'&( *#@ !G C .)= )= 1" ( C
&.(&. "& : <! "#>F
e r U(er ) e b
=l(eb)
sweep line
:chijrX{|Z 9s¦ [ÀA! Â,opbsoijg8g§chbAmZ{tZ&lm|cdgkb
>ng¥Z&jZ{G¢Nm|UXZ§twZm|t À! Âng¨bsgpmUsoOkZojb8v²tcdYXehZ3YgjbXgjmgjbschlc¹m~vQs{gkJZ&{wm~vk :Xgk{fZµXopYXehZj¢
cd! cqtx6Zehg¥ !
¢%m|UXZb [ÀA! ÂnY3oOvQx6Z§ZbAmch{|Zehv²opx6gjZ [ÀA!
Ângj{ m|UXZv²Y3oOv$x6Z§cdbAm|Z{|ehoklZ&±·?tZZ
4JijrX{|Zjº
e b
l(eb) e b’
l(eb’)
:cdikrX{ZX¦
+ ! # !
Z#UsoOkZm|glgkbstwcqZ&{> gjrX{ <8chbsXtHgj6Z&jZ&bkmtmUsopm,gllrX{H¥#UXZb[m|UXZnt¥Z&Z[edchbXZ#{|Z&ojlUsZ&tHopbZ&bs6gjchbkm
gjoªtwZ&ijYZbAmG Ú,ojlU¯ZkZbAmMlgj{|{ZGtw6gjbsst[mg«m|UXZ$chbstZ{mchgjbu¢Vm|UXZ$Z&edZmchgjb¯gpoªxXedrsZQgj{'oª{ZG
tZikYZbAm& Z¥#cdehe>bXg¥ Zµ8sehojcdb²cdb¾ZmopcheHUXg¥¡m|UXZ&tZDZ&jZ&bkmt op{|ZUsojbsehZ&N¢%ojbsQs{gkZmUJomfgjrX{
chb8op{|chojbkmt · X{|gj6Z{mchZ&t k¢s\¢XkºVoj{ZaY3opchbAm|opchbXZGN
3.2 Ê Î%GÐAO&Ï?Î-ª²OÐ8دGÐ Ë ÐÎ6
TVUsZ#bXZ¥±{ZGtwZ&ijYZbAm !%Vcht*}~rJt~m£chbstZ{mZ&[chb[mUXZehcqt~m£gj*oklmchjZV{|Z&tZijYZ&bkmtHopbsDm|UXZ#UXZ&oj [À!%Â
cqtVchbXc¹m|chojedc+ZG<ojt#opbZYm~vtwZm&
rX{Vchb8Ooj{cqopbAmt#op{|Z¶gkxA8chgjrstehv3YojcdbAmopchbXZ&N
3.) Ê Î%GÐAO&Ï?Î-ª©&ÐQGÐ Ë ÐÎ6
UsZb<mUsZDt¥Z&Z<edchbXZ{|Z&oklUXZ&t¤m|UXZZ&bs6gjchbAmngp£o3xXehrXZtZikYZbAm ! ¢6¥Z 4J{|twm¶ZmZ{|YcdbsZfmUXZ{ZG
tZikYZbAm ! mUsopmnedchZ&tH}~rst~m¶xJZ&edg¥mUXZZbs6gjchbAm&¤¬È ! g8Z&t#bsgpmchbkm|Z{twZGlm ! ¢smUXZ&b¨c¹m¶cht#chbstZ{mZG
chb [À!  mUXZ&{¥#cqtwZk¢X¥¤Zf{Z&Jgk{wmVmUschtncdbAmZ&{|tZ&lmchbXi3sojcd{ojbs<{Z&JZGom#m|UXZt|opYZ X{glZ&t|t#¥#c¹m|U<mUXZ
oklm|cdkZa{|Z&tZijYZ&bkmVm|UsomcqtVxJZ&edg¥ ! rsbkm|cde%¥Z {|Z&ojlU<o[{ZGMtZikYZbAmVmUsopmng8Z&tVchbkm|Z{twZGlm!
TVUXcqtaX{|g8lZ&rs{Z[lopb²xJZ¥#{|c¹mmZ&b$chbQmUsZD gkedehg¥#chbXi'¥VoOvj¢*¥#UXZ&{Z ! 0 Z&bXgpm|Z&t¶mUXZojlmchjZD{ZG
tZikYZbAmVmUsopmedchZ&tVchYY[ZGcqomZ&edvMxJZ&edg¥ !
Ê
Î&ÐAO
!
;!
UXchedZ=!
chbAmZ{tZ&lmt !
{|Z6gj{mVmUXZfchbAmZ{tZ&lm|cdgkb
! !
!@
¬®bstZ{mE! chb [À! Â
TVUXZchbAmZ{tZ&lm|cdgkbstaxJZm~¥Z&Zb !% ojbs¨m|UXZ3{|Z&$tZikYZbAm|t¶mUJomop{|Zx6Zehg¥ m|UXZedZôm Z&bs6gjchbAm gj
!#¤opbJ'opx6gjZ ®ÀA!#ÂHUsoOjZ¶xJZ&Zb{Z&Jgk{wm|Z&N¢Am|UXZ{|Z gk{Z¶X{gkJZ&{wm~v¨acqt,m|{rsZn gk{ !#
Z&tchZGt¢8cdm¤cqt#twmchede
ojedcq§ gk{VmUXZfgjmUXZ&{#xXehrXZtwZ&ijYZbAmt
~À!k¤{|ZY3ojcdbXZGmUXZt|opYZ gk{¶ZGojlU$oklmchjZDxXehrXZ[tZikYZbAm !DmUsZb$s{gkJZ&{wm~v¨\3Usoktnx6ZZ&b$Y3opchb»
mopchbXZ&u
e b
l(eb)
e r e
r -1
:chijrX{|Z;6X¦>¬®bstZ{mchgjbgp !
:chbsopehedvk¢%ehZmrstflUsZ&l<QX{|gj6Z{m~vQX3¬ÈE! chtfmUXZ3bXZ&¥YchbXcdYDrXY gj À! Â¥#UXZ&{Z ! &
®À! ¢
m|UXZb'gp%lgkrX{twZ#cdm¤g8ZGt,bXgjml{|gkt|t ~À!
Â| ,<gj{|ZgkZ{G¢mUXZ¶gpm|UXZ{¤twZ&ijYZbAm|t,gp [ÀA!
ÂY3oOv[gjbXehv[l{gAtt
À!
Â%gkbDm|UXZ#{|cdikUkm£gjJm|UXZt¥Z&ZedchbXZk¢pmUsZ{|Z gj{|Z¤mUXZ&v[l&opb[bXgpm¤l{|gkt|tcdm,x6Z gk{ZVm|UXZ#{chijUAm£Z&bs6gjchbkm
gj !
cqta{ZGojlUXZGQtcdbslZ!
edchZ&tax6Zm~¥¤ZZ&bQmUsZ&tZDtZijYZ&bkmt¶ojbs!
TVUsZ[l&ojtZ¥#UXZ&{Z !
cqtabXgpmamUXZ
bsZ¥YchbXchYrXY!gp [À!%£cqt¤m{|chAcqope¼
3.. Ð Ë¾Í Ï?Î-«$Ð8د&Ð Ë ÐÎ6
ng¥trXX6gktZ mUXZDtw¥¤ZZ&MehcdbsZ{ZGojlUXZGtmUXZ{|chijUAm#Zbs6gjchbAmgp£o[{ZG<twZ&ijYZbAm !% ZbXgjmZGtVmUXZ
YchbXchYrXYgp À!%OÂ>¬È !cdbAm|Z{twZGlm|t !%
@
¢jmUXZ&b'{|Z6gj{mHmUXZ¶cdbAmZ&{|tZ&lmchgjbu¢km{|vopikojcdb¥#cdmU !%
0
hh£rXbAmche%o3tZijYZ&bkm!
cqtV gjrsbsMmUsopmng8Z&t#bXgjmcdbAm|Z{twZGlm²¤TVUXZb cqt#YgjZG3 {|gjY [À!
Â
m|g À!
¢ojbs§¥¤Zatwm|oj{wmVopiAopchb§¥#c¹m|U'm|UXZabXZ¥ YchbXchYrXYÉgj [ÀA!
Â
mUsZ{|¥#chtZj¢8cdg8Z&t¤bXgpm
chbAmZ&{|tZ&lmE!
¢s¥ZftcdYXehv'YZ{|ijZ [ÀA!
Â,opbs [ÀA!
@
ÂHm|U8rstVY3opchbkmopchbXchbXiX{gkJZ&{wm~v§s
¬®bmUschtojedikgj{|c¹m|UXY<¢Ê Î%GÐAO chtVm|UXZ X{|g8lZ&rs{Z ¥¤Z XZ&t|l{|cdx6Z&Mcdbm|UXZfX{|Z8cdgkrst#tZ&lm|cdgkbu
UXchedZ
&
À[ÀA!GpÂw£chbAmZ{tZ&lmtE!%
@
{|Z6gj{mVmUXZfchbAmZ{tZ&lm|cdgkb
ZµAm|{|oklmÓ {|gjY [ÀA!%Â
Ê ÎGÐkO ! 0
[À! @ Â &
[À! @
 [À! Â
ojQcdbAmZ&{|tZ&lmchgjbstYoOv<x6Z{|Z6gj{mZG¨rX{|cdbXi§mUXcqtngkJZ&{|opmchgjbu¢6¥¤ZY3oOv}~rstwmfcht|loj{|m|UXZYÓx8v
lgjYsoj{chbXiDmUXZlgAgk{|XcdbsopmZGtgj%m|UXZfedZômnZbJJgkcdbAmt
ngpmZDmUJom ~À!#¤cqtnbsgpm¶rs6XopmZG$oôm|Z{aYZ{|ijchbXi[ÀA!G
@
ÂVopbs À!%OÂVcdb²gj{Z{nmg <kZZ&QgjrX{
m|cdYZlgjYXehZµc¹m~v'xJgkrXbsN
TVUXZfchb8op{|chojbkmtVop{|ZaY3ojcdbAm|ojcdbsZ&ojtVcqtVZ&ojtchedv'lUsZ&l<jZG'oktVcdbmUsZfX{Z&8cdgkrstVlojtZj
3.3 Ð Ë¾Í Ï?Î-«² &ÐQ&Ð Ë ÐÎ6
%Zm !#xJZ¤m|UXZtwZ&ijYZbAmmg x6Z¤{|ZYgkZ&ojbs!%
&
®À!#Â>nt>¥¤Z#tojchDxJZ gj{|Zj¢ ~À!#Âul&opb[xJZVgjxXm|opchbXZG
chbQ¿[À?ÃhÄjÅ
Âm|cdYZ {|gjY mUXZfYZ{|ijZGopxXehZaUXZ&ojXopm|otwm{|rslm|rX{Zam|Usomntwmgk{ZGt,m|UXZtZm|t[ÀdÂ
¬È !%[cqt[bXgpm[mUsZYchbXchYrXY gj [ÀA!Gp¢¥ZD}~rstwm{|ZYgkZ§c¹mG m|UXZ{|¥#chtZj¢,¥Z<t~m|cdeheV{Z&YgjZ !#
{|gjY [ÀA!%¢xsrmm|UXZbM¥ZabXZ&Z&3m|g[lUsZ&l<¥#UXZm|UXZ{¤m|UXZ¶bsZ¥ YchbXcdYDrXY Ógp À!%Â>cdbAm|Z{twZGlm|t !%p
¬Èc¹mg8Z&t&¢N¥¤ZXrstU©c¹mfg¥#b8¥¤oj{|stx8v¨{rsbXbXchbXi§m|UXZX{glZ&rX{|ZDÊ Î%GÐAG ·©¢!%
@
º¢*mUXZ&b${|Z6Z&om
m|UXZ¥#UXgkedZX{glZ&t|t#¥#c¹m|U¨mUsZbXZ&¥ YchbXchYrXYÓrXbAmchec¹m g8Z&tbsgpmnchbAmZ{tZ&lm !
¢6tg3mUJomnX{|gj6Z{m~vM
{|ZY3ojcdbstm|{rXZk
Zµ8m{ojlm!% {|gjY [À!%Â
¥#UXchedZ & À [À!%pÂw£chbst~m|Z{twZGlmtE!G
{|Z6gj{m¤m|UXcqt#cdbAm|Z{twZGlmchgjb
Zµ8m|{|oklm {|gjY [ÀA! Â
Ê Î&ÐAO ·²¢! @ º
bslZopiAopchbu¢m|UXcht¶X{|glZ&XrX{ZfY3oOvM{Z&Jgk{wmxsok<cdbAm|Z{twZGlmchgjb<¥#UXcqlUQlojb¨xJZ$4XµZ&<x8vo'twchYXedZ
m|Z&twm&
¬Èmnl&opbxJZfZGojtcdehv'lUXZGl<jZ&§mUsopm#mUXZfchb8op{|chojbkmtVop{|ZaY3ojcdbAm|ojcdbsZ&N
% 2
TVUsZ lgj{|{ZGlm|bXZ&t|t¤gpmUXcqt#opehijgk{cdmUXY°cd{|Z&lmehv3 gjehehg¥t¤ {gkYÉX{|gj6Z{mchZ&tafopbs<X
+
,QopchbkmopchbXchbXi[mUXZfgk{|Z&{ZG'{|Z&tZijYZ&bkmt#ehchtwm#m|o'<jZ&t¿[À
ÃdÄkÅ
£mchYZj %glopehc+chbXi3opb<ZbsXJgkcdbAmgj{
{|ZYg8chbXicdmmo <jZGt¤¿[ÀFÃdÄkÅ
Â|>TVUsZ¶gjmUXZ&{¤Jop{m|tgpNmUsZ ojedikgj{|c¹m|UXY/m|o <kZ¶opm¤YgktwmV¿[ÀFÃdÄkÅ
 gj{VZ&oklU
Z&jZ&bkmgk{{|Z6gj{mZ&©chbAmZ{tZ&lm|cdgkbu¢mUXZ&b«¥Zf}~rJt~mbsZZ&©m|gQX{|gjZ[mUJom[ojb¾cdbAmZ&{|tZ&lmchgjb«lojbXbXgjmxJZ
{|Z6gj{mZGam~¥#chlZj uZmB!%,ojbs=!#ux6Zm~¥g¶cdbAmZ&{|tZ&lmchbXintZikY[Z&bAm|t&#ôm|Z{{Z&Jgk{wm|cdbsimUsZch{chbAmZ{tZ&lm|cdgkb
gkbslZk¢ ®ÀA!# !% ,<gj{|ZgkZ{G¢6s{gkJZ&{wm~v²\MYZ&ojbstamUJom ®ÀA!#Ânl&opb¾gjbXehvQikg<Xg¥#b$m|UXZ§ehchtwmgp¤{ZG
oklm|cdkZVtZikYZbAm|t&¢jopbJD¥¤Z#bXZ&jZ{>m|Z&twmHmUXZ#chbAmZ&{|tZ&lmchgjb[xJZm~¥Z&Zb !#HopbsoftZikY[Z&bAmm|UsomedchZ&tHojxJgkZ
~À!#Â|
+ !$ ),+ 4
Ñ cdbslZ[mUXcqtopehijgk{cdmUsYcht Z&edZ&Y[Z&bAm|op{|v¨¥¤Z[gkbXehvQbXZZG²mg¨lgkbstwcqZ&{ oM Z&¥ Z&ijZ&bXZ{om|Z[l&ojtZ&ta¥#UXcqlU
m|rX{|bgjrmmg¶xJZ¤jZ&{vaZ&oktwvam|gnUsojbsehZjT¥¤g <8cdbJXtgpsZ&ijZbsZ{ojlvaY3oOv¶gl&lrX{G¦uZcdmUsZ{m~¥g¶Zbs6gjchbAm|t
UJoOjZ¶mUXZt|opYZfopx*lcqtt|o[gj{#opb<Z&bs6gjchbkm#ehcdZGt#gjb<olrX{|jZk
TVUXZF4s{twmVl&ojtZ¶lojb'x6ZftwgkedkZ&3x8v3Zµ8m|ZbschbXiDmUXZ soj{wm|chojeJgk{|XZ{¤gjb'Z&bs6gjchbAm|tVopx*lcqtt|opZ¶mgopb8v
m|gpmope6gj{Z{GTVUsZngj{Z&{¤tUXgjrseh3x6Z¶mUXZ tojYZ gj{¤Z&oklU§gjbsZngpNm|UXZ¶ gjrX{¤XeqopbXZ tw¥¤ZZ&st£¥¤ZaJZ&{w gk{Y
gk{#gpm|UXZ{|¥#chtZ twgkY[Z chbAmZ{tZ&lm|cdgkbstV¥¤gjrXeq'bsZjZ&{VxJZfikgAgN
¬ÈoJgkcdbAmehchZ&tfgkb©o¨tZikY[Z&bAm&¢%¥¤Za}~rstwmDlgkbstwcqZ&{ mUsopmc¹mcqtojxJgkZm|UXZ'tZijYZ&bkmrs{chbXi<m~¥¤g
t¥¤ZZJt¤ojbsxJZ&edg¥mUXZtZikYZbAmrX{|cdbXi[m|UXZ m~¥g3t¥Z&Zst¤¥#cdmUmUsZfvk»®oµcqtV{Z&jZ&{|tZ&N
M +¤ ³M#«M M #
* !$# 4 -!
ZfX{|Z&tZbAmo[ij{|gjrXXchbXiDmZGlUXbXcqykrsZ¶m|Usom¥#cheheuxJZfrJtwZ rXeuchbMm|UXcqttwZGlm|cdgkbojbs'chbmUXZfbsZµ8mgjbXZk
Ð Í OÐ Ë . % * % & 2 %'&( ) /% % *F C"F( "& *% %
C $0*- I *#7C' ( * 0 ;<
; (-C C 0
&
¿[À
Ç
6Â 1
(-J % # %'&( ) 1 B & . *A & 2 "
¿[ÀÀ
'Á
6ÂjÃhÄjÅ
ÂF )& $ 3 #/6
¬®b'o;4s{|twmtwmZ&u¢k¥¤Z¶ojXXehv[oOoj{cqopbAm£gj
Z&bAmehZvA» mmY3opb t£ojedikgj{|c¹m|UXY -ß\0/Jmg[twZm 3HÚ£ojlUm|cdYZ
ojbchbkm|Z{twZGlm|cdgkbfchtZm|Z&lmZ&u¢pgjbXZgpm|UXZchbAmZ&{|tZ&lmchbXintZijYZ&bkmtucqt{Z&Y[gkZ&N ,¨gj{|Z,X{|Z&lchtZehvj¢O¥#UXZ&b
m|UXZtw¥¤ZZ&XcdbsiMehcdbsZDZ&bslgkrXbAmZ{tnmUXZ
»Fm|U$Z&bs6gjchbkmG¢*m~¥g<loktwZGtnY3oOv<gllrX{&¬È£mUXZZ&bs6gjchbkm cqtao
ehZômVZbJJgkcdbAmE!*gptgjYZ¶tZikYZbAm k¢8¥Z lUXZ&l<[¥#UXZmUXZ&{?achbAmZ{tZ&lmtHm|UXZatZikYZbAm|tcdYYZ&XchopmZehv
ojxJgkZ#gj{¤xJZ&edg¥;!È>¬®bMtrslU§ol&ojtZj¢¶cht,{Z&Y[gkZ&3ojbs3¥¤ZX{|g8lZZGmgmUXZ¶bXZµ8mZ&bs6gjchbAm& UXZb
m|UXZ
»¼mUMZbsXJgkcdbAm#chtVo{chijUAmVZbs6gjchbAm !%¢¥¤Z lUXZ&l<¥#UXZmUXZ&{m|UXZ twZ&ijYZbAm|tchY[YZGchopmZ&edv§ojxJgkZ
ojbscdYYZ&XchopmZehv'x6Zehg¥ !% chbAmZ{tZ&lmGH¬®b¨twrslU¨o3l&ojtZj¢gjbsZagjm|UXZfm~¥¤gcdbAmZ&{|tZ&lmchbXi3twZ&ijYZbAmtVcht
{|ZYgkZ&3opbJ3¥Z¶X{|g8lZ&t|tHm|UXZ
»¼mUMZbs6gjchbAm& mUsZ{|¥#chtZj¢A¥Zach{ZGlmehv3X{|glZZG[mgDmUXZ[À
VÁ1@
ÂF»Fm|U
Z&bs6gjchbAm&>¤mHmUXZVZbJDgjsm|UXZ#tw¥¤ZZ&u¢mUsZ#{Z&YojcdbscdbXi tZijYZ&bkmt>op{|ZVtwmgk{ZGchb 0 ZVm|UXZb[{|Z6Z&om
m|UXZX{|glZGrX{|ZV gj{ 0 opbs[m|UXZ¶trXxstZ&yArXZ&bAm,twrXxJtwZm|t,
0
gj{
&
0; ;-
@ Ù edZGop{|edv
¥¤Z UsoOjZ[¦
Í6Ì ÐAO|Õ 3
1 ! 7 B= ! 7 (- B! 13!
I
¶tolgjbJtwZGykrsZbslZj¢s
0
ojbs§m|UXZ{|Z gk{Z &
¿[À
Ç
JÂ|
Ù gkbst~m|{rJlmchbXi ·
& @ ; ;
ºnm|o'<jZ&tf¿[ÀÈ
0 ÃdÄkÅ
0 Á ¤m|cdYZ¥#UXZ&{Z3 ·!#"$AºcqtamUXZ
b8rXYDxJZ&{¶gj£{Z&Jgk{wm|Z&<chbAmZ{tZ&lm|cdgkbstnXrX{chbXi'mUXZlgkbst~m|{rJlmchgjbQojbs$&%
&
a¬®bsZ&Z&N¢*chbstwZ&{wm|cdbsi'gj{
{|ZYg8chbXinopbDZbsXJgkcdbAm>cdb[o¶tgj{mZGfedcqtwm>l&opbx6Z¤XgjbXZ¤cdb¿[À?ÃhÄjÅ
ÂJm|cdYZj¢kopbsmUXZVlgAt~m>gpss{glZGttchbXi
ojbMchbAmZ&{|tZ&lmchgjb<lopbx6ZlUsop{|ijZG3mg[m|UXZfb8rXYx6Z{#gjchbAmZ{tZ&lm|cdgkbstmUsopmnop{|ZfZm|Z&lm|Z&N
' # - + ),+ 4
TVUsZ¨ijZbsZ{ope#loktwZQl&opb±xJZ$tgjehjZGªx8v½{|rXbXbscdbXi¾m|UXZ¨{|Z&8»ÈxXehrXZQopehijgk{cdmUXY gjb±ZjZ&{v½sopch{3gpftwZm|t
À- ;<
Â|£¡t~m|{|ojcdikUkm gj{|¥¤oj{|'opbsojedvtcht¤v8cdZ&ehst¦
( Í Í &&?OÕ( *) #%
& 2 2C'&( ) B" Q C C %
¿[À
Ç
ÃdÄkÅ
 &/¿[À
 '9 % K %; +
@
;08/321
(* #%#+, ' ´"M . s%#
TVUsZf{ZGtwrXedmgp>m|UXZX{|Z8chgjrst#tZ&lmchgjb¨lojbx6ZtedchijUAm|edvMcdYX{|gjZG'¥#UXZ&b<mUXZDtwZ&ijYZbAm|t#oj{ZfJtwZ&rsgp»
tZikYZbAm|t&
Ð Í OÐ Ë 3 7 & .
% ')('& % C"¶A *B$B "
¿[ÀÀ¼
Áª
ÂjÃhÄjÅ
'绂
JÂF + B % 81 ,
TVUXZA4s{t~msop{mMgjfmUXZ¾opehijgk{cdmUsY)cqt§m|UXZ¾lgjbJt~m|{rslmchgjb gjfmUXZ¾trXx6Xcd8cqtwchgjb gj Ñ Z&lmchgjb h
m|Usom lgkbst~m|{rJlm|tQtrXxstZm|t¶gpHbXgkbQchbkm|Z{twZGlm|cdbXiMtwZ&ijYZbAm|t&TVUXZ[twZGlgjbJ<sop{mncqtnopb$Zµ8mZ&bstwchgjb¨gj
ojb«opehijgj{|cdmUXY x8v Ù Usopb -ß0/,m|UsomtgjehjZGtamUXZ'{ZG8»¼xsedrXZ§l&ojtZj Z'okXopmfm|UXcqtopehijgk{cdmUsY twg<m|Usom
tZikYZbAm|t¤¥#cdmU<lgkedgk{|tVlojbMx6ZfUsojbsehZ&N
TVUXZ3ojedikgj{|c¹m|UXYÓchtfotw¥¤ZZ&$ehcdbsZ[ojedikgj{|c¹m|UXY<¬ÈmfrstwZGta¥#c¹m|bXZ&t|twZGt ojt¶cdbAm{|grslZ&²cdb - /¼D¬È,m~¥¤g
tZikYZbAm|t 0 ojbs8
chbAmZ&{|tZ&lm&¢sojbsc¹>mUsZfedZômnZbsXJgkcdbAmgp, 0 cqt#xJZ&edg¥
¢smUXZ&cd{chbAmZ&{|tZ&lmchgjbcht
okttZ{mZG<x8v<o'JgkcdbAmtopmcqt~ v8chbXi
+nopbs+ 0 3VchtalojedehZ&Qo@ ) V gj{¶mUsZDchbAmZ{tZ&lm|cdgkb
·FtwZ&ZF4sijrX{|Z8pº£gpm|chlZamUsopm#m~¥gchbkm|Z{twZGlm|cdbXistZrJgp»®twZ&ijYZbAmtojed¥VoOvtUJoOjZ o¥#c¹m|bXZ&t|t
1 S 2
S
t
:cdikrX{Z$88¦cqt#oD¥#cdmbsZ&t|t#gpmUXZfchbAmZ{tZ&lm|cdgkbu
2+ ()(# +. 4 $!$ * 6!
¶rX{|cdbsiQmUsZt¥Z&Zu¢>m|UXZ<ojlm|cdkZMtZikY[Z&bAm|t[op{|Z'chbstwZ&{wm|Z&«chb³o$edcqtwmncht[tgj{mZ&½ojllgj{chbXi¨mg
ojb¾gj{Z{f{|Zeqom|cdgkbu¢bsgpmZG f¢%m|UsomcqtlgjYsopmchxXedZ3¥#cdmU¾m|UXZ§¥#cdmbXZGttZ&tfmUsopmUsoOjZ3ojed{|Z&okv$x6ZZ&b
Z&bslgkrXbAmZ&{ZG«x8v¾mUXZ¨t¥Z&ZXchbXi$ehchbXZjª¬®b³ijZ&bXZ{ope¼¢>mUXcqtgj{Z&{cqt3c¹¸*Z{|ZbAm[ {gkYÆmUXZMm{|rXZgj{Z{
ojedgkbXiMmUXZ§t¥Z&ZXchbXiedchbXZk ,<gj{|ZDs{ZGlcqtwZ&edvk¢u¥Z¥#chede,{ZGykrscd{|Zm|Usomfm|UXZ gjehehg¥#cdbXiX{|gj6Z{m~v¨UXgjeqXt
m|UX{|gjrXikUQmUsZlgjrX{tZDgj,mUsZojedikgj{|c¹m|UXY</{Z&X{ZGtwZ&bAm|tnmUXZ§oklmrJope>gj{Z{ gj,mUsZtZikYZbAm|t opehgjbsi
m|UXZt¥Z&ZXchbXi[edchbXZj
Í6Ì ÐAO|Õ 5
& 9¿ 0 ;¿
;|¿ ; ;|¿
1 0 ;|¿
1 = * B *<) * + B9 7 "&
,¿ 0 1¿
1
B B% #0 && "& 1 & =& "& % 3;a¿7 @$ A +
¿ = @ % ?F fÀ¼¿ ;|¿ 0  (- C ¿ ¯¿¯¿ 0 M¿[À?ÃhÄjÅ
 )& G
+-!$+=!$() !
¶rX{|cdbsinm|UXZ#tw¥¤ZZ&u¢¥Z¤Y3opchbAm|opchb [tgj{mZ&edcqtwm|tgpJoklm|cdkZ¤tZikY[Z&bAm|t&¢gjbXZ gj{>Z&oklUtZm0 X{|g8XrslZG
chbªm|UXZ 4J{|twmtwmZ&u«TVUXZ&tZ'ehcqt~mtDoj{ZM{Z&X{ZGtwZ&bAmZ&«oktxscdbsoj{v«tZ&oj{|lU¾m|{Z&Z&t& ZojehtgQY3opchbAm|ojcdb½o
tgj{mZG$ehcqt~m n ¡lgjbAm|ojcdbstfojede£mUXZ'oklm|cdkZtZikYZbAm|tfsedrstfm~¥g¨Z&edZ&Y[Z&bAm|t&¢
Á ojbs
¢¥#UXcqlU
oj{Z¶{ZGtw6Z&lmchjZ&edv3Y3oµchY3ope*opbsMYcdbXchY3ope6 gj{¤m|UXZam~¥g[gj{Z{t,ojbsgDbsgpmVcdbAmZ&{|tZ&lmVojb8v§twZ&ijYZbAm&
ojtwmehvj¢% gj{opb8v$tZikYZbAm3! 7 ojbs² gj{ojede
$
&
¢%¥¤ZYojcdbAmopchb²m|UXZ§tZijYZ&bkmt [ÀA! ; ÂnopbJ
nÀ! ; £gp
m|Usomnoj{Z chY[YZGchopmZ&edv'ojxJgkZaojbsxJZ&edg¥ ! >cdbmUsZ gj{Z&{&
4 4$! + 4 4 . !
TVUsZagkbXehv§bXgjbX»Fm|{ch8choje6Jop{mVgj%m|UXZopehijgk{cdmUXY cqt¤mUsZfrX6somchbXi3gj %Zm¨x6Z o[ehZômZ&bs6gjchbkmgj
m|UXZ twZ&ijYZbAm¤mg[xJZ chbstwZ&{wm|Z&N :ch{|twm&¢8m|Usopb<tmgzH{|gj6Z{m~v§¢8¥Zaehg8l&om|Z <chb n¬È>Z&bXgpm|Z&tmUXZ
b8rXYDxJZ&{fgpVsopch{t gpVcdbAmZ&{|tZ&lmchbXi¨twZ&ijYZbAmtamUsopmUsoOjZ ½oktamUXZ&cd{;4J{|twmf¥#c¹m|bXZ&t|t¢¥¤Zl&opb¾{|Z6gj{m
m|UXZ&tZchbAmZ&{|tZ&lmchgjbstVopbJMrs6XopmZ ±chb¨¿[ÀÈ,ÃdÄkÅ
§Á
uÂHmchY[Zokt¤ gjehedg¥t&
%Zm 3ojbs*$xJZm|UXZ3m~¥¤g¨twZ&ijYZbAm|tcdYYZ&cqom|Zehv$x6Zehg¥ ojbs©opx6gjZ «mUJomop{|Z{ZmrX{|bXZG
¥#UsZb§tZ&op{lUXchbXi %Zm nx6ZmUsZ¶tZmgjutwZ&ijYZbAm|t atwrslUm|Usom ¶opbJ !%¢opbs§edZm
x6Zm|UXZ¶tZm¤gp%tZikY[Z&bAm|t¶trslU3mUJom? ¶opbs [·?tZZ:chijrs{Zakº :Xgk{ZGojlU ¢kmUXZatwZ&ijYZbAm|t
gj " lojb§xJZ¶ gjrsbs§chb¿[À
@HÁ
: " :êÂ>mchY[Zax8v¥¤oje<8chbXiDojedgkbXi twm|op{mchbXi[om[À- ;|Â
>nZbslZj¢X¥¤ZflojblgjYXrm|Z chb<¿[À
Á : :êÂHmchY[Zk
%Zm & opbs
&
¢6opbs<ehZm
xJZfm|UXZb8rXYDxJZ&{gp£tZijYZ&bkmtVxJZm~¥Z&Zb
ojbschb ¾Yrstwm£xJZ#{|Zgk{|XZ{|Z&gkb[mUXZ#chbAmZ{|ope9 ;(=Jojl&lgk{|chbXi¶mg m|UXZ#bXZ¥±gj{Z&{HcdbsXrslZG
x8v ZaUsoOjZ 0
c¹>opbs'gjbXehv3c¹ 0 1
gk{cd¥¤Z¶UJoj 0
}~rstwm#xJZ gj{|Z¶s{glZGttchbXi
Ñ gk{wm|cdbsi<·Fojl&lgj{chbXi[mgm|UXchtnlgjYsoj{cqtgjb{rXehZGºm|UXZ
tZijYZ&bkmtVcdb<m|UXZf{opbXikZ94;(=l&opb<xJZ
XgjbXZ chb¨¿[À
ÃhÄjÅ
ÂHmchY[Zk
%ZmHrst>x6gjrXbs
Z#gjxstZ{|jZ,mUsopmHojb8vtwZ&ijYZbAmHgp cdbAm|Z{twZGlm|t% ojbsfm|Usom0cqtmUXZ 4s{t~m
¥#cdmbsZ&t|t£gp%ojedesmUXZGtwZ¶cdbAm|Z{twZGlmchgjbJt ,<gj{|ZgkZ{G¢popb8vtwZ&ijYZbAm,gj94;(=JmUsopmcqtojxJgkZ[opbJ3bXgpm
chb chbkm|Z{twZGlmt opbs ¨cqt¤mUXZF4s{t~m#¥#cdmbXZGttVgjojede*m|UXZ&tZ cdbAmZ&{|tZ&lmchgjbst& >Z&bslZk¢
&
¿[ÀÈsÂ
¬ÈmV gkedehg¥tmUsopmnopehe*mUXZftgj{mchbXiktm|o'<jZf¿[À¼VÃhÄjÅ
Âm|cdYZfchbmgpmopeNopbJ§m|Usom#m|UXZflgjYXrXm|om|cdgkbMgjojede
m|UXZ fojbs [m|o'<jZGt¿[À
Á
6Â
&
¿[À
Ç
6Â>mchYZj
bslZ opbs UsoOkZ£x6ZZbDlgjYXrXmZ&u¢GZGojlU bXZ&¥«cdbAmZ&{|tZ&lmchgjbflojbfxJZ¤Zm|Z&lmZ&cdblgjbstwm|ojbAm
m|cdYZkTVUscht UXgkehsta gj{amUXZ3Jopch{|tagp 8 opbs$mUXZsojcd{ta¥#c¹m|U«otZikYZbAm 3cdb ojbs²o
tZikYZbAm,m|UsomV¥¤okt,chbXcdmcqopehedv3x6Zm~¥¤ZZ&b opbJ<opbs'{Z&YojcdbJt,opx6gjZ %£Ú£oklU3mchYZaojb'chbAmZ{tZ&lm|cdgkb
cqtf{|Z6gj{mZGN¢u¥¤Z3rX6somZm|UXZ§6gjchbAmZ{t [À ;
Âcdbªlgjbstwm|ojbkm m|cdYZjTVUXZb¾¥¤Z§lojb²chbstZ{mfmUXZ3ehZôm
Z&bs6gjchbAmgp¨chbmUXZsom|otwm{|rslm|rX{|ZachbQ¿[ÀuÂ
&
¿[À
Ç
6Â>mchYZj
. + + 4 4 -!
TVUscht <8chbsgp>ZjZ&bAmVgl&lrX{tV¥#UXZbm|UXZtw¥¤ZZ&XchbXiedchbXZf{|Z&ojlUsZ&tVo[{chijUAm#ZbsXJgkcdbAm&H¬Èmcqt#UsojbsehZ&chb
o[¥VoOv3jZ&{v'tchY[cheqop{¤mgm|UXZfX{|Z8cdgkrst¤gjbXZfojbs<g8Z&tVbXgjmVgj¸6Z&{opb8v'okXcdmchgjbsojeuc3§lrXe¹m~vk
) ! ! # + ( ! #
v©lgjbJt~m|{rslmchgjbu¢%mUXZ'gj{Z{gj m|UsomDchtDlgjYXrXmZ&¾¥#UsZb¾cdbJtwZ&{wm|cdbXi8½chtDlgjYsopmchxXehZ3¥#c¹m|U*%¢
c¼ZjojedemUsZ[oklmchjZtZijYZ&bkmt¶trslU$mUsopm 0
topmcqt~ v 0
,¨gj{|ZgjZ&{&¢JmUsZDgk{|Z&{¶gj
m|UXZ&tZ¶tZikYZbAm|t,¥#cdehe6lUJopbXikZcdb§mUXZatwZGykrsZe6c¹ojbsgjbsedvcdu¥Z¶cdbstZ{m¤ojb3ZbsXJgkcdbAm? JmUsopm¤t|om|cht4sZ&t
C 0 DTVUXZchb8Ooj{cqopbAm gp£m|UXZ3opehijgk{cdmUXYcqtamUsZ{|Z gj{|ZY3ojcdbAm|ojcdbsZ&©opbsN¢utchbslZZ&oklU$sojcd{
X e u e
L ud Ldu
d
m M
:chijrX{|ZfX¦ ! *Xom|cdbsi
gjchbAmZ{tZ&lm|cdbsi[JtwZ&rsgp»®tZijYZ&bkmt,UJojt#o[¥#c¹m|bXZ&t|t&¢mUXZfojedikgj{|c¹m|UXYÉ¥#chedeN{|Z6gj{mopehe*mUXZfchbAmZ{tZ&lm|cdbsi
Jopch{|t¤gpHtZijYZ&bkmt
-
M+¤
.
^achjZ&b§otZmgjuijZ&bXZ{opeXYgkbXgpm|gjbXZntZikYZbAm|t&¢j¥¤ZUsoOkZX{|Z&tZbAmZG3opehijgj{|cdmUXY3t>m|g{Z&Jgk{wm,mUXZ¶sopch{|t
gj¤tZikY[Z&bAm|tamUJomfcdbAm|Z{twZGlm& rX{ojedikgj{|c¹m|UXY3tarstwZgkbXedvQmUX{|ZZ3tcdYXehZ3X{|Z&cqlopmZ&t¶m|Usomojedehg¥¡mg
XZ&lcqZ[c¹m~¥g<tZijYZ&bkmtacdbAmZ&{|tZ&lm&¢Ncd¤oMJgkcdbAmfcqt edZôm gk{¶{|chijUAm¶m|g¨opbXgjmUXZ&{¶6gjchbAm&¢%ojbs$cd¤oM6gjchbkm
cqt#opx6gjZk¢8xJZ&edg¥gj{Vgjb<otwZ&ijYZbAm&HTVUXZGtwZam|UX{Z&Z X{ZGcqlom|Z&t#tZZ&Y mgx6Z mUXZftcdYXehZ&twmX{|Z&cqlopmZ&t
m|Usom¶edZGojMmg§trXx*ykrJoj{om|chlfojedikgj{|c¹m|UXY3t
rX{nopehijgk{cdmUsYt#oj{ZojedYgAt~m#gjXmchYoje%chb<mUXcqt{|Z&twm{|cqlmZG
YgZ&eNgpHlgjYXrXm|om|cdgkbu
ZlgkbslehrsZ¥#c¹m|UQtwgkY[ZfgkJZ&b<X{gkxXehZY3t :cd{t~mG¢Jl&opb<¥Zf{|ZYgkZ oÃhÄjÅ
?oklm|gj{chb¨gjrX{Vm|cdYZ
lgjYXehZµcdm~v§{ZGtwrse¹mt
TVUXZ/aÀ
Ç Á'
ÃhÄjÅ
Âuehg¥¤Z{Hx6gjrXbs[UXgkehst> gj{HikZbXZ&{|ojeXlrX{|jZ#tZijYZ&bkmt¬Èm,Usokt>xJZ&ZbJgAttchxXedZ
m|gXgMx6ZmmZ{ gk{ edchbXZ3twZ&ijYZbAmt-/È Ù ojb$¥¤Z[ojehtgMg<x6ZmmZ{a gk{agjmUXZ&{ftw6Z&lchoje>lrX{|jZtwZ&ijYZbAm|t
trslU$okt¶lcd{lehZtwZ&ijYZbAm|t& Ñ gjYZDX{Z&edchYcdbJop{|v{|Z&trXedm|tn gj{amUschl<<gjxX}~Z&lmtaUsoOjZx6ZZ&b²gkxm|ojcdbsZ&¨x8v
¶cdikbXZ{|gjbA-dO7/È
¬®b±mUschtsojJZ&{&¢¥Z<UsoOkZM{|Z&twm{|cqlmZG½gkrX{3ommZbAm|cdgkb½m|g©YgkbXgpm|gjbXZ¨tZijYZ&bkmt´TVUXcqtY3oOvªxJZ
oQ{ZGt~m|{cqlm|cdgkb«¥#UXZb¾m|UXZ'6gjchbAm|tD¥#c¹m|U½o¨jZ&{wm|chl&ope,m|opbsijZbAmDoj{Z§Xc3§lrse¹m[mg$lgjYXrm|Zj©TVUXchtchtDchb
Jop{mcqlrXeqop{mUXZ<loktwZ' gj{lch{|ledZGtDZ4sbXZGªx8v©mUX{|ZZJgkcdbAmt¥#UXZ&{ZMmUsZMX{|Z&Xchl&omZ'mUsopmlgjYsoj{ZGt
s»¼Zµ8m{|ZYZ JgkcdbAm|tVUJojt#Z&ij{|ZZ\ D[¥#UXchedZam|UXZfcdbAm|Z{twZGlmchgjbX{|Z&cqlopmZ Usokt#Zik{Z&ZDO\gkbXehvj
!##'+¤#
-d/ ¨ojbX
opeqopxJopbuNbMgjm|cdY3oje6ojedikgj{|c¹m|UXY gk{ 4JbschbXiDtZikY[Z&bAm,cdbAm|Z{twZGlmchgjbJtN¬®b B79BF C
7 ( 7 &"- & 2 " ) & h¢sJopijZGt#\j\X&X¢J&kk
-\/ s,
ZbAm|edZ&v²opbJ«T
mwm|YojbXbu¨ehijgk{cdmUsYtf gj{{|Z6gj{mchbXi¨opbJ«lgjrsbkm|cdbsi¨ijZgkYZm{|cql
chbAmZ&{|tZ&lmchgjbst&
& & " Ih¢
Ù
»~\pJ·?kº¦6'9k 6 9"88¢
Ñ
Zm|ZYx6Z{&8pX
- / s»D
gkcht|twgkbXbsopmMopbs X Ñ bXg8Zv8chb<*/Ú 3§lcdZ&bkm¨opehijgk{cdmUsYt3 gj{edchbXZ²ojbs lrX{kZ²tZikY[Z&bAm
chbAmZ&{|tZ&lmchgjb©rstwchbXi¨{|Z&twm{|chlmZG$X{|Z&Xchl&omZGt3¬®b EG C 7 ( 7 &0F & " I
) & h¢sJopijZGtV8D 8 8X¢6&kjs
-9'/ jZ&ojbȦojbXcdZ&e
gjcqttgjbsbsom#opbs :s{|ojbslg3zJzH{|ZJop{om|os£#gjxsrst~m¶XehojbXZt¥Z&ZM gj{chbAmZ{tZ&lm|cdbsi
tZikYZbAm|t& 7 K & " I£mg3opX6Z&oj{&
-/fT ,¾ Ù Usojbu%tchY[sedZ¶m{op6Z+&gjcq3tw¥¤ZZ&'opehijgk{cdmUXY/ gj{{Z&Jgk{wm|cdbsi{|Z&1xXedrsZntwZ&ijYZbAm¤cdbAmZ&{w»
tZ&lmchgjbst&>¬®b B79 C 0,F E- & " I ) & h¢XsojijZ&t\'6j \ 6kX¢J&j'9s
-6 /
Z&{bsoj{| Ù UJo +Z&edehZ$ojbs >#Ú£Z&ehtxX{|rXbXbXZ&{& nbgkmchY3ope¶ojedikgj{|c¹m|UXY) gj{'chbAmZ&{|tZ&lmchbXiªehcdbXZ
tZikYZbAm|tVchbmUXZfXeqopbsZj 7 ¨¢sks·wGº¦h X9J¢JGjk\X
-80/;@ £
Ù
ehoj{B<tgjb«opbJ«z
Ñ
UXgj{G¨XsedcqlopmchgjbstDgp#{opbsgkY t|opYXehcdbsiQcdb³lgjYXrXm|om|cdgkbsope
ikZgkY[Zm{|vj¢¬¬ $B% & 2 " )
& h¢9J¦k8 9A\Xj¢sGjkX
- /;:¤HYgj{|Zj¢£zH^[ :X{opbslcdgAtos¢ojbs½^[ uchgpmwmoX© {|gjxXrJt~m{|ZikcdgkbªopXs{gAojlU©mg$mUXZ<lgkYD»
srm|opmchgjb gj ikZgkY[Zm{|chlik{|ojXUst&¨b^[
cdeqop{c¼¢#^[ :¤,¬ÈmopehchojbXgs¢#VzHchZm|{|oklops{chbsoX¢VopbJ
^[*zHrsl&lc¼¢*Z&cdmgk{|t&¢ 7 =& 7 ¢NkgjehrXYZ39=6sfgp % "B % & " I Ch¢
JopijZGt 08j 6G'6s Ñ X{|chbXijZ&{w»HZ{|ehojis¢6&kjX
- / ehchAchZ{ ¶Z8chedehZ{t¢kehZµXopbsX{|oF:X{|gjb88cdehehZj¢
Z{|bsop{,¨gjrX{|{|ojcdb%¢popbs,<gjbschyArXZVT%Z&cdeheqoprsNJÚ>µXojlm
s{ZGchl&om|Z&t¤ gj{lcd{lehZfop{ltVop{|{opbXikZYZbAm|t&>opX6gj{mZ {ZGlUXZ{lUXZ jA\ 6X¢8¬~¶#¬~¢uGjkX
-hED / ¶gjbJopeq Ú@ b8rmU% 7 = &:/; ¢kgjehrXYZ 6'D'6±gj % "B & " I C
Ñ X{chbXikZ{»HZ&{eqopiJ¢>nZcqZehxJZ&{iJ¢J^aZ{|Y3opb8vj¢*Gjk\X
-hj/f^achrstZX6Z uchgpmwmoX¢:s{|ojbslgzfzH{|ZJop{om|os¢opbs#gjx6Z{mgTopY3okttchos #gjxXrJt~m©X{gOµchYc¹m~v
yArXZ&{chZ&t&¦ubfchehedrstwm{om|cdgkb¶gjXZik{Z&Z»®{chjZ&b¶ojedikgj{|c¹m|UXYZGtwchijbu " 7 & 2 " h¢p\js·?Aº¦'6'9
kjs¢N&jkX
-hG\/
Ñ
vAehopchbªzHchgjbu¨¬®bAmZ&{oje,op{|c¹m|UXYZm|chlj¦[ojb«Z%3§lchZbAmchYXedZ&YZbAm|opmchgjbªopbJ«opbªopsXedcqlopmchgjb©mg
lgjYXrmom|cdgkbsopeNikZgjYZm{|vj>¨b.7 7 =H ) . # ) 7 )0; 0 %&
1I ( B)ê¢XsojijZ&tVk 6kEDs¢JGjjs
-h& / :¤[z[zH{|Zsoj{|opm|o opbs ,¾¬ Ñ UsojY[gAt & 2 " ) # ) & )+ 7 ( BG ( ) s
Ñ X{chbXikZ{»HZ&{eqopiJ¢Aj{|MZ&cdmchgjb%¢ lm|gjx6Z{f&k'Ds
-h9'/ Ñ Ñ lUXch{{oXH#gkxXrstwmbXZGttVcqt|twrXZGt>¬®b j{|ip»® schijZ&{ Ñ okl<'opbJ jgj{|ijZ ! {|{rXmcqoX¢Z&Xc¹m|gj{t¢! 1)(
*7" $ & 2 " ) ) ) & % +¢kÚHeqtZ8cdZ&{ Ñ lchZbJlZ#zHrXxXehcqtwUXZ&{|t
pngj{mU»>gkedeqopbJN¢jY[»
twmZ&{|sopY<¢N&kjX
-hG/ nbAmgjchbXZ nchijbsZ{|gjbu«ehijgk{cdmUsY[ZGt[`eh`YZbAm|ojcd{|Z&tJgkrX{{Z&Jgk{wm|Z{[edZGt[cdbAmZ&{|tZ&lmchgjbst[gjx}~Zm|t
lgjrX{|x6Z&t&>nopX6gj{mZ ¶Ú, opehijgk{cdmUXYcqyArXZj¢z>op{|cht&¢:X{opbJlZj¢NGjkX
-hE6 / Ù
,opuHTg¥¤oj{|XtZµXoklmVijZgkYZm{|cql¶lgjYXrmomchgjb%F & 2 " )
& & +
7
'1h¢28·wGº¦
\jX¢NGj8
-h080/ Ù 0@,ojuj#gkxXrstwm%ijZ&gjYZm|{cqlHlgkYXrm|opmchgjbuj¬®b joklgjxfÚ#^ag8gY3opbopbJ pgAtwZ&XU gjrX{<jZk¢
ZGcdmgj{t&¢ 1,*7"$ %B / & " # ) & )+¢AlUsopXmZ{HA¢sojijZ&t!6kj "6'6js
Ù Ù zH{|Z&t|t Ù ¢
gloopmgkbu¢: £¢uGj8