HAL Id: inria-00072609
https://hal.inria.fr/inria-00072609
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-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
How do we perceive the Eye Intrinsic Parameters ?
Thierry Viéville, Jacques Droulez, Chin-Hwee Peh, Alexandre Négri
To cite this version:
Thierry Viéville, Jacques Droulez, Chin-Hwee Peh, Alexandre Négri. How do we perceive the Eye Intrinsic Parameters ?. [Research Report] RR-4030, INRIA. 2000, pp.39. �inria-00072609�
ISRN INRIA/RR--4030--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 3
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
How do we perceive the eye intrinsic parameters ?
Thierry Viéville, Jacques Droulez, Chin-Hwee Peh and Alexandre Négri
N° 4030
Octobre 2000
%&'(&') "$+*
,.-/1032 2#465%/1798/;:1:;03<>=@?BADC3EGF(0IHJK2MLBF:10INPOQ=@R-/1S"T#UWVW0I0YXZ0I-\[ADS]
^ :10`_ABS(]2M0Yab7dce2M/
fgih#jWklnm oqpdrsk#tsu`vrswyx9pzgix`j.jWk|{qj.u`v}giwypik`~
wyj.u`9k# ~ix9ppi k# ~Dv x9ppu`wy su>piv k#
t x>krZx9wx`w
Zu`iBx9trikt k v}gk t v}gkpd9l9m v#r x9itk>99nm l9!u`9k#
K#M¡ i¢`d£¤
kbiwv ¥i!gik tk¦gixQ§¨r xzwyik pdrswª©¬«gixQ§®u¯3w¥u`°@«3r k j±kMu`°yn§wªrsg
rsgikbwpdrstwp wyvvMu`°ywtsu>r wx9p²u`tsu>j.krsk#t x`©r gik¦¯3wy ¥u`° k#pi x9t#~w;³´k`³µrsgikbk#«9k>~¶wp²r gik
vMu`k'x>©jWx`r wx9p%Bk t v#k r wx`p·³
oqpv#x9jWi¥r k t¸¯3wy wyx9p·~i¥iv}gu`pu`it xdu`v}g¦wu>°tkMu`«.§k °y°¹ik¯9k °yx9Bk ·~i©ºtx9j»ex>rsg
u9k x9jWkrst wyv%u`pi¼u`°y9k#itsu`wyvBx9wpdr .x>©Z¯3wyk#§³½o¾rWwy!pixQ§®§k °y°k r}u`°wy gik#¿wypÀrsgiwy
Á k °y²rsgiu>rv u`°ywit u>rswyx9pj.k#v}gu`piwy jW!tk °u>rsk#rsx%rsgikbvMu`jWk#tsuµwypdrstwpiwv¦u`tsu>j.krsk#t
u`°°yxQ§Ârsxu>u`r ¥iv}gYu¦Bk tv k Prs¥u`°(«3r k j+r xKwypiv x9jWwypiwyp©ºx9t j.u>r wx`pu`pix`r}u`wyp
ut x9i¥rZu`piw9pwÁ v u`pdrZk r wj.u>r wx`pµx`©"r gikÃr t ¥ivrs¥itknu>pij.x>rswyx9pu`tsu>j.krsk#t x`©
rsgik«3rsk jÄwªrsk °ª©u`p%r gik v k#pik`³
fgk!u` ¥ijWrswyx9px`©rsgiwy§x9t Å%wZrsgu¡rrsgkiwyx9°yx99wyvMu`°"¯3w¥u`°Bk tv k#rswyx9pj¥r
u`°xBk%u` k#6x9p¿ ¥iv}g6j.k#v}gu`piwy jWu>pi¿§kµgiuQ¯`kzwyjWi°k#jWk pdrsk#Ærsgik%it x>rsx3v x9°y
§giwv}gzu`°y°xQ§Çr x!¯9k twy©¬«r giwyZgd«3Bx`rsgik# wy ³
ȵÉdʹËÌÍ ¡Î(`£Ï x`rswyx9p%ek#t v#k r wx9p¹³¸Ðu>°wyitsu¡rswyx9p
ÑÒºÓÔGÒºÕ@Ö¡×·ØKÙÚÛÖÛÜ#ÝÙ#Ü#Þ@ßQàáÛâÛãåäÛÖeæIç èyéëêqêqìíïîPèyéëð èòñòñóéô¹èòõ>êqèyöí÷¾ê
ø(ù
àúûúóüýþ\ÿQþ¾äëþÖÛØiä;ã ÖÛö¡éëí@êQñóéô}ñòñóé é #é÷¾êïö}õ!¾é í÷¾ê
"
Óä#;ãûà¡äú%$Ûã'&þ( ;ã)#+*à,ßQã-Ûý ä|á`àïþÖIéñóé/.Ûéëç0ëç>ô·õ10íòé2¡í-
#W"( $ $ qW6*
£ piwy v ¥Prsk¸wyv w3v x`j.jWk#pIr\j.kr r t k¸k#p#¯3wyik pv k°uZ©1u#x9px9pdr\°yk¸«3rsh jWk
¯Pwy ¥k °9h tkn°yk Zu`t u`jWh#rstk Zik'vMu`°ywtsu>r wx9p%wypIr t wypi hI¥ik Ãik'°óx3k wy°"u`¥%piwy¯`kMu`¥zikn°u
ek#t v k#rswyx9p%i¥%j.x`¥¯9k#j.k#pdrM³
p6¯3wy wyx9pÆu>t¦x9tiwpiu>rsk#¥it ~v#k#rrsku>iit x3v}gikµk rKiYiwk#pÆi¯9k °yx9iB k>~Zrsu`pdr
i¥ ex`wpdrik¯3¥ik` x9jW#r t wI¥ikI¥ ûu>°9#it wI¥ik`³ oq°k#rk p k"!¹kriwyk p #rsu`i°yw#I¥ik
ik itxPv#k ¥iZikÃvMu`°ywyitsu>r wx`p%tk °u>rswª©ºZu`¥%$Ku`tsu>j.hrstk wpdr t wypi h3¥k k'°uvMu>j.#tsu
ek#t jWk#rrsk pdru`¥W«3rsh jWkek#t v k#rswª©·kóu`u`r k t@u`¥WjWwk#¥%$.u`¥!rq«3BkZ&ówyp©ºx9tjbu¡rswyx9pi
3¥w3°y¥iwIk r©ºx9¥t piwdkr ´x9r k piwyt'°uZ©ºx9wy(¥pikk#r wj.u>r wx9pt x9i¥rsk@k#r( wy9piw
Á
v u>rswª¯9kk
u`tsu>j.hrstk I¥iwGvMu`t u`v#r twk pdr°ykj.x`¥¯9k#j.k#pdrkr°uWrst¥iv#r ¥it kx9 k t¯9#kx9ik t¯9`³
()´gd«PBx`r gih k'kÃv krst uM¯`u>w°·k#rI¥ikðu!ek#t Bk v#r wx`p¯3wy ¥ik#°°ykiwyx9°x`9w*I¥ikix`wyrk#°°yk
u`¥i w,+rst kzu` #k ¥it¦ikzrsk#°¦j.#vMu`pwj.k#k#rx9p i¥r¦ikjWk#rrst kk#p i°u`v kk
it x`r xPv#x9°yk I¥iwGi¥iwy k pdr°ykjWk#r r t k'k p%#¯3wyik piv#k`³
- Í
Û
Ë
¢.
£
(k tv k#rswyx9pz¥µjWx9¥¯`k jWk pdr ³Ðu`°ywtsu>r wx9p
,e!"(. #n
-u`Wrsgik%it u`wp¿u`°ª§uM«3KuYit k#v wy k/.0213 4657 8:9; < = +>! "+?1w1³´k`³Æx9PrswyvMu`°3@
!$4$&' ( ) ~§gik p Bk t©ºx9t jWwypiÀu²Bk tv k#rs¥u>°r}u`ÅD~Z¥iv}g u`<?º k#kBAól>i~¸l7CED©ºx`tKu
t k#¯3wyk#§F@jWx`rswyx9pek#t v#k r wx9pHG
fgwbjbuM«¼°yxPx`Ž°wyÅ`kµurst u`pi9k I¥ik rswyx9pJIK-xQ§¸k¯9k#t ~wª©Ãx9pikv x`pi wyik tbpix`rbu
iwx`°x9`wv u`°;~i¥rµu`p u`trswÁ v wu`°¯3w¥u`°Ã«Prsk#j%~r giwy%Bk v#x9jWk µu`p wpik k#Çitwj.u`t«
u`piÀ©º¥ipiu>j.k#pdr}u`°I¥ik rswyx9p·~w;³´k`³¼ix§k%giuQ¯`k%r xL4$1M>NE4$( nrsgikwypIr t wypi wyv%x9rswyvMu>°
u`tsu>j.krsk#t x`©rsgikï3w¥u`°" k#pi x`t rsxWek#t©ºx9tjÄu`pd«Bk tv k Prs¥u`°·rsu` ÅOG
P ¥iv}gQ! "Wiu`tsu`jWkrsk t¦v#x9t tk Bx9pi¹~©ºx`tbwypirsu`piv#k`~rsxYrsgikzuQ¯`k t u`9kµ©ºx3vMu>°
°k#pi`rsg²x`©rsgk¦x9r wv u`°@ k p x9tSR x9t'rsgikbit wypiv wyu`°@Bx9wpdrTex` wªrswyx9pBUWVSXZY[\V6])^V_
?1w1³ók>³ rsgikwypdrsk#t k#v#r wx9p x`©Ãr gikx9r wv u`°u $wy¦§wªrsg rsgkt krswypu`°°u`pk`~Zu`°y xÀvMu>°°yk
` ©ºxQ¯9k ua}~D k#kbwyi³cCd@³oqpµ©1u`v#r ~ewªrgiu`Bk k#pk"$PBk t wyjWk pdr}u>°°ª«ik#jWx9pir tsu>r k BAeCQl~BlgfD
rsgu>r ~©ºx9t¥ ¥u`°Ãv u`jWk t u`%u`pi v x9piwk t wypi ` t x`ex`r wv#haz¯Pwy ¥iu`°Ãek#t v#k r wx9p¹~rsgik# k
rsgitk ku`t u`jWk#r k t %i¦ik# v#t wyekitx9ek#t °ª«½rsgkit x>k#v#r wx9p x`©u lgj ¥iv °ywkMu`p¿ex9wypdr
k XlYmH]no]pq_¸x9pdr x¦u!t krswypu`°j Bx9wpdrrUZXsY[t]4^!_¶rsgitx9¥i9gvu
w [ X [\VyxKRFz
^ X ^V{x|R~}
§wªrsg
z X m'6p
} X n6p
?:Cd@
v x9piwk t wypiÀu ©ºtsu`jWk%x`©tk#©ºk#t k pv ku¡r r}u>v}gik 6r xr gikµtk#r wpuP~¸§giwyv}gÆx9t wy9wyp6wy.r gik
k#«9kÃx`rswyvMu`°Bv k#pdrst k?ºwy©u>pI«`~igik#piv kr gikwyjbu>9kpix3u`°Bex`wpdr@|~Pr gikx9r wv u`°eu $PwBk wypi
u`°wy9pik#§wªrsgYrsgikp{ëu $Pwyu>pigx9t w x9pdrsu`°@u`pi¯`k trswyvMu`°@wj.u``kbv#xPx9tiwypu>rsk#Bk wypi
u`°wy9pik#z§wªrsgzr gikm u`pi<nÃ{ëu $Pk t k ek#v#r wy¯`k °ª«9~"u`w°y°¥rst u>rsk#wyp<bwyi³cC9³fgkÅ`k«
ex9wypdr¶gik tk?1 k#kACddDi©ºx9tu`pWwypdrst x3i¥ivrswyx9p\@\wy¶r gu>rr giwÅIpixQ§°yk i`kwy¶jbu>piu>r x9t«nrsx
(%4eg
þ{ ;ã-#;þ &þ #ïà Bä>ÿä2#;ãëþ eã'r`àúåÿúóþ# #;þ( Ö6ä2#ïãëþ( 0`þã'Ûý ;ã-# #ïþrGã-#;â
ä|áÛã-#ïäú>úûþ##;þ( %·âÛþ(ÿ>ä|ú eà,%&þ #;à eäïþ ïþáïþ( ;þ #ïþÿä #;â¡þ #ïä1 1á`à ;þ"à,Pä&#þ( #ïà·ä¡ÿ ä|úåä
ã'ã)#¾ä|úûã'Ö0#;âÛþ¶ÿQà#>áïàMÿd1 #>þã-¡ý ïã-# #;þ(Ãä 0!ä>ÿ #;â¡þïà áïàMÿd1 #0 Fào¡g¢>£y\¤·âÛþ
ãåÿQþ(#ïã)#+*¥ä2#;㧦ã o ïã-# #;þ(¨
þ ïþáïþ( ;þ # #ïâÛþ àá>àÛþ # ¶à,"ää#ï㩦à@ä &þ(#;ào1 ;ã-Ûý
d1 ïãûá # ÖQþý«ª\¬<®°¯±²>¯g³E²>¯´µ \¶"þ(;þ{·þ ä1 c¸c¹O ,ªà ;àþo¸Sº+»d
¼
Ò/ ëàád #;þ( &Mã' ;ãóàIÖàïþ\½ úûä ;ãä|ú©¾àQÿQþú À¿ªþýÁ«ÂÝÃMÄ!1 ;þ #¹à\ä#ÿÛÿQã-#;ãûà¡äú½/#ïþ(¾â¡àúûàýãäúž
á>ä¾äþ#ïþ hÀÆ ÃàÛþ¶â>ä¡ÿ¿yãÇÄ #;âÛþ¶â¡àïã§Èà #ïäúMÉ2&þ#;ãäú ä|úûþ ïä#;ãûà2GâÛþÃáÛ㧦Qþú ä;þ Ûà# "Ê>äïþ
#˽ ;þ #¾äÛýÛúûä¾qÖGâÛþ(;þ ä ã-#eâ>ä \`þþ( ;âÛà Á§ÂsÞMÖEÌÌÃ!#;â¡ä# #ïâÛã' Bá¡ä¾äþ#;þeã' eä|ú«¹ä * à1 #ïä #
ã'áïä #ïã'þÖ #;â61 äZþ ä ;ãóú'*2>þ ;þ# #;àà¡þÖ61 ;ã-Ûýä 1ã«áÛúûþ à àïÿÛã-¡ä#;þ #¾ä1 ,ªà"ä2#ïãóà=ËÆ #;âÛþ
à#;âÛþ(@â¡ä¡ÿ;¿yãóãMÄ #ïâÛþ Íþ% ,yä #ïàÎGã-#;â ëàïþ( ;á>à¡ÿ 2Á«Â)Ï4à #ïà¹ã' ïä|úûãóýdþ #@à, #;âÛþàá #ïã'ä|ú·ä¦Qã'
Gã-#;â ïþ( ;á`þ( # #;à #;â¡þ ïþ#;ã'¡äúDáÛúåäÛþ#Ö7GâÛþ(;þ ä ã-#@ã Öiã'!á¾ä #ïã'þÖiä|ú«¹ä * Ûþýúûãóý#ã§Ûúûþ ,ªà¶úóà½ä>ÿ
â¡ãóý#âÊE¡ä|úûã)#+* äþ¾ä 2Á«Â ÞQÖ!ÌÌÃÐnÒ/.ä#ÿÛÿQã-#;ãûàIÖ #ïâÛþÚá¡äïä¸þ#;þ( àQÿQþú¹ã &þ*þ:Ñ ëãûþ #ÒGã-#;â
ïþ( ;á`þ( # #ïàä #ïà/ä|úûã«ïä#;ãûàþ +#ïã§ä#;ãûà¥ÁÌÂ|Ö>Þ|ÜÃ3
k#¯>u`°y¥u>rsk#¶¥v °ywik u`p9k#x9jWk#rstwv u`°&I¥u>pIr wyr wk# ¥iv}g%u`¸iwr}u`pv k #~u`pi`°k# ~¯`k °yxPv#wyr wk# ~
k#rsv>³³y³
0 Focal
f fronto−parallel plane
Optical center
Retinal plane
axis Optical
length point (fovea)
Principal m m
x y C
z
M
bwy9¥it kCguojk v#t wyrswyx9pµx`©wypdrstwpiwv'u`t u`jWk#r k t Zx`©u¯Pwy ¥iu`°"«3r k j³
r wv u`°wrsx9trswyx9piu`tku`°y xÀrsx½Bkr}u>Å`k p wypdrsx¼u`v#v x9¥ipdrvA´l ~¸lddD;³ oqp v#x9j{
i¥rsk#t¯Pwy wyx9p·~P9k x`j.krstwviwyr x9tr wx9pguM¯`kBk k pbjWxPik#°w k#·~I§gik tkMu`@x`rsgk t@x9rswyvMu>°
x9¥itv k#\x`©k#t t x`t 2?ºiw!¹t u`v#r wx9p¹~du`v}gitx9j.u>rM~`u`r w9j.u>r ~9k#r v`³y³³«@u`tk¸pik#9°yk v#r k !x9tr}u`Å>k p
u`u ` t k# wyi¥u`°Manpix9wy k`³oqp©1u`v#r ~e¥iv}gYu¦v x9tt k#v#rswyx9pv u`pBkkMu> wy°y«µix9pik¥i wypiv ¥k
¥iv}gµu`t k#v#r wª{¾°wypikMu>t wªrq«/A!D"u`piKr t kMu¡rsk %wypu!©ºtx9pdr{¾k pijWxP¥i°k>³fgik#«§wy°y°·pix`rBk
v x9piwk t k#zgik#t k>³
b¥itrsgik#t jWx9tk`~ir}u>Å3wypiWrsgikÃk«9knu`pu>r x9jn«Kwpdrsxbu`v#v x9¥ipdr ~wyrZwyZx`I¯3wyx9¥irsgiu>r ¥iv}g
u`tsu>j.krsk#t ¯>u`t«©ºtx9j r gik©ºxÛ¯`kMuÃrsxrsgikBk twigk t«`~P¥ru`°x'§wªrsgWrswyjWkZu`pi.k"$3rsk#t{
pu`°G©1u`v#r x9t #~§giw°yknr gik wytk#r wj.u>r wx9pµ©ºt x9j®r gikwtk v#rx9ik t¯>u>r wx`pzx`©\r gik'k pd¯3wtx9pP{
j.k#pdr?¬¯9k#t« °ywÅ>k °ª« wyr 'tw`w r t ¥ivrs¥itk )@wywpik k ²u x9°ª¯>u`i°yk.t x9i°yk j AûfDï~Gk¯9k p
©ºx9t¯>u`t«3wypiYu`pi¼¥ipiÅIpixQ§p ©ºxPv u`°¸°yk pi>rsg u`pi¼itwpiv#wiu`°ex`wpdr+Aó`D;~@¥rWpI¥ijWk t wò{
vMu`°y°y«bwy°°ò{qv#x9piiwªrswyx9pik#/AûgdDï~P¥ipi°yk u>v#rswª¯9kÃek#v wÁ vÃiwy i°u`v#k jWk pdrsu`tk¥ik ?1j.u`wypi°ª«
t x`rsu>rswyx9pu`°eiwy i°u`v k#jWk pdrsA´lP~0CQDÐ@³
o¾r¸wrsgI¥iu°yk `wyr wj.u>r k,I¥ik rswyx9prsxu>r r k jWrr xÅIpixQ§ §gik#r gik tr gikit u`wypKix3k
pix`rrst«rsx ` uM¯9x`w7a!¥i wypiz¥iv}g²v u`°ywit u>rswyx9pÅ3pxÛ§°yk 9k`~(k wªrsgik#tnBk v u`¥i kbuM¯>u`w°u`i°yk
¯Pwy ¥iu`°u>rsuzj.uM«½pix`rek¦ ¥.v wyk pdrWrsxk °ª©{qv u`°wyit u>rsk<AeCf~\gdD¸x9t¥ir!Bk vMu>¥i kwªr
wbx`©¬rsk#p6Bx9 wi°ykzr x²Bk t©ºx`t j ¯3w¥u`°Zx9tW¯Pwy ¥x>{qjWx`r x9tbrsu` ÅIHAeCP~¶3~¸dDZ§wªrsgx9¥r
vMu`°ywtsu>r wx9pzx9t§wªrsg ub¯`k t«u`iitx$wyj.u>rsk!ÅIpixQ§°yk i`k!x`©rsgik#wt¯`u>°¥ik#AC D;³ ¤ gik p
pix`rvMu`°ywtsu>r k ·~u¯3w¥u`°¹«3rsk#j®wy¸pix>rx9ik t¯Pwypir gik#¥iv#°wyikMu>p9k x9jWkrst«bx`©·r gik
v k#pik`~d¥r\x9p°y«'wªrst x>k#v#rswª¯9k¸x9tubpk¸9k#x9jWk#rst«9~du>u`t u`jWk#r k t wy k#!wpA´dP~` DI©ºx9t
wpir}u`pv k`³¶uv x`pi k3¥k piv#k`~ ` vMu>°wyitsu¡rswypiax9t\pix>rrsgk¯Pwy ¥iu`°i«3rsk j w\kI¥iwy¯>u`°yk pdr
x`©v}gix3x9wpi¿u¼9wy¯`k p ?ºk wªrsgik#t ¶¥v °ywik u`p x9titx>k vrswª¯9k6@%9k#x9jWk#rst« rsx t k#it k# k pdr
u>r w u>°"u`piÅIwpik#j.u>rswyv'¯3wy ¥u>°wyp©ºx9tj.u>rswyx9p·³
P
¥it itwwp9°y«Wk#pix9¥i`g·~Iv x9piwk t wypi'©ºx9twpr}u>piv kZek#t v#k r ¥u`°Dk"$PBk t wyjWk pdrs¥i wypi
u`pu`jWx9tigixg#wpi9°u` k AeCgCED;~Gwyrgu`ek#k p k r}u`°wy gik#r gu>rr gik!¯Pwy ¥iu`°«3r k j wy
u`i°ykrsx!tk{qv u`°ywit u>rsku!Zwyk j.u`pipwu`pjWkrst wyvu`u>rsk#r xrsgkÃ9° u> k#ik#©ºx`t j.u>rswyx9pi#~
rsgiwy!i°u`r wv%u`u>r}u>r wx`p¿u>°°yxQ§wpiYuv x`t t k#v#rWBk tv k Prswyx9p¿x>©Zr gik v#k pik ¥iv °ywyikMu`p
9k x9jWkrst«`³Wfgik.u` ¥ijWrswyx9pr gu>rÃr gikWitsu`wyp v x9`piwyr wx`pwyj.x`t k
`
¥iv#°wyik u`pr gu`p
u.pik!x9tÃitx>k vrswª¯9ka!wÃtk wyp©ºx9t v#k Yd«ut k#v k pdr'k"$PBk twjWk pdr'Aû>D\wyp§giwv}gwyrgu`
ek#k p²gixQ§p²rsgiu>r¥iPk#v#rsit k©ºk t k#pdrswu`°°ª«Yv x9p wyik t'rsgikbx9tr gix99x9piu`°wªrq«v#¥ik<?¬rsgI¥i
u`p ¥iv °ywyikMu`p%v ¥k@t u>rsgik#t¸rsgiu`pKu`t u`°y°k#°wy j©ºx9t¸wypirsu`piv#k?º§giwyv}gwu>pu.pikv ¥k@
wp%x9t k trsxWBk tv k wª¯9k'rsgikx9twk#pdr}u>r wx9p%x`©u! ¥it©1u`v kit uQ§pµ¥i wypi.v ¥it¯9k# ³
u.v x9pikI¥ik#piv k>~·t k# kMu>t v}gik#Ãrs¥«3wpi¦k wªrsgik#tiwyx9°yx99wyvMu`°"jWx`rswyx9pµek#t v k#rswyx9p
Aól`P~"l!CED¶x9tÃk¯9k#prsgkbBk tv k Prswyx9px>©¸rsgk.v k#pikbrst¥iv#r ¥it kW©ºt x9j+j.x>rswyx9pv#¥ik '?1k>³ó³
Aû7C>~¤DÐ@u`°y§¸uM«3u` ¥ijWkrsgu¡rZr gikitsu>wp¥ik u.v u`°wyit u>rsk#%¯3wy ¥u`°G«3rsk#j³
fgk p·~r gik#I¥ik#rswyx9pwypix`rr xÅIpixQ§ wy©"rsgkitsu>wpK¥ k ¥iv °ywyikMu`pBk tv k Prswyx9p·~iwªr
ixPk# ~Ii¥r2 rt&$¤$657 >! "4$1M>NE4$>"!$%$&' ( G%oqp.u`trswyv ¥i°u`t#~
v x9piwk t wypiWjWx`rswyx9pBk tv k Prswyx9p·~§k§¸x`¥i°°ywÅ>kÃrsxrs¥i«¦wpr giw¸u`Bk t ~¤
Ò) 4$Î:'$;!$=57 ¥:9¹ ¹>! "2!$%$&' : ) #G
fgwZwr gik9xdu`°·x>©(r giwu>ek#t ³
oqpr gikpik $3r k#v#rswyx9p·~d§kt k¯3wwyrr giku`t u`jWk#r k t wysu¡rswyx9p!x`©rsgkt krswypu`°t x>k#v#rswyx9p
x`©(r gik'lgj j.x>rswyx9pµx>©\u`px93k v#rwyp%r gikvMu`k'x>©wypdrst wypiwvnu`tsu>j.krsk#t¯>u`twu¡rswyx9pi ~wyp
x9t ik#trsx.eku>i°kÃr xW©ºx`t j.u`°ywk'x`¥itit x9°k#j%³
fgk p·~§¸kbiwv ¥igixQ§r gikb¯>u`twu>r wx`pix`©¸r gik¦wypdrst wypiwvbu`tsu>j.krsk#t 'v x9¥i°y½Bk
ek#t v k#wy¯`k ·~Gwypzr gikv u` k§gk t kk wªrsgik#trsgikek#t v k#rs¥iu`°(«3rsk#j+wyÃu`i°yknr x¦r}u>Å`kwypdrsx
u`v v#x9¥ipdrZ ¥iv}g%¯`u>t wu`i°yk x9tpix`r ³
btx9j®rsgwZiwy v ¥ wyx9p·~Bu`t u`iwy9jWu>t kit x`ex9k µwp%x9t k trsxbu`pu`°ª«3 k'gixQ§ÂgI¥P{
jbu>p%v#x99piwªrswyx9p%wwyp ¥ik#piv k#d«wypIr t wypi wyv'u`tsu>j.krsk#t ³
$PBk t wyjWk pdr}u>° kr{¾¥i u`pi¿t k# ¥i°ªrs¦u`t kµit k# k pdr k u`pi¿iwy v ¥ k# wp rsgikµ°u`r
k vrswyx9pi#³
&&') "( $" (#& Â) #n & #
("krZ¥itk#¯3wwyrZrsgknu>tsu`jWk#r k tw u>rswyx9pzx>©rsgkntk#rswypu`°"itx>k vrswyx9pzx>©rsgklgj jWx`rswyx9p
x`©¶u`px`Pk vrM~·u>ik¯9k °yx9Bk wypBAó3~yCMD(©ºx`tpik#¥it x¡{qigd«3 wyx9°yx99wyvMu`°(j.x3ik#°x>©jWx`rswyx9p
ek#t v k#rswyx9p·~G¥rpixQ§ v x9piwk t wypiµu`p¥ipP{¾vMu`°ywyitsu>r k ¯3w¥u`°\«3rsk#ju`Ãik¯9k#°x9Bk
©ºx9twpr}u>piv k'wpvAó`lD;³
fgk kik¯9k#°x9j.k#pdrsu>t kÃu`k x9pf(uQ«3°yx9tk"$Pu>pi wyx9piZx`©"rsgikÃtk#r wpu>°¹¯9k#°x3v wªrq«
Á k °y·~r gik%lgjį9k °yx3v wªrq«¼ek#wpiu` ¥ijWk rsx Bk%°yx3vMu`°y°y«½v x9pir}u`pdrbu`pi¼rsgk%¥it©1u>v k
°x3vMu`°y°ª«i°u`pu`tx9t I¥u>itsu>r wv>³
¤
kÃrsgI¥iZu`t u`jWk#rsk#t wy ku
rsgik%lgj t wy9wy¿jWx`r wx9p¿x`©Ãu Bx9wpdr k X Ym8]no]pq_¿rsgitx9¥i9g wªrs.wpir}u`pdrsu¡{
pik x`¥iÃrstsu>pi °u>rswyx9p ¯9k#v#rsx`t X Y]])_Àu>pitx`r}u¡rswyx9pu`°¯`k °yxPv#wyrq«ZX
Yg]]_ zx.r gu>r¹u !
k X" x k #$
w;³´k`³Àu %&
'&( !
m X)#x|p*,+Bn-
!
n X.ox m/0+ p1
!
pX)2 x n-3+ m43
?;g@
rsgik!°yx3vMu`°v k pkbrst ¥v#rs¥t k!rsgitx9¥i9gr gikW k v#x9pix9t ik#tÃfuM«3°yx9tk"$Pu`piwx`p²x`©
rsgik!t x$PwjWwªrq«9~w1³´k`³rsgk.wypd¯9k#t kbx`©@rsgik!ik#rsg¹~(x%r gu>rM~·©ºx`tÃwpir}u`pv kW¥i rsx
k#v x9pi%x9tik t#~i§¸k'x9rsu`wp/u
5QX 6
p
X.57yx z859x }:59x z;59Òx zF}5<ox }=;>5<? ?1l@
§gik tku
?1w3@rsgik'u.pikitx$wyjWwyrq«©ºx`trsgiwytk#r wpu>°·ex`wpdrZw35@M~
?1wywÐ@r gik'lgj ¶¥v °ywik u`pz¥it©1u>v kpix9tjbu>°GwBADC Y5<]5<]E57_ "~
?1wyww3@§giwy°k!r gikWx`rsgik#t 3¥iu`pdrswªrswyk 'Y59]59]5<?E_u`tk!t k#°u>r k rsxrsgk.¥it©1u>v k
v ¥it¯>u>rs¥itk`~óu` v}gik#j.u>rsw k#wp bwyi³DP³
N Z
Surface Patch
Retinal Plane
m
x y
M
C z
bwy9¥itk'7uÎjk v twr wx`pzx`©rsgik°yxPv u`°·v k pknrst ¥v#rs¥t k`³
btx9j r gik knpix>r}u>r wx9p ~¥iwpiWrsgkr k jWBx9tsu`°¹ik twy¯>u>r wx9p%x`©#?hCd@~u
w !
[ X !
[\V x !
R~zx|R
Y !
mD+ z !
pq_
!
^ X !
^V x ! R~}qxKR
Y ! n +B}
!
p~_
?º=@
u`piYu`ii°ª«3wpµpix>r}u>r wx9p'wypdrstxPi¥v k wp Aó`lD\©ºx`tÃ¥ipP{qv u`°ywit u>rsk#tk itk k pdr}u>r wx`p²x`©
rstsu>pi °u>rswyx9p%u`pi%t x>r}u>r wx9p/u
w
X )
X .
§wªrsg X R [\V
R ^V
6
? g@
¥i wypi8?hCd@|~c?ïg@|~c?1@u>pi ? g@|~§¸k'vMu>p%§twyr ku
%&
'&( !
[ X R !
x
[ x
Y 3+B[ _ x Y +B^ _ x !#"
!
^ X R !
$
x
^ x
Y ,+B^% _ x Y+ #x [& )_Zx
$ $
!#"
? @
§wyr g u
!#"
X
'
Y( :+B^V) _7+
$ $
Y 3+B[¤V) )_³
иx`pi wyik twpµu>°x¦rsgikn°x3vMu>°(rst ¥v#rs¥t kx`©r gikv k#pik`~"u`u`t u`jWk#rsk#t wy k#Ywyp ?1l@
rsgiknk"$Pit k# wx9p x`©r gikk v#x9pi x9t k tk $Pu`piwx9px>©rsgiknt k#r wpiu`°(¯`k °yxPv#wyrq« Á k °yj.uM«
ekçt wªr r k pvu
w !
[ [\V x [ [ x [
; ^ x [ [ ; x [
;
[~^ x [
; ;
^ ;
!
^ ^V x ^ [ x ^
; ^ x ^ [ ; x ^
;
[~^ x ^
; ;
^ ;
?g@
§giw°ykrsgik'it x$Pwyj.wªrq«9~B©ºt x9j ?;l@u`pi ?hCd@|~DwZ9wª¯9k pµwpzubtk#rswypu`°·©ºt u`jWknx>©\t k©ºk tk piv#k`~
w;³´k`³Àu
5JX 6
p
X yx [
x ^
$ x [;
x [F^
$ x ^ ;
$ $
?>f@
§wyr g u
%&
'&(
;
7V
V $
$
; $
7V
$ V
$$
$ $
; 7V
V $ ;V 7V
V $
;V
$ $
§giwy°yk`~Bu>©¬rsk#t x9jWknu`°y9k it uWu`pi%¥i wypi8?hCd@|~c?ïg@|~c?fg@~c?º=@u`pi ? g@|~§knx9Pr}u`wyp/u
%&&&&&&&&&&
'&&&&&&&&&&( [¤V X R !
x > x + [\VtY>^V) 3+B[¤V) E_%gR ;
[ X
+ > yx!
x x Y>^V) 3+"{[\V Îx [\V^V _4gR
;
[ ; X $
3+ x [\VtY( 3+B[\V _%gR
;
[ X +#
x x Y( +B^V _4gR
;
[ ; X +#
$ x $ + Y( +W[¤V) _%gR
;
[
; ;
X x $ $
?1@
u`pi/u
%&&&&&&&&&&
'&&&&&&&&&&( ^V X R ! $
x > + + ^V Y^V) 3+B[\V 6_4gR ;
^ X
ox + ^V Y ,+ ^V) _4R
;
^ ; X
+ > yx!
$ x Y$"{^V 3+B[\V +B[\V^V) _%gR
;
^ X x
^
; X +#
x $ x Y( +B^V _4gR
;
^
; ;
X +#
$ x
$$
+ Y( +W[¤V) _%gR
;
?hCM@
fgk kÃkI¥u>rswyx9pi¸ik v t wyBkrsgiktk#r wpu>°eiwy °u`v#k jWk pdr ~w;³´k`³\§gu¡rvMu`p¦ek°yx3vMu`°y°y«
ek#t v k#wy¯`k v?º¥ibr xnrsgk k#v x9pKx9t k t@x9p.rsgikZt krswypun§gik#pKx9ik t¯3wypi!u't wy9wbx9Pk#v#r
wp%jWx`rswyx9p·³
Zx`rskrsgu>r¸wpKr gik ` vMu`°ywtsu>r k vMu> karsgkt k#°u>r k k3¥iu>rswyx9piv#x9t tk Bx9piKr xu`të{
rswyv ¥i°u`tv u` k!x>©\rsgix` k§wªrsg<[\VFX ^VqX ~ÀR/X
6
~¹ xbrsgu>rrsgikt k#r wpiu`°(v#xPx`t iwypu>r k
Y>[c])^7_v x`t t k# Bx9pirsx8Yzc])}!_~3§wyr g
X u`pi
X.b³
(Gkr¥u>pu`°ª«Pk'r gik kkI¥u>r wx9p ³
%&Ã ¸&n (¸ i& %(&Ã" &'$"*
oqpbrsgk9k#pik t u`°ev u` k>~3©ºtx9j ?;g@@u`p/?hCQg@~d§¸kguM¯9kSCQk3¥iu>rswyx9pi#~i¥rrsxnekZt k#°u>r k
rsx Y>[\V6])^V6]R\]
!
[\V6]
!
^V6]
!
R_|~
~
u`pi|Y 6]$
]$
$
]$
]$
$ ]$
$ $
_w;³´k`³;Cf¥ipÅ3pxÛ§p ³ ¤ k
rsgI¥iZu`t kwypzu>pµu>°9k#itsu>wv wªrs¥iu>rswyx9p§wyr g ` rsx3x!jbu>pI«¥piÅIpixQ§pihanu`pµj.uM«<u
¥i k'giwy9gik#t{qx`t ik#tZkI¥u>rswyx9piAeCEDï~§gik#pzv#x9pi wyik#t wypiK k#v x9pP{qx9tik tZkI¥u¡rswyx9pi
?º§wªrsg ¯9k °yx3v wªrswyk Ku>piÆu`v#v k#°k#tsu>r wx9p ~w1³ók>³Æ¥iwpi½u>r.°k u`r.rsgitk k¯3wk§@Wr gik
jWx`rswyx9pu`t u`jWk#rsk#t Zek#v x9jWk ` x`i k#t¯>u`i°yka
§¸k §wy°°k"$PBk twjWk pdrwª©r gikYj.x>rswyx9p Bk tv k#rswyx9pÇv#x9¥i°yÇBkiu` k#Çx9p ¥iv}g
giwy9gik të{qx9tik t!v#x9pirstsu>wpdrs#~i¥r.¥i¥u`°jWx3ik °ybwyp iwx`°x9`wv u`°¯3wwx9p¿u`tk%px`r
u`k x9p ¥iv}g v ¥k ek#vMu`¥ kÀrsgik«Âu`tkÀpix`r v#°k u`t °ª« x9 k t¯9k# wyp pk ¥itx>{
igd«3 wyx9°x``«vAól>~Dl!CD
iwyt k vrs°ª«¥ikrsgik ` x9ik t¯>u`°kaw1³ók>³'r gikjWx`r wx9p Á k#°iu`tsu`jWkrsk tu`pirsgI¥i#~
u`wpdr t x3i¥iv k#Çd« A´l9D'u`pi AeCdD;~x9pi°ª« v x`pi wyik trsgk²u.pik x9titx>k vrswª¯9k
9k x`j.krstwv u`°Git x`ek#trswyk x`©rsgk'v k#pikb~ir giwyZwyvMu`°y°yk ` §¸k u`Å9{qv u`°wyit u>rswyx9p7as~
wpdr t x3i¥iv k@u`iwyr wx`pu`°9v#x9pir tsu`wypdrs?1v#x9pirsu`pdrvMu`°ywtsu>r wx9pu>tsu`jWk#r k t ~¡Å3pxÛ§°ò{
k i`k'x>©rsgk ` gix9tw°#x9p7aw1³ók>³@r gik'i°u>pik'u>rZwp Á piwyrq«`~eiu`tr wv#¥i°u>tZwi°u>v k jWk#pIr ~
k#r v`³y³³«@®wp¿x9t k tWrsxYekµu`i°yk%r x²k u` wy°y«¿u`piu`°y«3k%r gik kzu`t u`jWk#r k t #~rsgu¡rbwy
v x9p wyik tW Bk v#w
Á
vµwyr ¥u>rswyx9pi ?1u`!«3pdrsgik# wy k# wpAËCEDÐ@~¶rsgiwy!wWv u`°°yk
`
k °ª©{
vMu`°ywyitsu>r wx`p7a}~
u>Zk#¯9k#°x`ek#zpixQ§³
bDtx9jÇkI¥u>rswyx9p'? @|~`u>wv ¥i k wp Aû>lDï~Û§k@j.uM« Á trv x9p wyik tjWx`rswyx9p'ek#t v k#rswyx9p
v °yx9 kÃrsx!r gikit wypiv wyu`°"Bx9wypIr ~w;³´k`³r gik©ºxQ¯9kMu>°Bk tv k#rswyx9p·³
@« ` ©ºxÛ¯`kMu`°Maek#t v#k r wx9p¹~P§kjWkMu`pKr gu>r¸r gikr k tj§wªrsg
!"
wpik `°wy9wyi°k wypiv kÃwªr
wt k#° u¡rsk rsxWk v x`pizx9tik tr k t jWwyp Y%Y>[ +B[¤V_4gR\]dY>^+B^V_%gR_|³
É¾Í ¢( ÍÉ Í (Î6 É É . Í Q Í o¾©§kv#x9piwik#tZr gik
¤) "! \x`©GrsgikZrst u`pi °u>r wx9p
?;u`°y xÅ3pxÛ§pu`r gik ` ©ºx3v ¥¸x`©Gk $Pu`piwx9pah@
u`pi x`©Ãr gikt x>r}u>r wx9p
~r gik kkI¥u>r wx9p ~©ºx`t Á $Pk wypdrst wypiwv u`t u`jWk#r k t #~
guM¯9k'k"$u`vrs°ª«rsgiksu>j.k©ºx9tj u`wypµrsgk'v u`°ywit u>rsk#µv u` k>³¸fgiwywZwy°°y¥ir tsu>r k
wpHbw³¹l³fgiwwpx`rr t ¥ik>~Gu>rr gik
`
k#v x9pix`t ik#tha}~ew;³´k`³§gk prsgkrsk#t j
!#"
wpixWjWx9tk'pk 9°yw`wi°yk`~Du>¯3wy wyi°yknwyp ?1@u`piW?hCM@³
t r
z / Z z y
t / Z + rx y t / Z + rx
b(w9¥t k'l!u ek#v#r x`©rsgk'tk#r wpu>° Á k °yµv#°x` krsx!rsgikéºxQ¯9k u³
oqpWu`tr wv#¥i°u`t¶wyr@wy@x`I¯3wyx9¥i¶rsgu¡r§¸kZkMu`w°ª«!iwyr wp9¥iwy g @u`pi ¶i¥r@pik wªrsgk t
©ºt x9j Ãpix9t ©ºtx9j d³
oqpYr k tj.nx`©v#x99piwªrswyx9p·~yjtx9¥i°ykEHA´D;~(¥i99k#r uek#t v k#rs¥iu`°jWx3ik °\©ºx`tr gik
itsu>wp'rsx`k#r(t wyx`©Prsgiwy\u`jniwy9¥iwªrq«9³¶Ð¸x9jWi¥Prsk t¯3wwx9pu`°`x9t wªrsgijW(¥i k k v#x9pi
x9t k trsk#t jWrsxWiwysu`jniw`¥u>rsk'?1k>³ó³A´dlD@|³
É
Í
Í É
"
Í
o¾©@§k!k °ywyj.wypu>r k
©ºt x`j ? @Z§k!x9rsu`wp uKv#x9pirstsu>wpdr
x9prsgikv u`°wyit u>rswyx9pµu`pijWx`rswyx9p%u`tsu>j.krsk#t §wyr gµj.uM«Bk§t wªr rsk#pWu
!
U
!
U x|U
#"
U X ?hCCd@
§wyr g8U X Y>[c])^\]
6 _ ~
%$ X '&($ ~i§gw°yk u " X ³
fgiwyKv x9tt k# Bx9pii¦r x²v#x9pdrswyp3¥x9¥iK©ºx9tj Aól9Dx`©rsgik ` ©º¥ipiu>j.k#pdr}u`°ZjWx`rswyx9p
kI¥u>r wx`p7aÃx`©'(Gx9pi9¥ikr{(-wy99wypi¹Aû¡D"u`pi Ï uM«3u`piÅ9{(bBu>¥i9k t u`AûfDï³
)
·âÛþ ïþ#;ã'¡äú#á`àã' # à;þ 1á`à¡ÿÛã-Ûý #ïà #ïâÛþ¹á;à+*ºþ(#;ãûàà,1#;âÛþ #¾ä1 ;úåä2#;ãûà¿yã°þ #;â¡þeþáÛãóá`àúûþtÁ«Â)Ï)ÃMÄ
à;þ 1á`à>ÿ #;à #;â¡þ ;þ#;ã'¡ä|úÛúûà ä#;ãûà¸à, #;âÛþ &}äÛã' ;âÛã'Ûýá`àã'# ,ªà #;âÛþ(ÿQã-ïþ(#;ãûàà,!#;â¡þ #¾ä1 ;úûä#;ãûà=
oqprsgikéºxQ¯9k u`°"vMu`k'?º k#k'A´l9DG©ºx`tuW`k pik#tsu`°Gk $t k wyx9p\@§¸kx9rsu`wyp u
" X + "
+ "
+ "
§wªrsg
%&&&&&&&&
'&&&&&&&&( X
X Y( 3+ R !
$
_ tx ) 3+
X Y( xKR
!
_ yx Îx
XlY x :+Y ; x ; _4 7_4
?hCQg@
xrsgu¡rrsgikj.u>rstw$ "
wy@u©º¥piv#r wx9p.x`©
?¬§wyr g.rq§x'ik 9tk k#¸x`©B©ºt k#k ix9j wypiv k
wWx9p°y«²ik Á pk ¥i r x u vMu>°k©1u`vrsx9t)@u`pi¼x`©Zrsgitk kx`r gik t!wypiik#ek#piik pdr
I¥u`pdrswªrswyk ¥Y \]]_~r gI¥iZ©º¥piv#r wx9p%x`©wpik Bk piik#pdru`t u`jWk#r k t #³
'uv x9pikI¥ik#piv k>~©ºx9t©ºxQ¯9k u`°¶j.x>rswyx9pBk tv k#rswyx9p§k.x9pi°ª«pik k#9k pk t wyv
ex`wpdrsu>¦©ºx9t¦r gikvMu`°ywyitsu>r k vMu> k ?1k>³ó³ AeCddDÐ@b§gik tkMu`²9k#pik twvBx9wypdrs
u`t knt k3¥wtk ©ºx9t°u`t `k t Á k#°Ãx`©¯3wk§'?º k#k;Aól9D©ºx9tuKik#¯`k °yx9ijWk pdr'wpr gik
v x9pdr wpI¥ix9¥v u` k6@³
bt x9jrsgiwy©ºxÛ¯`kMu`°GjWx`r wx`pµk3¥iu>rswyx9p·~i§knk u` wy°ª«v x9jWi¥Prsku
?1w3@rsgikéºx3v ¥ix`©(k"$Pu>pi wyx9p
~
?1wywÐ@wª©\rsgknwypdrst wypiwviu`tsu`jWkrsk tu`tkv x9pir}u`pdru`pi X !rsgiknit x¡k v#r wx`pzx`©
rsgiktx`r}u>r wx`p
?º§giwy°yk'wª© ÒX §knx9p°y«tk v#xÛ¯`k t @³
-xQ§k#¯9k#t ~§k u
?1wyww3@!pik#¯`k t¦v u`pÆv#x9jWi¥rsk%rsgik%wypIr t wypi wyvu>tsu`jWk#r k tH?¬rsgik«¿x`pi°y«¿u`ek u`tbwyp
k"$3rst u¡{¾©ºxQ¯`kMu`°DjWx`rswyx9p.rsk tjW ~P§gwv}gj.uM«Wk"$Pi°u`wyp¦rsgikpI¥ijWk t wyvMu>°eiw.v ¥i°ªrq«.rsx
k rswyjbu¡rskÃrsgik#j wypµit u`v#r wv#k?ºk`³´i³AógDÐ@4@u`pi§ku`°y x
?1wª¯!@npik#¯`k t!v u`pÀv x9jWi¥PrskKrsgk wyt¯>u`t wu>rswyx9piO?1w1³ók>³rstsu>v}Årsgk j©ºt x9jx9pikKwyj{
u`9k!rsxµu`pix>rsgik#t@Ã wypiv k.rsgik# kbr git k#kb¯>u`tw u>r wx`pi Y"R
!
]R
!
$
]
_'u`tkWx9pi°ª«
v x9prst u`wypik %d«rq§¸x.kI¥u>r wx`pi ³
KÉ ¢ ÍBÉ Ê Í #Q ¢` É (Î Í Í P
wypiv#kzwªr¦wybpx`r¦Bx9 wyi°k%rsxYk!ek vrswª¯9k °ª«
x9°ª¯9krsgikj.x>rswyx9p¦k3¥iu>rswyx9piwyp¦r gik9k#pik t u`°BvMu`kd«bk#°wyjWwpiu>rswypi
~3§kj.uM«
¥i kÃr giku`tsu>j.krsk#t wysu>r wx9p%x`©rsgikv k pkr t ¥ivrs¥itk'u`pi% x9°ª¯9k'?;@u>pi ?:CQ@|³
oqprsgiwyv u` k>~Du`¯3wwi°yk'wpW?;@u>pi ?:CQ@u`p%j.u`k'k $Pi°wyv wªrwpµu`iBk piw$ n~
§¸kv u`pµk#°wyjWwpiu>rsku
?hCd@r gikr t ¥ivrs¥it u`°Gu`t u`jWk#rsk#t ~ $
~
$ $
~
~ $
u`piWY 0+
_u`p·~
?ïg@r gikt x`rsu>rswyx9pu>°¹u`t u`jWk#r k t ¥Y( +
!
$
_~tY ox
!
_u`pi M~
jWw$Pk §wªrsg wypdrst wypiwvu>tsu`jWk#r k tK¯`u>t wu>rswyx9p·~ wypiv#krsgik«6x9Bk t u>rskz°wypikMu>t °ª«
wprsgikbkI¥u>r wx9p ~\§wªrsg½v#xPk bv#wk#pdrs!x9pi°ª« ©º¥ipivrswyx9p½x`©r gik¦r tsu`pi°u¡rswyx9p X
Y ]# ]# _³
unv#x9pi kI¥ik piv#k`~i§ku`°y xnixpx`r@t k v#xQ¯9k trsgikwpdrstwp wyvu`t u`jWk#r k t wp.rsgiwy
vMu`k`~§gik tkMu` $>$>"B3 ) 4<$ ) $>"! ; >%$! 1$"B
(¤u`iwy v#¥i k wyprsgik kI¥ik °1³
fgik kkI¥u>rswyx9piu`tk wypi9¥i°u`t¶x9pi°ª«!wª© ~X rX Ãi¥Prwyp. ¥iv}g¦uv u` k>~d§¸kZix
pix`rtk v#xÛ¯`k t
~
$
~
$ $
i¥r¶rswy°°ix`r}u`wyp ~ $
~Y =+
_u`pi Y( 2+
!
$
_|~
Y( x
!
_|~ M~§gik#t k u` ZwpixWjWx9tk'k¯>u`°¥iu>rsk#µwyp%r giwyZv u` k>³
fgikWu`t u`jWk#rsk#t x`©@rst u`pi °u>r wx9pYj.uM«ekWx9Pr}u`wypik ©ºt x`j rsgk.jWx`r wx`pk3¥iu¡{
rswyx9pB?:CgCd@u`Ziwv ¥i k#itk#¯3wyx9¥i °ª«9~ei¥r§¸kn gixQ§ wpu`iBk piiw$ r gu>rZrsgik«
j.uQ«bBkÃu`°y xv#x9jWi¥rsk#iwtk vrs°ª«¦©ºt x`j»r gik u`jWkÃkI¥u>r wx9p wypiv krq§x!°ywpik u`t
v x9prst u`wypIr k $wyr Bk#rq§k k pr gik wytZv x9jWBx9pik pdr ik
Á
piwp¦rsgikrst u`pi°u>r wx`pz¥i
rsx.u!j.u`9piwªrs¥ik©1u>v#rsx`t ³
KÉ
.1 Í Ì
¬
É
¢9
ÉÍ
"
É
QÛ (.1 Í Í
b
Í·Í
oqpÃrsgikv u` k§gik#t k
§¸kguM¯9kÃuO?ºwÐ@¶i¥itkrst u`pi °u>r wx9p¦x9tuO?ºww3@@v u`jWk t un©ºxPv u`°ò{q°yk pi`r g¦¯>u`t wu>r wx9p¹~Pw1³ók>³
u~ x3x9j~\?¬rsgu>r\w¶u¯>u`twu¡rswyx9p!x`©rsgkwpdrstwp wyvu`tsu>j.krsk#t ~d°¥i¶ur tsu`p °u>rswyx9p
x`©rsgk!x9rswyvMu>°v k#pdrst k6@§¸kWu>°xguM¯9k
!#"
X ³ ÃuKv#x9pikI¥ik pv k`~"rsgk k+$)
!$"1$¥4$ S:996 4$17 ) (¤>"³
oqpµr gikjWx`r wx9pzkI¥u>r wx9p
" X .©ºx`tu.i¥it k'rst u`pi °u>r wx9p¹~B§giwy°yk wypiv k X
wp ?:CÛg@~
"
gu`Zu! wyj.°k#tZ©ºx9tj ©ºx9tu x3x9j³
KÉ .1 Í Ì ¬ É ¢9 ÉÍ É " .% º eW Í Q Í oqprsgkvMu`k\§gik tk
X qX bw1³ók>³rsgiknt x`rsu>rswyx9p%gu`Zek#k pv u`piv#k °y°k#k"$Pv k#rwypµr giki°u`pik'x`©
rsgiknt krswypu~D§¸knrswy°y°\u`tknwypzrsgksu>j.kn wªrs¥u¡rswyx9pu`u.©ºxQ¯9k u`°(Bk tv k#rswyx9p·~·¥r
§wyr g X [\V'u`pi X ^V M³!fgiwy'jWk u`piÃrsgu¡rnwª©: X §k.x`r}u`wyp
u`trswyv ¥i°u`tkI¥u¡rswyx9pirsxWk rswyjbu¡rsk[¤VZu`p ^Vwª©rsgik«%u`t kÃv#x9pirsu`pdrM³
Zu!v x9p kI¥ik#piv k>~er giwwyZu`°x $!$>"1$4$ ¥:9{96 4$17 ) (¤"³
fgiwygu`KBk k#p%u`°tkMu`P«bj.u`kk"$Pi°ywv#wyrwypujWx9tk Bk v#w
Á
và wªrs¥u>r wx`pI«8Aó`lD;³
Í 1Î É É É ¢ Í Í É e Í É É " . É .;Πиx`pi wyik twp ` ©ºxÛ¯`kMu`°Ma
ek#t v#k r wx9pjWkMu`pi'¥iwpizu%t u>rsgk t j.u`°y°¯Pwy ¥iu`°
Á
k °y·³¦oqpY ¥v}gÀu%v u` k>~r gik
pI¥ijWk t wyvMu>°Pr}u`iwy°ywyrq«'x`©Prsgik@k rswyjbu¡rswyx9px`© k v#x9pix`t ik#tr k t jWwp?;@"u`pi'?hCM@
?1w1³ók>³ [ ~ [ ; ~ [
; ;
~ ^ ~^ ; ~ ^
; ;
@¸j.uQ«px`rek¥.v wyk pdrM³
uv x9pikI¥ik#piv k>~3§kj¥ir¶u`pu>°y«3 k§giu>rv x`¥i°!BkBk tv k wª¯9k#bwª©§ku`pu`°ª«3 k
jWx`rswyx9p.ek#t v k#rswyx9pK¥ wypix9p°y«!r gikv x3k .v#wk#pIr ¹?1w1³´k`³ [\VM~ [ ~[ ; ~ ^VQ~ ^ ~ ^ ; @
x`©rsgik'u.piku`trx`©rsgikt krswypu`° Á k#°¹³
oqp²x`r gik t§x9t in j.u`°y°@¯3wy ¥u>° Á k#°u`tkbrsx%Bk¦jWxPk °yw°#k ½d«u.pik¦tk#r wpu>°
jWx`rswyx9p Á k °yiA3~D`lD
oqprsgiwyvMu> k`~9§kixÃpix>rt k#v xQ¯9k#t ~ $ ~ $ $
i¥Prr w°y°ix9rsu`wyp ~ $
~\Y ) +
_(u`piOY( + !
$
_~\Y gx
!
_|~ M~`wpv k`~dwypr gikkI¥u¡rswyx9pi\j.u`ikk"$Pi°ywv#wyr¶wyp
u`iBk piw$ n~I§kZx9pi°ª«!¥i ku.pikZv x3k .v wyk pdr ¸x`©ersgikt krswypu`° Á k#°.rsx'v x9jWi¥Prsk
rsgik# k I¥u`pdrswªrswyk #³
b¥itr gik tj.x`t k`~ewy©\wyrwpx`rBx9 w°kr x¦k#rswyj.u>rskr giknr tsu`p °u>rswyx9pz©ºtx9jÄrsgik# k
kI¥u>r wx`pi ~wªrZvMu`p%r w°y°Gekk#rswyj.u>rsk#·~D¥i wypiWrsgikjWx`r wx9pk3¥iu>rswyx9p·³
fgik tk#©ºx9tk`~"¥iwpi% j.u`°y°\¯3w¥u`° Á k#°nix3k Ãpx`rt k#rst u`wypY§gu¡rnv#x9¥i°yBk!t k|{
v xQ¯9k#t k#µ©ºt x`j ?;g@u`piW?hCQ@|³
-xQ§k#¯9k#t ~Zwy©& X X P~
u`pi $
u`t kµpix½jWx9tkzx9ik t¯`u>i°k>³ fgwKwy
rsgI¥iuµiw!¹k#t k#piv k¦§wªrsg²j.x>rswyx9p½Bk t v#k r wx`p½¥i wypik v#x9piP{¾x9t ik#tnr k t jW#³µoqp
u`trswyv ¥i°u`t!wª©x9pk%`k pik#tsu>r k .uYrswyj¥i°y¥i.x`©u °u>pIr k i°u`pik¥ipiik#t 9x`wpiu
rst u`pi °u>r wx9p u`°yx9pi¼rsgik¡{qu $Pwµx9pi°ª«9~Ãu`v#v x9tiwypi¿r x r giwµjWxPik#°;~rsgikYi°u`pik
x9t wyk pdrsu>rswyx9p¦wy¸x`i k#t¯>u`i°ykwª©u`pi¦x`pi°y«.wª©GrsgikZ¯3w¥u`°BixPk#¸pix>rx9pi°ª«.¥i ku.pik
v x9jWBx9pik#pIr x>©er giktk#r wpu>°
Á
k °y·³fgiwyj.uM«!v#x9jWi°krskZk"$Pwyrswypiv#x9jWu`twx9pi
©ºx9tr t ¥ivrs¥itku>piWj.x>rswyx9pbBk tv k#rswyx9p¦Bk#rq§k k#pK°u`t 9ku>pibj.u`°°i¯3wy ¥u>° Á k °y·~
u`°yt kMu>« k"$PBk twjWk pdr k Âwyp iwyx9°yx99wyvMu`°¯3wy wyx9p ©ºx`twpir}u`pv kYI« A©fD;~§giwy°ku
t k#v k pdr§x9t ÅHAó~CQD·u`°tkMu`P«Kik#j.x`pir tsu>r k rsgu¡rgI¥ij.u`pj.x>rswyx9pek#t v k#rswyx9p
wj.u`wypi°ª«t k#°u>r k rsx!rsgwu.pikv x9jWBx9pik#pIr ³
. º(Û Í Í É Í Í 6¢ Í #QÛ eÛ oqpdrst wypiwv¶u`t u`jWk#r k t "k#rswyj.u>rswyx9p
©ºt x9jrsgik%jWx`rswyx9p6v#x9pirstsu>wpdrsH?1w1³ók>³Æk °ywyj.wypu>r wp²ik#rsgÆwyp¿r gik%kI¥u>r wx9p@
wwpik k ¼Bx9 w°kvAófD¸k#¯9k#p ©ºx`tn¯>u`t«3wypi u`piÀ¥piÅIpixQ§p ©ºx3vMu`°@°yk pi`r g u`pi
it wypiv#wu>°\ex`wpdrAó`D?;u`°ªrsgix`¥i9gwyprsgu>rvMu`kWu>r°ykMu`rwj.u``k ~Gx`tkI¥iwª¯>u¡{