Efficient Linear Elastic Models of Soft Tissues for Real-time Surgery Simulation
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Efficient Linear Elastic Models of Soft Tissues for real-time surgery simulation Stéphane Cotin , Hervé Delingette, Nicholas Ayache. N° 3510 Octobre 1998 THÈME 3. ISSN 0249-6399. apport de recherche.
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(52) 6×wrxgiky1rxwzy1p jLp y1t [spy"ðÇy1tv.j ðÇj.j.T+u1Mv :wz j. f:gjI xwi5dîMu?rxwzy1pÐ xíT xrjLi dj jLîzy12j.D!í Iw y1#p "$ %'&)( >* ì &QA '?+rRu :j. wpDry!u1v.v.y1dpsrrxgj y1îdikjLrxtxwzv:p+u?rxdtjÑy?ð-rxgj®y1txsu?p 6×wrxgkudj'ð²y1txiu?rxwzy1pÙîM+u 6 djLtxw j."ð²ty1i u"iu1 0}À xdtxwpd|ikyJdjLî +ägdwîzj»rxgj. j¼u?ddtyu1vgj. »u?îîz(y 6wpDrjLtu1vLrxw j5tu?rj. TrxgjLíÐ.j Tgdwdwru îMu1;v :y?ð tju?îwz xi wpv.j!rxgjLíGtjLdtj. jLpsrIuÙ y1îwz{u1
(53) uÙTwz vLtjLrj¼ jLry?ð iu1 j. prxgj
(54) y1rxgjLt®g+u?pBsrxgjv.y1psrxwpDddi ikj.vg+u?pdwzv. Ñrxgj.y1txíkj. xru?dîwz xgj. :u¼ jLr®y?ð2tjLîMu?rxwzy1pT} gdw 2jL r 6®j.jLpÐrxgj xg+u?jy?ð¶u!2yDTíDv.y1p xrxtu?wpsr ×u?pGwpsrjLtxp+u?î2dj'ð²y1txiu?rxwzy1p/ 6×wrxgdwpÕrxgdwz y1îwzÓu?pÓrxgjÐ.j TrjLtxp+u?î:ð²y1tv.j. ku?ddîwzj.*ryÝw r Onqp+u?txrxwzvLdîMu?tÑrxgjÕ jÐy?ðIjLîMu1 xrxwzv y1îwzd wz 6×wzdjLîíÜdj. vLtxw2j. wpÜrxgjÙîwrjLtu?rxdtj > Pñ×ñ:d ì Id]ñ ,2d ì j' ? -»jLpjLtu?îîí u?txw¬} y1
(55) ikyDTw #+vu?rxwzy1p »u?p{ wi|dîw #+vu?rxwzy1p# FÐ)j ®îwpju?txwz Yu?rxwzy1p Fu?tj¼wpsrxtyJTv.j.{rytj.Tv.j v.y1i|ddru?rxwzy1p rxwikj{y1try&wpTv.jÝu+u?txrxwzvLdîMu?tG2jLg+(u Jwzy1tu1 kwrkwz x+u?îîí rxgjvu1 j{wp dtxwzvu?î xwidîMu?rxwzy1p nqpÕy1tdjLtÑry5iu?wpDru?wpÐtju?î¬}~rxwikjtu?rj. :y?ð¶dj'ðÇy1txiu?rxwzy1p]TrxgjIu?drxgy1t g+u1Ó y1ikjLrxwikj. Ùtj.v.y1dt jÐry{ðÇy1txi5dîMu?rxwzy1p y?ð
(56) rxgjÐdj'ð²y1txiu?rxwzy1p îM(u 6 rxg+u?r|u?tjGð~u?tk+u 6Ñuí ð²ty1i rxgj.j du1vLr¾y1p)j -×vLrx+u?îîí 6×g+u?rj jLtÑrxgj
(57) u?ddtyu1vgÕrRu :jLp]swr® x+u?îîíÙt/j .+j.vLr Ñu!pj.j. ðÇy1t:d.j #pdwpdÙv.y1i|dty1i|wz j. ×jL,r 6®j.jLpikj.Twzvu?î2tju?îwz xi u?pÕtju?î¬}~rxwikj»v.y1p xrxtu?wpsr Ay1t:wpT} ru?pv.j P×u?w4p JwîîzVj > Pñ×ñ:R ?+d.j #pj. rxgj×j y1îdrxwzy1pGy?ðBu» jLr y?ð-txwwzGu?p|dj'ð²y1txiu?dîzj y1îwzd dpdjLt¾rxgjw0p .jLpv.j
(58) y? ð u?txwzy1 ðÇy1tv.j. nÀpkrxgdwz ®vu1 jrxgj
(59) dj'ð²y1txiu?rxwzy1pÕîM+u 6*wz ¾tjLdtj. jLpsrj. DíÝu|gDíD2jLt0}ÀjLîMu1 xrxwz%v D+u1 xw¬}À xru?rxwzvikyDdjLî~2u1 yJvLwMu?rj.{ryÐ%u #pdwrjÙjLîzjLikjLpDrIikjLrxgyJ{ð²y1trxgj pDdikjLtxwzvu?î]tj. y1îdrxwzy1p 1pTðÇy1txrxdp+u?rjLîídrxgj!vu?îzvLdîMu?rxwpdkrxwikj. F j jLtu?îBi|wpsdrj.
(60) FkiRu :j rxgdwz u?ddtyu1vg{ jLîzj. :ðÇy1t xdtxwzvu?î¶ xwidîMu?rxwzy1p dtIikjLrxgyJ wz ryÐ y1ikjÙ.j JrjLpDrB+u1 j. y1p rxgjÙy1dy xwrj"tju1 y1pdwpd 2&j 6×wîî xru?txr 6×wrxgÙu xwi|dîzj®ikyJdjLî}-wp"y1tdjLtÉryy1dru?wp"y1drxwiu?îTv.y1i|ddru?rxwzy1pÙrxwikj. ]}Bu?p¼rxgjLp 6®j 6×wîî dty12y j| y1ikjkwi|dt y jLikjLpsr"wpÞy1tdjLt¼ryrRu :jÙwpsryÝu1v.v.y1dpDr"u1dTwrxwzy1p+u?îÑvg+u?tu1vLrjLtxwz xrxwzv. u?pÙikj.Twzvu?îu1 xj.vLr ì wpv.j 6Ñjdy1p %r :Jp(y 6 rxgj.j u1vLrp+u?rxdtjy?ðrxgjdj'ð²y1txiu?rxwzy1pîM+u 6 y?ð iky r y?ð`r:rxwz j. 6®j!g+ u j!+u1 j.y1dtikjLrxgyJy1p{u"îwpju?t
(61) jLîMu1 rxwzv»ikyDdjLîwpdy?ðrxgj! y?ð²r rxwz xj. ì vgÐu!ikyJdjLî+wz ®wpdj.j.Gu!yJyJu?ddt(y Twiu?rxwzy1py?ðÉu!îMu?txjpDdi52jLtÑy?ð]iu?rjLtxwMu?îz 6×gjLp{rxgj»tu?pdj5y?ðTwz xdîMu1v.jLikjLpsr tjLiu?wp
(62) xiu?îî f:gj¼dty1dîzjLi4y?ð¾v.y1i|ddrxwpdGrxwikjtj.TvLrxwzy1pÜg+u1
(63) 2j.jLpÜ rxTwzj.sQí P®ty?} wzjLîz jLp u?p ñÑy1rxwp > P ñ: '? xwpd&uÜv.y1pdjLp u?rxwzy1p*rj.vgdpd w Dj >*,-wz3j '?
(64) u?ddîwzj.Óry&u #pdwrjjLîzjLikjLpsr ikjLrxgyJ 4äwrxgG xvgÐu?pGu?ddtyu1vg]Jrxgjv.y1i|ddru?rxwzy1pGrxwikjt(j Ddwtj.Ùð²y1t¾rxgjdj'ð²y1txiu?rxwzy1p . . . 1. . .
(65)
(66) !" #%$'&(*)+, -.$'! /&01 !23. . ?y ðu y1îdikjLrxtxwzvÜikyDdjLî»vu?p jÝtj.Tv.j. ry&rxgj v.y1i|ddru?rxwzy1pàrxwikjÜy?ðuÞikyJdjLîIy1pdîí wp y1îDwpd!rxgj× dt0ðÇu1v.jÑpyJdj. y?ðrxgj:ikj. xg nÀp >ÒñÑï »R?Y6®j:g+u ju?îz yIdty1y j.|uikjLrxgyJ ðÇy1tÐtju?î¬}~rxwikj wpsrjLtu1vLrxwzy1p 6×wrxgu y1îdikjLrxtxwzvÞdj'ð²y1txiu?dîzj ikyJdjLî!y?ðÙu?py1txsu?p f:gdwz ikjLrxgyJBd+u1 j.y1p{uÙ jLry?ð dtj'}Àv.y1i|ddrj.Ý(j sdwîwdtxwdi y1îdrxwzy1p ×wz jLtxíÐj.0vLwzjLpsr 6×gdwîzj pyÙry12y1îzy1ívg+u?pdj. Iu?tj!jLt0ðÇy1txikj. P®drÉry1y1îzy1ívg+u?pdj. x+u?îîí¼yJv.vLdt 6×gjLp"rxgj®rxwz xj ikyJdjLîJwz É xddi|wrxrj.ryIvLdrxrxwpd y1t»rju?txwpdy12jLtu?rxwzy1p ¼f:gj. jÙvg+u?pdj. !iRu :jÙ jLîzj. Irxgj"ikjLrxgyDd »+u1 j.Üy1p&u?pDQí :Jwp y?ðÑdtj'}Àv.y1i|ddrxwpdÝy?ðÑrxgjÙw4p jLt jky?ðÑrxgj| xrxw 52pj. »iu?rxtxw wpv.jÙrxgjÙikj. xgikyDTw #+vu?rxwzy1p wpTv.j. u
(67) ikyJTw #+vu?rxwzy1pÙy?ð-rxgdwz iu?rxtxw >ÒñÑy1r '? f:gjÑvg+u?îîzjLpdj:wz rxgjLp"ry2jÑu?dîzjÑrydty?} 2y j×u
(68) ð²y1txidîMu?rxwzy1py?ð2uIdj'ðÇy1txiu?dîzjjLîMu1 xrxwzvÑikyJdjLî+u?îîz(y 6×wpd¼tju?î¬}~rxwikjdj'ðÇy1txiu?rxwzy1p % vLdrxrxwpd 1ÓðÇj 6Opsdi52jLt y?ðtj. ju?tvgjLt g+(u j®ðÇyDvL j."y1p"rxwz xjÑvLdrxrxwpd!wprxgj®ð²tu?ikj 6®y1t : y?ð2 xdtxjLtxí wi5dîMu?rxwzy1p ì y1pd5u?pÙ
(69) j.dTí > ì R ?dg+(u j×dj. vLtxwj.kurj.vgdpd w Dj:ðÇy1tvLdrxrxwpd îwpju?t»jLîMu1 xrxwzv"y1T0j.vLr Id.j #pj.u1 #pdwrj"jLîzjLikjLpsr»ikyJdjLîz #JI(y 6Ñj jLt2rxgdwz
(70) rj.vgdpd w sj 6Ñu1 y1pdîíu?ddîwzj.r%y jLtxí xwi|dîzj»,r 6®yGTwikjLp xwzy1p+u?îÉy1Txj.vLr py1rxgjLtu?ddtyu1vg] 6®jLîî¶ dwrj. ry»rxgdwz% :Dwp|y?ð2dty1dîzjLiDv.y1p wz xr wpk xwpd!iu1 x}À xdtxwpd!ikyDdjLîz PÑíÙv.y1p xrxtxvLrxwzy1p]rxgj. j ikyJdjLîz Ñvu?pG2j
(71) ju1 xwîíikyDTw #+j.Gry"u?îîz0y 6äry12y1îzy1ívg+u?pdj. %JI(y 6®j jLtTrxgjLík.j TgdwdwrÑpy1p tju?îwz rxwzv!jLg+ u Dwzy1tTdtxwpddj'ð²y1txiu?rxwzy1pÜy1tvLdrxrxwpdGy12jLtu?rxwzy1p
(72) d x D7 :jLí{wz xj"rxg+u?r¼uÕ xdtxwzvu?î xwidîMu?ry1t!i5 xr5u1dTtj. »wz IryÕtju?îwz xrxwzvu?îîíÜikyDdjLîrxgj"2j'} g+u Dwzy1dty?ð y?ð`rÑrxwz xj. ÑwpÐtju?î¬}~rxwik)j dt jLíGy?ðÉ y?ð²rÑrxwz xjIikyDdjLîwpd|vu?pÐ2j
(73) ðÇy1dpÕwp >Öï»jLîMR ? JI(y 6Ñj jLtE 6®jÙdtxwz/j .í{dj. vLtxwj"rxgjÙ xwi|dîzj. xrIdwzy1ikj.vg+u?pdwzvu?î®ikyDdjLî O
(74) rxgj & " % "& % $
(75) ikyDdjLî f:gj×dgsíT xwzvu?î+2jLg+(u Jwzy1t®y?ð]u! y?ð²rrxwz xjikyDdjLîiuí|j×v.y1p wzdjLtj.Gu1 îwpju?tÑjLîMu1 xrxwzv×w¬ð rxgjÑTwz xdîMu1v.jLikjLpDr ¾u?ddîwzj."ry
(76) wr tjLiu?wp| xiu?îKî > Adp+@l ?B`îzj. ¶rxg+u?p y?ð+rxgj®ikj. xg" 2w 90j D u1 ®rxgjITwz xdîMu1v.jLikjLpsr ×wpvLtju1 jDrxgjîwpju?t×jLîMu1 xrxwzv»u?ddt(y Twiu?rxwzy1pj.v.y1ikj. Ñîzj. :u?pîzj. u?îwz 1
(77) îrxgy1dgwrwz uIiuX0y1t¾Twz Yu1! 1u?pDru?jy?ðîwpju?t®jLîMu1 xrxwzvLwrÀísrxgj×wpsrjLtu?rxwzy1pÕy?ð-ðÇy1tv.j ðÇj.j.T+u1Mv : wp&rxgj xwi5dîMu?rxwzy1pîwi|wr ¼rxgj|tu?pdjky?ð×dj'ðÇy1txiu?rxwzy1p 5ry iu?îî dj'ð²y1txiu?rxwzy1p f:gdwz Õwz Gj.vu? j{rxgj{ð²y1tv.jÝwp rxgjÝ xdtxj.y1p Ö Õg+u?p 6×wîîwpvLtju1 j u1 GgjÝwpvLtju1 j. Õrxgj dj'ðÇy1txiu?rxwzy1p]Irxgs Ðdtj jLpDrxwpd*îMu?txjdj'ð²y1txiu?rxwzy1p ñÑy1p (j DjLpsrxîíIrxgj dj'ð²y1txiu?rxwzy1p tjLiu?wp tju1 y1p+u?dîí xiu?îî py1rxgjLt5wpsrjLtj. ry?ð×îwpju?tjLîMu1 xrxwzvLwrqíwz ¼rxgj|2y xwdwîwrqíry v.y1i|ddrj u?psí ikj. xgädj'ðÇy1txiu?rxwzy1p ð`ty1i rxgj :Jp(y 6×îzj.TjÜy?ðu #pdwrjÜ jLrGy?ð5jLîzjLikjLpDru?txí dj'ðÇy1txiu?rxwzy1p +u1# 6®j 6×wîî¶ j.j»wprxgj»p.j Jr j.vLrxwzy1p AÉwt r0 6Ñj¾d.j #pj®u×tj'ð²jLtjLpv.j y1îdikjLrxtxwzv®u?p+u?ry1i|wzvu?îJikyDdjLî 1v.y1txtj. y1pTwpdIry wr tj. xr2y xwrxwzy1p 1pdjLt
(78) .j TrjLtxp+u?îÉv.y1p xrxtu?wpDr dðÇy1twp xru?pv.j"u| xdtxwzvu?î]wp rxtxdikjLpsr-rxgj u?p+u?ry1i|wzvu?îdikyJdjL î swz dj'ðÇy1txikj. 4&j®tjLdtj. jLpsrrxgj:dj'ð²y1txiu?rxwzy1pky?ð2u y1îdikjLrxtxwzv . 1/1. =WYXZ+[\.
(79) $ !3 !3()+ 2
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(85) If:gjTwz xdîMu1v.jLikjLpDr j.vLry1t.
(86) g+u1 rxgdtj.j¼v.y1i|y1pjLpDr (O
(87) /). 0.
(88) 132-4. 5
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(90) . f:gj¼Twz xdîMu1v.jLikjLpsr j.vLry17t
(91) :dyJj. py1r
(92) vg+u?tu1vLrjLtx2w 9j"rxgj¼dj'ð²y1txiu?rxwzy1p y?ðrxgj ?u p+u?ry1i|wzvu?î×ikyJdjLî<Ay1tÙwp xru?pv.jdpdjLtku{rxtu?p xîMu?rxwzy1p984y?ð
(93) rxgjGikyDdjLî rxgjÐTwz 0} dîMu1v.jLikjLpDr j.vLry1t»wz .
(94) :);8Gddr»rxgj5ikyJdjLîdyDj. Ipy1r»íDwzjLîz u?psíÜdj'ð²y1txiu?rxwzy1p A+y1tu{îwpju?tkjLîMu1 rxwzvÐiu?rjLtxwMu?î~ rxgjjLîMu1 xrxwzvjLpjLtxí=< >,?A@CBEDGFGHikju1 dtxwpd&rxgju?iky1dpsry?ð dj'ðÇy1txiu?rxwzy1p{y?ð A"%$I&( JTwz
(95) d.j #pj.Ýu1V >Òñ®wMu '? O B00 D <J>,?A@CBEDGFKH/)ML N OQPSRTVUW+YXZOQPSR[U 6×gjLtj j ]_^ ]Ð xíDi|ikjLrxtxwzv"iu?rxtxw JR B :Jp(y 6×pÞu1 Irxgj Itj.jLpT} ì .r ` jLp+u?psr5 rxtu?wpÜrjLp y1t D \ rxg* wz d.j #pj.{u1V O Ra)cNd b efg+=efihj BÇ D X&u?tj»rxgj5ò¶u?ik¼v.yJ.j 0vLwzjLpsr Ivg+u?tu1vLrjLtx2w 9.wpdÕrxgj¼ xrxw 52pj. ×y?ðu|iu?rjLtxwMu?î \ L u?pk s+u?rxwzy1p :Dp0y 6×pu1m lnoqp'"r & %ts D xg0y 6 :rxg+u?r®rxgjIjLîMu1 xrxwzv»jLpjLtxíy?ð u5dj'ð²y1txiu?dîzj y1Txj.vLr×wz
(96) $u D+u1Ttu?rxwzv»ð²dpvLrxwzy1p{y?ðrxgj¼Twz xdîMu1v.jLikjLpsr j.vLry1 t nÀp rxgjð²y1îîz0y 6×wpd{ j.vLrxwzy1p 6Ñj 6×wîî v.y1p xwzdjLt!rxgj"ð²tu?ikj 6®y1t : y?ð #pdwrj|jLîzjLikjLpsr u?p u1 xdikjÙrxg+u?r5rxgjky1Txj.vLr¼wz !tjLdtj. jLpsrj.&Dí&uv.y1pTð²y1txiu?î®rjLrxtu?gj.Ttu?î¾ikj. xg GnÀr!wz ¼rxgjLp 2y xwdîzjryÙv.y1i|ddrj5u?r×ju1vgpyJd.j u u¼ðÇy1tv.wj v F v.y1txtj. x2y1pTwpd|ryÙrxgj»djLtxw 1u?rxwzy1py?ðÉrxgj jLîMu1 rxwzv¼jLpjLtxí 6×wrxg{tj. xj.vLr
(97) ryÙrxgj!pyDdj»2y xwrxwzy1yp x F O < îMu1 rxwzv v F ){z BÇl D x F PÑj.vu? j¼rxgj5jLîMu1 rxwzv5jLpjLtxíwz s+u1Ttu?rxwz v z 6×wrxg{tj. x2j.vLr
(98) ryrxgj5Twz xdîMu1v.jLikjLpDr j.vLry1t rxgj®ðÇy1tv.j.W v F u?tjÑîwpju?tð²dpvLrxwzy1p y?ð+rxgj×Twz xdîMu1v.jLikjLpDr j.vLry1t y?ð-ju1vg|pyDd:j x}| f:gjLtj'} ðÇy1tj1wrvu?pÙjÑ xg(y 6×p|rxg+u?ri|wpdwi|2w 9.wpd¼rxgj:jLîMu1 xrxwzv:jLpjLtxí"y?ðurxgdtj.j'}ÀTwikjLp xwzy1p+u?î+y1Txj.vLr t(j Ddwtj. ×rxgj¼ y1îdrxwzy1p{y?ðu"îwpju?t xíJ rjLi y?ð rxgj»ðÇy1txi O 6×gjLtj O. ~ '. ). . 1. . .
(99)
(100) !" #%$'&(*)+, -.$'! /&01 !23 . wz Ùrxgj $ Giu?rxtxw ÓtjLdtj. jLpDrxwpd&rxgjÐry1y1îzy1í u?p* xrxw 52pj. Ùy?ð»rxgjTwz 0} vLtjLrxw29j.y1T0j.vLr \ tjLdtj. jLpDr rxgj¼Twz dîMu1v.jLikjLpsry?ð u?îî]pyJdj. \ 5v.y1i5dwpj. Iu?îî¶.j TrjLtxp+u?îBð²y1tv.j. u?py1dpu?txív.y1pTwrxwzy1p nÀpÐ j.vLrxwzy1p |u?pT 6®j! xrxTí v.v.j. xw jLîíÕu % /$ %$
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(102) ®ikyJdjLîBry v.y1i|ddrjrxgjkdj'ðÇy1txiu?rxwzy1pOy?ðu?pÞjLîMu1 xrxwzvGy1T0j.vLru?p&rxgjktj. xdîrxwpd{ð²y1tv.j. A wp+u?îîíC 6Ñj g(y 6#wp j.vLrxwzy1p |u?pÐgsíJdtxwz{u?ddtyu1vg 6×gdwzvg{v.y1idwpj. ×rxgj¼u1! 1u?pDru?j.
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(175) . . . b. &Ýy1tj.y jLt+j( D+u?rxwzy1p BÇ DÑwpTv.j. ×rxgj»ðÇy1îîzy(6×wpdGtjLîMu?rxwzy1p xgdwO ~ ?C B @ ) ~ BD?AC @ h nÀrwz wi|2y1txru?psr ry»py1rxwzv.j:rxg+u?rrxgj. j× rxw 5 pj. iu?rxtxwzv.j. y1pdîíÙdjL2jLp|y1p|rxgj®iu?rjLtxwMu?î vg+u?tu1vLrjLtxwz xrxwzv.
(176) 6×wrxgdwpu¼rjLrxtu?gj.Tty1p F|rxgdty1dgÐrxgjòÉu?ikIv.yJj.0 vLwzjLpsr L F u?p X F Fu?p rxgj»j.y1ikjLrxtxíy?ðrxgj»rjLrxtu?gj.Tty1p N F u?r×wr tj. xr g+u?j) Ay1t®uIw jLp jLtxrj.fx F rxgj×jLîMu1 xrxwzv:ðÇy1tv.j:v F wz rxgjLtj'ðÇy1tj×rxgj× xdi y?ð]u?îî+v.y1psrxtxwddrxwzy1p v |%$ ð²ty1i u?îî®u1?u1v.jLpsrrjLrxtu?gj.Tty1p N F &jkvu?p&rxgjLtj'ðÇy1tjj.Tdtj. !rxgjjLîMu1 xrxwzv|ðÇy1tv.j h '". u1 (O. v F. v F ). 6×gjLtj. L. ~ K x F x F + |. . 8 " $. L. ~ B x | x:|. B D. zw ×rxgj¼ di y?ð rjLp y1t ~ ? u1 yJvLwMu?rj.Ðry|rxgj!rjLrxtu?gj.Ttuu1?u1v.jLpsrryfx F ~ B wz :rxgj! xdi y?ðrjLp y1t ~ ? u1 yJvLwMu?rj.ÐryÙrxgj!rjLrxtu?gj.Ttuku1?u1v.jLpsr
(177) ry|j.Tjf !u% u?p x F ¾wz ×rxgj¼îwz xry?ð(x F pjLwB gDy1t ~ G. . . . . /1!# G! GD &? &D 0 !. #". %$. $. »w jLpÐu5rjLrxtu?gj.Ttu?îBikj. xgy?ðÉu" y1îwzI F|wpÐy1dt:vu1 jIu?pu?p+u?ry1i|wzvu?î] xrxtxvLrxdt.j F 6ÑjIddwîz u u?ruÞ rxtxvLrxdtjÝwpv.y1tx2y1tu?rxwpdÓrxgjpy1rxwzy1p#y?ð jLtxrxwzv.j. j.Tj. u?pärjLrxtu?gj.Ttu! A+y1t Þ ju1vg jLtxrj.2 6®j! ry1tj $ 0"2% %
(178) "
(179) fs $ !$"$ % 0" %drxgj»vLdtxtjLpDr×y xwrxwzy1pkx F JrxgjItj. xr 2y xwrxwzy1p x FL u?pÝrxgjrjLp y1t ~ .Ay1t»ju1vgj.TjK6®j" xry1tj¼rxgjr 6Ñyu1?xu1v.jLpDr jLtxrxwzv.j. u1 6®jLîî u1 IrxgjrjLp y1t ~ B SA wp+u?îîíð²y1t»ju1vgÜrjLrxtu?gj.Tty1p]E 6Ñj xry1tjÙutj'ð²jLtjLpv.j"ryGwr ðÇy1dt jLtxrxwzv.j. ku?pwr Ù xw Þj.Tj. ku1 6ÑjLîî×u1 "rxgjÐò¶u?ikÐv.yJ.j 0vLwzjLpsr L F X F u?prxgjGðÇy1dt j.vLry1t F d.j #pj.{wp(j s+u?rxwzy1p B² j.j¼u?djLpTw D. &. . ('. *. . . 1/1. =WYXZ+[\. &. )&.
(180) . . $ !3 !3()+ 2
(181) C- R . &=K0CD ? !@ $ G0 D ! G $ $. . . . &j j:rxgj j6×ry1pdwMu?pTw 52jLtjLpDrxwMu?î-j( D+u?rxwzy1p7B Du1 rxgj
(182) j( s+u?rxwzy1pky jLtxpdwpdrxgjiky1rxwzy1p y?ðrxgjÑîwpju?t¾jLîMu1 xrxwzvÑikyDdjLîfÉy!Twz vLtjLrxw29j×rxgj:Tw 52jLtjLpDrxwMu?î+j( D+u?rxwzy1p|wp|rxgj:rxwikjÑdy1iu?wp] 6Ñj jÝuÜð²y1dtxrxgäy1tdjLtGdpdj'} »drxruÞikjLrxgyJ >Ö%Af `»T+? äf:gdwz ikjLrxgyD tj( sdwtj. Gry j u?î+u?rjÙðÇy1dt¼rxwikj. ¼rxgjkð²y1tv.j. u?ddîwzj.&ry F wpÞy1tdjLt¼ryv.y1i|ddrjrxgjkp.j Tr!y wrxwzy1p F Od+ JI( y 6Ñj jLt 6®j"g+(u jÙv.y1i|+u?tj. rxgdwz !u?ddtyu1vHg 6×wrxgÜrxgj"iky1tjvLîMu1 wzvu?î dîzjLt ikjLrxgyJU b 6×gdwzvgäy1pdîí t(j Ddwtj. y1pjÝj 1u?î+u?rxwzy1p y?ðIðÇy1tv.j. #dtGv.y1pvLî xwzy1p wz |rxg+u?rGrxgj
(183) dpdj'} »drxru ikjLrxgyJ u?îîz(y 6 rxgjÐ jÐîMu?txjLtkrxwikj xrjL FOu?y1dr|rjLp rxwikj. îMu?txjLt F rxg+u?pÝrxgj5dîzjLtIikjLrxgyJ
(184) ñÑy1p (j sjLpDrxîírxgj!ðÇy1dtxrxgÝy1tdjLt
(185) dpdj'} »drxrukikjLrxgyJîzju1d ryÜu{ j.j.J}~dÞðÇu1vLry1tÙy?ðu?y1dr" r 6Ñyd u1 yJvLwMu?rj.G 6×wrxg uîzj. 5txwz :T 5y?ð
(186) Tw jLtx)j Üf:gdwz wz +u?txrxwzvLdîMu?txîíÐwpDrjLtj. xrxwpd 6×gjLp{dj'ðÇy1txi|wpdG rxw 5Üiu?rjLtxwMu?î]t(j DdwtxwpdG iu?îîBrxwikj¼ xrjL . . . . = 5@6! 6$ 8 < G B$ K!$&9 0C!"D $ ". $. $. .
(187). . pj×y?ðrxgj:+u1 xwzv:ru1 :wpk xdtxjLtxí xwidîMu?rxwzy1pv.y1p xwz xr wpÙvLdrxrxwpd"u?pÙrju?txwpd5 y?ð²r rxwz xj) wrxgrxgj¼TíJp+u?i|wzv!îwpju?t
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