Interactive axial deformations

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(1)Interactive axial deformations Francis Lazarus, Sabine Coquillart, Pierre Jancene. To cite this version: Francis Lazarus, Sabine Coquillart, Pierre Jancene. Interactive axial deformations. [Research Report] RR-1891, INRIA. 1993. �inria-00074780�. HAL Id: inria-00074780 https://hal.inria.fr/inria-00074780 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 AUTOMATIQUE. Interactive Axial Deformations Francis Lazarus, Sabine Coquillart and Pierre Jance`ne. N˚ 1891 Avril 1993. PROGRAMME 4 Robotique, image et vision. ISSN 0249-6399. apport de recherche. 1994.

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(110) {7\_q fQ`{GX/q9giWiX/nk`y \_Wi`gY|5`¤yx`G¥X/WY^3\7gYhnX/q5p"X/fg}\_hnqx`y `hngi|5`W¦fu?fR`q5y5hkq5ZHX[W¦pigiWY`Ggi{}|5hkq5Zgi|5` \*Thkp21{Gs *<hnZ/l5WY`bDX/W \pihk^w5n``G5\_^w5n`(4]9Ävgi|5`Ww,\_W}\_^`Ggi`WYp({7\_qfR`vy5`Gxqx`Gy¤hnqX/Wiy5`W gYX"`G9gY`q5y gi|x`v5vgi`{}|xq5hnjTl5`/QtTX[^`LX_gi|x`^ \*WY`pYl5Z/Z[`pigi`yhkq3gi|5`LDX[knX7zhkq5Z

(111) w,\_W]\*Z/W}\_w5|xpN -?`#|R\Go`gi|Tl5p1\_W"w5WYX/wRX[pY`y¢\3f,\_pihk{

(112) o`WYpihkX[q¡X_gi|5`5zgi`{}|5q5hkj9l5`/. 9:9. ß,;<>=?<.

(113) . !

(114) 5

(115) . 1. . 5

(116) !. . .

(117) 5/. .

(118) 5 . . . " . t9{7\_n`

(119) \*qxyHgEzhkp©gZ/W}\_w5|xp{7\*qH`7\*pYhkufR`¦\_y5y5`yHgYXgi|5`¦vh¥\_v`GDX/Wi^\7gihkX[qgi`{}|5q5hkj9lx`[ |5`"gEzhkp©g 21WY`pYw¨pY{7\*k( ` 4o/\*kl5`wR`Wi^hngipLgi|5`gzhnpigihkq5Z 2MWi`pYw¨pY{7\*khnq5Z 4X[gY|5`X/fr`{Gg\_WiX/l5q5y¡gY|5`3\*Thnp7 |x` X/fr`{GgYho`"hnpbgiX#yx`Gxq5`fRX[gi|H\gEzhkp©g\*qxy\

(120) pY{7\*k`1\_{]gYX[WL\*gz`7\_{}|wRX/hnq9gzX[<gY|5`\*Thnp7ÅTgi|5`pY`Lî\_{GgiX/WYp \_Wi`"gi|5`q%lxpi`yDX[W{X/^w5l5gihnqxZ

(121) gi|x`"y5`GDX/Wi^`yHX/fr`{Gg pvo`GWigihk{`pN *5WiX/^ gi|5`l5pY`W pzwRX/hnq9gvX[ oThn`Gm \"gzhnpig 21Wi`pYw¨5pY{7\_n(` 4 î\_{GgiX/Wb{7\_qfR`y5`G,q5`yH\*gv\*q9u¤wRX[hkqgbX[gi|x`\7hnp7 |x`vo/\*kl5`\_nX/q5Z¦gY|5`\*Thnp hnp

(122) gi|9lxpX[f5g}\*hkq5`y f9u hkqgY`GWYwRX/¥\*gihnqxZHgi|5ò[\_nlx`p#y5`G,q5`y fu¢gY|5`l5pY`W7 *<hnZ/l5WY` 8%hknnl5pigiW]\*gi`pgi|5` gzhnpig1\*{GgiX/W"f9u?\3pYhn^wxn`¦`G5\_^w5n`/ |5`l5q5y5`GDX/Wi^¤`Gy¡X/fr`{Gg¦hnp{X[^wQX[pY`yPX[ gE XWYX9y5ppY|5X7zq hnq *hnZ/l5WY$ ` 8N£Mn`G¥gNÅ(\pigiW]\_hnZ[|9g

(123) \*Thkp"|,\_p¦fR``q¢wRX/pihngihnX/q5`y fR`GgE `G`q¢gi|5` 8WiXTy5p7 *<hnZ/l5WY` 87£MWihkZ[|9ghnp X/fg}\_hnqx`Gy%zhgi|xX[l5g{}|,\_q5Z/hnq5Z3gi|x`pY|,\_wR`¦X_gi|5`#\7hnp7mrlxp©gfuP\*yxy5hnq5Z\#gEvhnpig1\*{GgiX/WX[ 13\7gX[qx` `q5y%\*qxy\_q5X[gi|5`WvX[( /È 1¦yx`GZ/WY``p\*gbgY|5`X_gY|5`Wv`Gqxy¨. . . . ¼ q¤X/Wiyx`WgYXhk^w5WYX7o/` gY|5`LvTh¥\_B`G¥X/WY^3\7gYhnX/q

(124) gi`{}|xq5hnjTl5`/m,\ X[qx`X_hnq ,l5`q5{`{7\_qfR`hnq9giWiXTy5l5{`y giX¦y5`G,q5`vgi|x`zwRX/WigihnX/qX[¨gi|5`L / pYw,\_{`vgiX¦fR`vy5`GDX[WY^`y¨ ¼ q3X[lxWhn^wxn`^`qg]\7gYhnX[qm*`v|,\o/`g}\('/`q \_yo/\*q9g}\_Z[`vX[xgY|5`by5`GDX/Wi^3\*gihkX[q\*hnpgiXyx`],q5` gY|,\7g pYw,\_{`/[pYhn^w5k` piX/nlgYhnX/q{X/q5pYhnpigip X[Qy5`Gxq5hkq5Z gX X/q5`pX[ hn q xlx`q5{ ` ¼ ^hnq\_q5. y ¼ ^3\7\*pZ[`q5`W]\_<{Gunhnqxy5`WYpL\*WYX/l5q5ygi|5`¦y5`GDX/Wi^3\*gihkX[q%\7hnp7m ¼ ^hnq¤fR`hkq5Z

(125) hkq5{nl5yx`y3hkqgY X ¼ ^3\7B |x`¦o/`Wigihk{G`pX[ gi|x`

(126) X[frs`{Gg"uhnq5Z3hnq5pYhnyx

(127) ` ¼ ^hnq 21WY`pYw¨QX/lgipYhny5 ` ¼ ^3\* 4 vhnnfR`¦Dl5kuyx`G¥X/Wi£ ^`y 21WY`pYw¨Rvhnnq5X[g"fR`

(128) y5`GDX/Wi^`y?\7g¦\_n 4] `Wigihk{`phnq5£1fR`Gg``q¡zhknfR`#w,\*WigihD\*kuyx`G¥X/WY^`y%fu hnq9gi`WYwRX/¥\7gYhnq5Z

(129) gi|5`y5`GDX/Wi^3\*gihkX[qw,\_W]\*^`Ggi`WYp7m9pYl5{}|H\_p gzhnpigNm5pY{7\*k`\_q5yH\*g©g]\*{}|HwRX/hnqg7 \_{}| GX/q5`LX[¨hk q xlx`q5{`hkpy5`G,q5`yfugi|5`gE X

(130) W]\*y5hkh 3\_q5 y

(131) X_`7\_{}|{hnWY{l5¥\_W{WYX/pipi£ pY`{GgihnX/qH\_nX/q5Z¦gY|5`\7hnp7 |5`pY`W]\_y5hnh\_Wi`y5`G,q5`yhnq3gY|5`p]\*^`^3\_q5qx`GWz\_pzgi|5`LgzhnpigbX[Wbgi|5`pY{7\*k` w,\_W]\*^`GgY`GWYp7 *<hnZ/l5WY`3 hknnlxp©giW]\*gi`pgY|5`¤lxpi`3X_zgi|5` X[q5`3X[zhn. q ,l5`q5{`/ ¼ q *<hkZ[lxWi`3 *£Mn`G¥gNm \w5¥\_q,\_W¦pilxW©1\_{` hnppY|5X7zq \_q5y¡gY|5`\*Thkp"l5pY`yPgiXHy5`GDX[WY^ gi|x`#pYl5Wi1\*{`hnpLohnpYl,\_nh `y¡hnq?f5D\*{ 'Q |5`#gEX X[qx`GpX_ hnq xl5`qx{G`3\_WY`3\*kpiXoThkpil,\_nh `yPl5pYhkq5ZgiW]\_q5pYw,\*WY`q5{hk`pN Ã ¼ ^hnq¡hnpgY|5`hnqxq5`W X/q5`z|xhnn ` ¼ ^3\*Phnp. ß*ÞîÙ½Ø.

(132)

(133)

(134) . /.

(135) !. *<hnZ/l5WY` 89Ã. vhnpigihnqxZ¦gE X

(136) pigiW]\*hnZ/|9gzWYX9y5p7. gi|x`X/lgi`WX[qx`[Äq5u?gY|5`3o/`Wigihk{G`pX[gi|5`X/frs`G{Gg¤nuThkq5Z%hnqxpihky5` ¼ ^3\7?vhnnfR`y5`GDX[WY^`y¨ |x` y5`GDX/Wi^`ypYl5Wi1\*{`hnp pY|5X7zqhkq *<hnZ/l5WY`L *£1WYhnZ/|gzz|5`WY`gY|5`y5`GDX[WY^`y\*ThnpzhnpbpY|5X7zqhkq3fx¥\_{ 'R. . . . {X[qxpi`jTl5`q5{`?X[¦gi|5`¡hnq5y5`wQ`Gqxy5`q5{`?X[¦gi|5`?vxz"gY`{}|5q5hkj9l5`¢\_q5yKgi|x`PZ/`X[^`GgYWihk{%^X9yx`GLhnp gi|x`"{7\_w,\*{hngEuX[ `GTw5kX[hngihnqxZ

(137) gi|x`"y5`GDX/Wi^3\*gihkX[qgYX9X/w,\*W]\_y5hnZ/^H |x`"y5`GDX[WY^3\*gihnX/qgYX9X/<^#l5pigDl5ku y5`pY{WYhnfR`gi|5`y5`GDX/Wi^3\*gihkX[q ¼ qv5vsm,\¦y5`GDX/Wi^\7gihkX[q¤giXTX[hkpb{X/^wRX[pY`y¤X[sÃ. 9:9. ). gEX¤\*T`p7Ãgi|5`l5qxy5`GDX[WY^`y¨mX[Wzhnq5hngih¥\_¨\*ThkpNm,\_q5y3gi|x`y5`GDX[WY^`y¨m5X/W xqR\*¨X/q5`/m ). gEX X/q5`pbX[hnq xlx`q5{`/m ¼ ^hnq\_q5y ). \"gEzhkpig Z/W]\*wx|H\*qxyH\pY{7\*k`Z/W]\*wx|¨. ß,;<>=?<. . ¼ ^\7Bm.

(138) (8. !

(139) 5

(140) . 5

(141) !. .

(142) 5/. .

(143) 5 . . . *<hkZ[lxWi`L Ã RX/q5`LX_hnqxl5`q5{`/. pgi|5`vy5`GDX[WY^3\*gihnX/qhnp Dl5nnu#y5`pY{WihkfR`y#f9u

(144) gi|x`y5`GDX[WY^3\*gihnX/q

(145) gYX9X/Mm/gi|5`p]\_^` y5`GDX[WY^3\*gihnX/q{7\*q fR`"\*wxw5nhk`y3pY`Go`W}\_RgYhn^`pgiX

(146) yxhQ`WY`qgbX/fTrs`{Ggip7mxpYhn^¤w5u

(147) fu¤lxpihkq5Z¦gi|x`p]\_^`y5`GDX[WY^3\*gihnX/qgYX9X/M ¼ q *<hkZ[lxWi`zagi|5`vp]\*^` yx`G¥X/WY^3\7gYhnX[q#giXTX[Qhnp\*wxw5nhk`y

(148) gYXgE X"y5h Q`WY`Gq9g X[frs`{Ggip7Ã5\pigiW]\*hkZ[|9gWYhnf5fRX/q\_q5y gi|x`gzhnpigi`yWYX9y5pzw5WY`pi`q9gi`yhnq *<hnZ/l5WY` 89 |5`hkq5hgYh¥\_B\7hnpbhkpz\pigiW}\_hnZ/|g nhkq5`v|5hnk`bgY|5`Lxq,\_<\*Thnp WY`w5WY`pi`q9gipv\

(149) q5XTy5`/5 gzhkp©gb|,\_p\*kpiX#fR``Gq%\*yxy5`ygiX#gi|xhnpbyx`]DX/Wi^\7gihkX[q y5`GDX/Wi^3\*gihkX[q¢giX9X/b{7\_q \*kpYX%fR`\*y,\_wgY`y giXPgi|5`X/fr`{GggiXPz|5hk{}| hng¦zhnkfR`\_w5w5nhk`y¨ ¼ q *<hkZ[lxWi`H¾H `3|R\Go`P\*wxw5nhk`y gi|5`q5X9yx`G£1y5`GDX/Wi^3\*gihkX[q giXTX[zgiX \_q X/fTrs`{Gggi|R\7g¤hkp#qxX_gpigiW}\_hnZ/|g7m \ |5X/WYpi`pY|5XT`/ |xhnpb|,\_pvfR``q^\*y5`wRX/pYpihkf5n`LpYhn^w5nuf9u\_y,\_wgihkq5Zgi|5`hnq5hngih¥\_<\7hnpzgiXgY|5`"pY|,\_wR`X_ gi|x`bX/fr`{GgN ¼ q *<hnZ/l5WY`b¾7£Mn`G¥g gi|5òThkpilR\*kh `y\*ThkphnpgY|5`bq5`G hkq5hgYh¥\_X[q5`/ |5` y5`GDX/Wi^`y¦|5X/WYpi`pY|5XT` hnpzpi|xXNvqhnq *<hnZ/l5WY`¾N£MWihkZ[|9g giX/Z[`Ggi|x`WvzhgY|¤gY|5`Lxq,\_<\*Thnp7. ß*ÞîÙ½Ø.

(150)

(151)

(152) . *<hnZ/l5WYàxÃ. . %$.

(153) . 7 .

(154) !. |5`Lp]\*^`Ly5`GDX[WY^3\*gihnX/q\_w5wxnhn`y3giX#gE Xy5hQ`WY`qgbX/fr`{GgYpN ¦. ³#% v %7 #. %&#. e ¦ ¤$. X?pYhn^w5k`¤`G5\_^w5k`p#X[Lyx`G¥X/WY^3\7gihkX[qxp#X[ `n '9qxXNvq X/frs`{Ggipfu gi|5`HvxzLgi`{}|5qxhnj9lx`%\_WY` nh knl5pigiW]\*gi`yhkq *<hnZ/l5WY` Èz\*qxy 9(*<hkZ[lxWi`Èw5WY`pY`qgipgE XbgY`7\*wRX[gip7[ÄqLgiX/w¨m*\_q\*Thnp<|,\_pfR``q"yx`GpYhkZ[q5`y f9u"gi|5` lxpi`Whkq5pYhny5` gY|5` piwQX[lg X_,gi|x`lxq5y5`GDX/Wi^`ygi`7\_wRX_g7 |x`y5`GDX/Wi^`ygi`7\_wRX[g(w5WY`pi`q9gi`y

(155) fR`kXN WY`pilxgipb¥WYX/^ \y5`GDX/Wi^3\*gihkX[qX[<gY|5`\*Thnp7 ¼ q *<hnZ/l5WY` Tm*\by5`GDX[WY^3\*gihnX/qLgi|,\*g<hkq5{nlxy5`pfRX[gi|

(156) \ y5`GDX/Wi^\7gihkX[qX[gi|5`\*Thkp\_q5y¦\ ^X9y5hnx{7\*gihnX/q X[gi|5`pY{7\*k`Z/W}\_w5|hnpz\*wxw5nhk`y¤gYX

(157) gY|5` - ¼ q X/l5W

(158) hn^wxn`^`qg]\7gYhnX/q¨mB{X/^w5l5gihnq5Z%gi|5`y5`GDX/Wi^3\*gihkX[q X[gi|5`^X/p©g¦y5`Gg}\_hnk`Gy X[frs`{Gg 2897È 1 o`W©gYhn{`5p 4 g}\ '[`p\_w5w5WYX7Thn^3\*gihno/ù 1T½ 1

(159) pY`G{X/q5yX[q\_q ¼ d ¼ t T 1 Á ¦.

(160) . 9:9. ß,;<>=?<.

(161) !

(162) 5

(163) . a. 5

(164) !. .

(165) 5/. .

(166) 5 . . . *hnZ/l5WY`"¾TÃ5y,\_wgihnqxZ¦gi|x`Lyx`G¥X/WY^3\7gYhnX/q¤gYX9X/BgiX#gi|5`X/fr`{Gg7. gY|5X/l5Z[| pYwR`{h,{\_q5y¢pYhn^wxn`/mQgY|5`¤yx`G¥X/WY^3\7gYhnX/q5pgi|,\*g

(167) {7\*q fR`3{GX/q9giWiX/nk`y¢vhgi|¢gi|5`3v5v gi`{}|xq5hnjTl5`L\_Wiò/`Wiu#{X/^^X[q\_q5y3^X[pig l5pY`G¥lxM *5l5Wigi|5`WY^X/Wi`/mv5v¨X Q`WYp gY|5`vDX[knX7zhnqxZ¦\_yo/\*q5£ g}\_Z/`pNÃ 9t hnq5{`gY|5`y5`GDX/Wi^3\*gihkX[q?hnphnq5yx`wR`q5y5`q9gX[bgi|5`X/frs`{GggYXv|5hn{}| hg"hnp

(168) \*wxw5nhk`y¨mhg¦{7\*q fR` Wi`G£Ml5pY`y3giX#yx`G¥X/WY^ X[gi|x`GWvX/fr`{GgYpN ). 9t hnq5{`3v5vzhkphkq5y5`wR`qxy5`qg

(169) X[zgY|5`¤{WY`7\7gYhnX/q¢gY`{}|5q5hkj9l5`/m hg{7\_q fR`l5pY`y?giX¡y5`GDX[WY^ `GThnpigihkq5Z

(170) X/fr`{Gg7 ). v5v<{7\_q`7\*pYhnnu¤fR`{X/^

(171) f5hkq5`y3zhngi|X_gY|5`Wzyx`G¥X/WY^3\7gYhnX/qgi`{}|xq5hnjTl5`p7mxpYl5{}|\_p ). ). \_qu. * *(. v5v({7\*qPfR`

(172) \_w5wxnhn`ygiX^3\_quy5h Q`WY`Gq9gZ[`X/^`GgiWYhn{^X9yx`npvpilx{}|¡\_ppiwxnhnqx`pil5Wi1\_{G`p7mRwRX[£ uTZ/X/q,\*¨pYl5Wi1\*{`pzX[Wb|xhn`W]\*WY{}|5hk{7\*¨pYl5Wi1\*{`p7. ß*ÞîÙ½Ø.

(173)

(174)

(175) . *¾.

(176) !. ). v5v<{7\_q`7\*pYhnnu¤fR`hkqgY`GZ/W]\7gY`yhnq9giX^X[pigZ/`q5`W]\*\*qxy3hkqgY`GW]\_{Ggiho`^X9yx`nhnqxZpiuTpigi`^p7 ). v5v<hnpbo/`Wiu¤hnq9gil5hngiho`\_q5y¥lxnu¤hkqgi`W]\_{Ggiho`/. |xhnpbw,\_wR`Wb|R\*pzpi|5X7vq¤gY|,\*gzv5v<hkpv\"oh¥\_f5n`Ly5`GDX[WY^3\*gihnX/qgY`G{]|5q5hnjTl5`LDX/WbZ[`X/^`GgiWYhn{L^X9yx`]£ nhkq5Zx ¼ gZ[WY`7\*giuhnq5{WY`7\*pY`p gY|5`"{D\*pYpbX[(y5`GDX/Wi^3\*gihnX/q5pzX[f5g}\*hkq5`yf9u3^3\_q5hnwxl5¥\*gihkq5Z\

(177) y5`GDX[WY^3\*gihnX/q `q9gihgu \_q5y ^3\('/`p¦`7\*pYhn`W¦gY|5`{X[q9giWYX[ X[vpYX/^`y5`GDX[WY^3\*gihnX/q5p¦l5pYl,\_nnu¡y5`G,q5`y l5pYhnqxZHgi|5` * * gi`{}|xq5hnjTl5`p7 |5`WY`¦\_Wi`|xXN `Go/`WL\#qTl5^

(178) fR`WvX[(`q5|,\_q5{`^`q9gipL\*qxy`GTgi`q5pYhnX/q5pvgYXv5vgi|R\7g ` pY|5X/l5nykh '[`giXhnq9o/`pigihkZ\*gi`/Rt9X/^`X[gi|5`^ \_WY`/Ã ). {}|5`{ 9' hkq5Z#¥X/WbpY`¥£MhnqgY`WipY`{GgihkX[qX[gi|5`yx`]DX/WY^`y3pYl5Wi1\*{`/m ). \*lgYX[^3\*gihk{7\*nnu M2 X[Wzpi`^hn£E\_lgiX/^3\*gihn{7\_nu4y5`pYhnZ/q5hnqxZ¦gi|5`L,Wipig\*Thkp X[qgY|5`pYl5Wiî\_{`/m hn^w5k`^`Gq9gihkq5Z\_q \*yR\*wg]\7gYho`3pYl5fRyxhohnpYhnX/qPgY`{}|5q5hkj9l5`pYl5{}| \_pgi|,\*g#X_LÁWihk`pipY^3\_hnW`Gg\*E ÀÇÁLV /7ÂhnqX/WYy5`WgiX^3\*hkqg}\_hnq\*q%\*{{`wg]\*f5k`Wi`pYX/nlgihkX[qX[gi|5`pYl5Wi1\*{`/m ). ). \*knX7zhnqxZpY`Go`W}\_<\7hnp gYX#fR`lxpi`ypYhn^

(179) lxg}\_q5`X/l5pYu

(180) DX/Wzyx`]DX/WY^hnq5Z#\_qX[frs`{GgN. *5l5Wigi|5`WY^X[WY`/mRy5l5`

(181) giX3gi|5`

(182) pYhn^¤w5nhn{hngEuX_. Yg |5`#yx`G¥X/WY^3\7gYhnX/q`q9gihgu 2îgi|x`\7hnp54}mQhngLpY``^pnh '[`nu gi|R\7gzgi|5hnp\*wxw5WYX\_{}| X[lxny\*kpiX#fR`Lo/`Wiu\*g©giW]\*{GgYho/`"DX[W\*q5hk^3\7gYhnX/q\_w5w5khn{7\7gYhnX/q5p7 vxzhnp w,\_WigX[ ¼ Ä /m\"Z/`q5`W]\*Bhnq9gi`W}\_{Ggihno/`^XTy5`nhkq5ZpiuTpigi`^³y5`Go`GkX[wQ`Gy¦rsX/hnq9giufu t5ÄÁ ¼i \_q5y ¼ d ¼ ¦. . . . % z$. . À + \_W]*a[Â. ¦ÇÆ" + \_WYWN<ÁvkX[fR\*B\_q5y ¨X9{7\_Bv`GDX/Wi^3\*gihkX[q5p X[t9X[khny¤V(WYhn^hgYho/`p7 ,

(183)

(184)

(185) !]m¨N 2E 4}Ã 8T 1Tm_l5u%7/*ax. À +/+ ]Â. V\*lx +X/WiWY`Q\*qxyX[^hkq5hnjTl5` +`{}|5^3\_q5qv`GDX[WY^3\*gihnX/qX[q£Ehn^`qxpihkX[q,\_BÄfr`{GgYpN ¼ q

(186) !

(187)

(188) 5

(189) :

(190) !

(191)

(192) 6

(193)

(194) !}m w,\*Z/`p /¾TT /È[x §?m _l5q5`7[/. 9:9. ß,;<>=?<. / . .

(195) !

(196) 5

(197) . 7È. . 5

(198) !. /X f5f !

(199) >

(200) ! V(|, gi|x`pihkpNmzqxho`GWYpYhguX[ bg}\_|¨m _l5q5`7/*ax. À X[fR*a[Â. +v¿tR. À X9j51*Â. t\*fxhnq5` X9jTl5hnk¥\_W©g7 .

(201) .

(202).

(203) 5/.

(204) 6

(205) . !

(206) & . . .

(207) 5 .

(208) !

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(241) Unite´ de recherche Inria Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 Villers Le`s Nancy Unite´ de recherche Inria Rennes, Irisa, Campus universitaire de Beaulieu, 35042 Rennes Cedex Unite´ de recherche Inria Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 Grenoble Cedex 1 Unite´ de recherche Inria Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex Unite´ de recherche Inria Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex. E´diteur Inria, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) ISSN 0249-6399.

(242)