Removing Degeneracies by Perturbing the Problem or the World
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Removing degeneracies by perturbing the problem or the world Pierre Alliez — Olivier Devillers — Jack Snoeyink. N˚ 3316 Décembre 1997. THÈME 2. ISSN 0249-6399. apport de recherche.
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(37) . . Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France) Téléphone : 04 93 65 77 77 - International : +33 4 93 65 77 77 — Fax : 04 93 65 77 65 - International : +33 4 93 65 77 65 à partir du 01/01/1998 Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65.
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(434) ]mf`cb;n{W
(435) bev]Ubeh
(436) ² Wg`kijlU]Ii·o¿h{vx^kij]Ud;lU`cijh{^cbe]¾ij`cdeb;v{^_h{SmV|n ÃÆb;v]Ub;]¶Wgd;Wg]UW
(437) vxijh{W&~Ib;^_]4h&n{Whn ^c]Àh{SUW7~m`cie]UW½fgie]¾rIW ^cV|~U`cWgV|Wg]4h{W5&lmnp^c]Ud hzbhxW
(438) nphxn.
(439) !& Oh{W
(440) nphxn°SUWhxSUWgvh{SmWh{vx^cie]Ude`cW 9&\^kn?bevx^cWg]4h{W5 fgbelU]4hxWgv{fg`_bf q^cn{W£YÀh{SIiKhw^cn
(441) »4SUWhxSUWgv ^knh{b|hxSUWa`cWéhbjÃOh{SUWab;v{^cWg]4hxW
(442) `_^c]UWÄÃÆvxbeO V h{b &q² "F& whxW
(443) nphxnSUWgh{SUW
(444) v Ä`c^_W5nbe]h{SUWfg^_vf`cW\h{SUvxbelmdeSh{SUWÄhxSUv{W
(445) WÄ~Ib;^_]4hxn $» &q » q² mbevV|b;nphij`cdeb;v{^_h{SUV n
(446) » "F& ?^knfgie`_`cW
(447) be]U`coÂSUW
(448) ].
(449) !& °^knh{vxlUW;² v{onxUij`cW I vxoeL K4b;v ;W
(450) belh$hxSUW°W
(451) deW
(452) ]UWgviKhxW.f
(453) ien{W
(454) nGÃÆb;vÅh{SmW°^c4^kW°ie]mafb;]mt4lUWgv$fgbeV|~UlhiKhx^_b;] beÃ.h{SUWWg`kijlm]mi·oÂhxv{^kij]Ud;lU`kiKh{^cbe]ÃÆbev~Ib;`_oqdeb;]mij`AV Wgh{vx^cf
(455) ng»sij]mÃÆb;lU]m+^_h]UW
(456) fgW
(457) nxn{iev{oÂh{bÂlmn{Wi fg`_W
(458) eWgv rmlU]m`c^_]mdhxW
(459) fSU]m^ct4lUWh{b|Wg]InplUvxWh{SmijhhxSUWÄV W
(460) v{d;W\nph{W
(461) ~hxbqb Ân{lUr¶t;lIieviKhx^cfh{^cV W;² lU]m`c^_]md^kn]Ubjh ]mW
(462) fW5n{nxijvxo ^_ÃOhxSUWa~Ib;^_]4hn{Wh^kn~IW
(463) vphxlUv{rsW
(464) $² h^kn|Wg`c`È q]UbK]¹hxSmiKh hxSUW7Wg`kijlU]mi·o¿h{vx^cie]UdelU`kiKhx^_b;]mn|ijvxWÇlmij`h{!b wbevxbe]mbe^^kijdevijV n h{SUW nph{lIoÂbjÃwijrInzhxvxi;f£"h wbevxbe]Ub;^GU^ciedevijV n°SIien^kWg]4h{^_mW5~mv{^cV|^Èhx^_;W
(465) n^c]&i;U^_h{^cbe]h{b # °ij]I {»V|b;nphabjÃSU^cfSºSmie]m`cW rU^knpW5f£h{b;vxn\ÃÆb;vÄh{SU%W wb;v{b;]Ube^ I GÄ`cW &e' K² ,vxWg`cWgKie];hW
(466) eW
(467) `_b;~UV|Wg]4ha^cn hx $SU W
(468) Ã2i;f£hh{Smijh\W
(469) deWg]mWgviKh{Wf
(470) ien{W
(471) nfgij]ÇrsW Smij]mU`_W5+^ÈhxS"i ~Uvx^cV ^_h{^ceWh{Smijh\fgbeV|~UlhxW
(472) nhxSU'W wb;v{b;]Ube^ b;vW
(473) `cielU]mi·oÂbeÃ.ie]4ode^ceWg]&npWghbjÃOmeWa~sbe^c]4hxn I G O OÇe K¥² O+b;lU]4h\ie]m ieie`ÈÃÆW
(474) `c I O (& &LK.d;i·eWiÂ]UW
(475) iev{`co4f
(476) ij]Ub;]U^cf
(477) ij`Gi·oÂh{b&fb;V ~m`_Wgh{WahxSUWWg`kijlU]Ii·o&^kiK d;vxieV¼beÃOfbefg^_vflm`ciev?~Ib;^_]4hnlInp^c]Udih{vx^kij]Ud;lU`cijh{^cbe]|hxSmiKh`_Wg´^cfgbedevij~mSU^cf
(478) ij`c`_oV iK´^_V|^cÉgW5n?hxSUW;W
(479) fh{bev hxW5Smn{nxiKijh\vx^c`_`_^konphxnh{SUh{W SUW|~IW
(480) ijvp]Uhxd;lU`_vxW5rmniKbehx^_ðb;ij]7`c`$bjh{ðvxi^kij]Ud;W
(481) `_`cW5ienlU^c]m]½i·o]Ub;h{]vx^cieU]UW
(482) deflUvxW
(483) `kiKi;hxnp^_^cb;]U]Sd|beI vW
(484) ;v
(485) L²K¥»$RieSm]m^cÇnh{^knvx^cienph{]U^cde`c`Olm]U`cijb;h{]^cbe¶fg]Çij]m^knbe]U]U^kbefghie`Å]UW5ÃÆbef£v ~sbe^c]4hnpWghxn^_h{SÇn{o4V|V|Whxv{^cW
(486) n
(487) »Un{lmfS&i;nh{SUWa;Wgv{h{^kfW5nbeÃ.i vxWgd;lU`cieUv =G¥d;be]$² ¸?UWg`knprUvxlU]U]mWgvAie]m ORQmf ;WvxWg~sbev{hÅh{SIiKh$^cV ~m`_W
(488) V W
(489) ]4h{^c]Ud IU^_vxW
(490) fh{`colU]mUWg*v )+)6I ?¸ O7e K zh{lmv{]mnb;lh.h{b\rIWiv{W5ij`q~mie^_] rIW5fgijlInpWbjÃsV ^_´W
(491) ~UvxbqUlmf£hn#h{ Sm ij hwie~U~sW
(492) ijv5² W^_`c`Un{WgWh{SUW5npWV|^È´W5 ~mv{blmfhxn^c]b;lUv~IW
(493) vphxlUv{rIiKh{^cbe]A»4SmWgvxWhxSUWgoÂ^c`c`9»^c]Ã2i;f£h5»SUWg`c~ ;WgW
(494) ~hxSUWaij`cdeWgrmvxie^cfUWgdevxWgW\bjÃOh{SUW hxW
(495) nphw`_bKa².¸?W
(496) `cn{rUvxlU]U]UW
(497) v?ij]mU ORQmf ;WvxW
(498) fgbeV|V|Wg]mh{Smijhwh{SUW # ÅhxW
(499) nph.mvxnphwrIWhxvxie]mnpÃÆbevxV W5 . . . 0". . . . . . . .. . . . . . . . . -. . 02. . . . ) . ) . . . . . . . . . . èÆ÷ç$è©ò.
(500) ¡£¢¥¤¡
(501) ¦£§©¨qªÂ¢9«q I¡{¬4¦£c ® ¬K¡a¢9«q ¯°¬K¡£c±. . hxb ij]&bevx^_W
(502) ];hiKhx^_b;]Âh{W5nzh^c]+be]UWÄSU^cdeSUW
(503) v^_V|W
(504) ]mnp^cbe]+rqo hxSUWanzhij]mUievx`c^Èéhx^_]Ud|V ij~A»USU^kfS`c^ÈéhnW
(505) i;fS ~sbe^c]4hÄh{bÇi&~mijvijrsbe`cbe^k7beÃvxWgeb;`_lUh{^cbe]$² RSUWg]Dh{SmWgo½n{lUded;W
(506) nph\~sWgv{h{lmv{rU^c]Ud&h{SmW `_^_éh{W5"~sbe^c]4hxn\hxb&vxW V|bK;W
(507) ºWgd;Wg]UW
(508) vxi;f^cW
(509) n
(510) ²Ç³n{^_]Ud½h{SU^kn~IW
(511) vphxlUv{rIiKh{^cbe]A»SmbKW
(512) eWgv5»Ab;]UWf
(513) ij]¿fb;V|~Ulh{W&i+h{vx^kij]Ud;lU`cijh{^cbe]$» n{lmfSÇienh{SUWb;]UWa^c] O^cdelmv{W ;»h{Smijh^kn]Ubjhh{SUWWg`kijlm]mi·o h{vx^kij]Ud;lU`cijh{^cbe]&bjÃ?ij]qo~IW
(514) vphxlUv{rsW
(515) +n{Whbeà ~sbe^c]4hxn^c]&h{SUW~m`cie]UWe²?RSqlmn
(516) »~Uvxbe~sWgv{h{^cW
(517) nbjÃh{SmWW
(518) `cielU]mi·oe»Un{lmfS+ieni;fof`c^cfg^Èhzo I u?m K»mfgie]U]Ubjh rsWienxn{lUV|W
(519) ÃÆbevh{SmWÄh{vx^cie]UdelU`kiKhx^_b;]mnbjÃ~IW
(520) vphxlUv{rsW
(521) &~sbe^c]4hxn
(522) ² ¶] ifb;V|~U`_Wgh{W
(523) `_oa^ sW
(524) v{W
(525) ]4h.ie~U~Uvxb;iefSA»5^_`c`cWg]mfgbelUv{hie]m qV|^_h{S I e K4SIi·eW°d;^_;Wg]inp^cV|~U`_Wij]I fgbeV|~U`cWhxWdevij~US¥h{SmWgbevxWhx^cffSmievxi;f£h{W
(526) v{^cÉ
(527) ijh{^cbe]bjÃsh{SUWhxv{^kij]mdelU`kiKhx^_b;]mnOh{SmijhijvxW·ie`_^k|fb;V ~m`_Wgh{^cbe]mnwbeà i Wg`kijlU]mi·o|hxv{^kij]mdelU`kiKhx^_b;]$²³np^c]Udh{SUW
(528) ^_vvxW
(529) n{lU`_h
(530) »qh{Sm6W g K¤¨ ±K§
(531) ªj¡ j®bjÃin{WhbjÃnp^_h{W5n¶Y¾SU^kfS fgbe]mn{^knzhnObeÃmhxSUWW
(532) d;W
(533) n.^_h{Sh{SUWW
(534) V|~hzof`cb;n{W
(535) f^cvf`cW~mv{b;~IW
(536) vphzo;Y fgij]rIWhxv{^kij]Ud;lU`kiKh{W5^c] iV ie]U]UWgv rmfgbe^_~m]mien{vp^knzhxhx^ÈhxWgWe]4»?hie]m^ȵhxS½V n{bei V|;WWanp~IlmW
(537) v{vpWhxlUh{vxSmrmijiKh|hx^_n{b;be]&V|^cW]7i;`c^_U]mW
(538) W
(539) ij¿vh{W
(540) ^cV d;WWyzbefgbe^c]m`cbenv]UhxbSUWW
(541) n^kbjijÃd;vxUie^V GWgvxWg^_]4h{h SÇfhzbe`cbÂbevfn
(542) b;²½`_b;RvxnSU^k^_nÃ?^È^khn^ci n rsW
(543) ielh{^_ÃÆlU`GvxW
(544) n{lU`Èh5»;rmlh^_hUb4W5n°vxW
(545) t4lU^cvxWfb;V ~mlh{^c]UdhxSUWWg`kijlm]mi·o U^ciedevijV¼mvxnph q^_h^kn°]mbjhn{lU^_hxierU`_W ÃÆb;vfgie]Ube]U^kfgie`sh{vx^cie]UdelU`kiKhx^_b;]beÃ.oq]mieV|^cf\bevvij]IbeV,~sbe^c]4hnpWghxn
(546) ² +* À½A#%O + g #()& &*g #( - .* WW5n{fgv{^crIW\hxSUW\h{W
(547) fSm]U^ct4lUWabeÃOd;Wg]UW
(548) vxijh{^c]Ud~sWgv{h{lmv{rmijh{^cbe]mnbeÃOhxSUWÄ~Uv{b;rU`cWgV ie]m~sWgv{h{lmv{rmijh{^cbe]mnbeà hxR SUW»K~Ibeb;vx`_`coqd;rqbeo]mieia`9»·n{ijWgvx]m^_W5n.¸wbjlmÃGfW`c´U^kijW
(549) V|ie]~U`cV|W
(550) nWhxhxSmv{^kiKfgh?n
(551) »K~U^_vx]bed;v{^_V|W5n{W
(552) nO]mÃÆnpvx^cbebeV ]In.npXb;vpijhx]m^_]Ud\Uhx² b WW
(553) `cijie`klUnpb]mi·V|oaW
(554) hx];v{hx^k^_ijb;]U] d;lUe`kiKh{^cV|^cbe]IWg]mnOn{^cbe^_h{]mS n T ie]m&ijvxvxie]UdeW
(555) V W
(556) ]4hxnbjÃ.`_^c]UWn{WgdeV|W
(557) ];hng² ).. .. . + . . . . . ". . 02. )+* . >W187=+/AL 56>?= /T78@ : J B > >W1 =+/ 2<7C-. ;b]Un\WÄV|npSUieb;]qlUo+`kfgberUV|vxW
(558) ~Ui l|hiKh{hx^cW
(559) ^_b;n]mrqijoÂ`Odefb;Wgb;V|V|~mWijh{vxW
(560) ^_]mvxn dq]UfgbKb4ab;»Ivxh{Ub+^_]mnpijb;h{vpW5hng².i&Rn{SUWh^knbe~IÃb;`_^k^kfnzhxo ^_]IÃÆb;f£hÄvf~Ibeb;V|^_]4~mhxn\iev{rq^c]U o d
(561) D~Ib;fgbq^_]4behxvnf
(562) ^_]IijiK]&h{W5rIngW » ^c]4h{W
(563) v{~UvxWhxW
(564) ^_]&hzb U^ GWgvxWg]4h
(565) »mrUlhfb;]mn{^cnph{W
(566) ];hi·ong² RSUW7mvxnphÂ^knieniºde`cbermie` g¡£¢9¤¡
(567) ¦ j¢9§2¬j¨ ¬ 7¢2« 7¯°¬j¡£_±+hxSmiKhvxWg~m`ci;fW
(568) Cn " rqVo " V $ x² b;V|~mijvx^_]md h{SU W
(569) U¶fbqb;vx^c]mijh{W
(570) nbjÃ?~sWgv{h{lUvxrsW
(571) 7~sbe^c]4hx-n ij]I & ]UWg;Wgv\~Uv{blIfW
(572) n hxSUWW
(573) deWg]mWgviKh{WÄfgi;npW V& V " & b;]U`_o^ÈÃ.~Ib;^_]4h(n +ij]m &iev{WÄ^k Wg]4hx^cf
(574) ij`¥² npW5fb;]m ^c];hxWgvx~UvxWhxijh{^cbe]Â^knieniaU ¡£¢¥¤¡
(575) ¦ K¢¥§2¬K¨½ ¬ ¢9«q I¡{¬4¦£_ g® h{SIiKhnpb;vphn?hxSUW~sbe^c]4hxnrqo h{SUW
(576) ^_v ~mv{beyzW
(577) f£hx^_b;]mnbe]4h{b hxSUWa`_^c]UW V
(578) Çvxijh{SUW
(579) vh{Smie]&b;]4h{b h{SU W
(580) 7ij´^cn
(581) ² RSU^kn.bevW
(582) vObeÃm~UvxW
(583) n{Wg]4hxijh{^cbe]^kn.W´Uiefh{`coÄrmief qijvUnÅÃÆvxbeVSIiKh.W^_`c`lmn{W^_h{S V|bevxWfgbeV|~U`cW´ ~mv{b;rU`_W
(584) V ng²³n{lmij`c`_o;»eWm]m ^Èh?W5ien{^_W5nzh.hxbÄfgbe]mfgWg^ceWbjÃGi\~IW
(585) vphxlUv{rIiKh{^cbe]|ien.~sWgv{h{lmv{rU^c]UdÄhxSUW~UvxberU`cWgV RSm^cn?n{lUd;deW
(586) nphxnwiÄ~sWgv{h{lmv{rmijh{^cbe]h{bie~U~U`coh{bÄhxSUWbevx`cqYÀ^c]hxSUWf
(587) ien{WbeÃGn{bev{h{^c]UdI»eiÄ]UWgÀiK´^knw^kn?fSUbj n{Wg]r4o ie]Â^_]Um]U^_h{W
(588) n{^cV|ie`ÅiKÁ ]UWhxvxie]mnpÃÆbevxV|ijh{^cbe] beÃÅh{SUW\~UvxberU`cWgV+»qij]m|hxSUW\^_]qeW
(589) vxn{Whxvxie]mnzÃÆb;v{V ijh{^cbe] ^kn?ie~U~U`c^_W5 h{bie`_`U~sbe^c]4hxn?h{ba~sWgv{h{lUvxr|h{SUWb;v{`kŲ vx^Èhx^_]Udh{SUWWg´~Uv{W5n{n{^cbe]ÃÆbevwhxSUWh{W5nzhien°iÄ~sbe`coq]Ubj V|^kij`G^c] Vie]mÂ^c]4h{Wgvx~UvxWhx^_]Udh{SUWafgb4WgÁÂf^cWg]4hxndeW
(590) beV|Whxv{^kfgie`_`co|de^ceW5n°lmnb;lUv^_V|~U`cWgV|Wg]4hiKh{^cbe]~Ib;`_^kfo ^_Ãh{SUW^ GWgvxWg]mfgW\^c ]
(591) 7fbqbev^c]miKhxW
(592) n^knÉgW
(593) v{bI»;hxSUWg]ÇfgbeV|~UlhxWÄh{SUW^ GWgvxWg]mfgW\^c ] fbqb;vx^c]mijh{W
(594) n
(595) ² . . . . . . . . "3 . . 0 . . . ". çOçú 5ë. 76
(596) 898 :.
(597) ©Ý§2
(598) 5§©©_ ¡ I¨G¬
(599) K§©¨. &. . hzb b;~U~sb;n{^ÈhxW ]Ub;] lU]U^kt;lmW f^cvp Æà belUv fbf^cvxfglU`kijv fb;v{]mWgvn W f~s`cbeW ^c];hh{n Smv{b;lUdeSh{SUvxWgW I~ b;^_]4hn deW
(600) ]UWgviKhxW rU^knz n{W
(601) f£hxbev ^cdelUvxWÄX ?W
(602) deW
(603) ]UWgviKhxW\f
(604) ien{W
(605) nÃÆbevhxSUW R-T,V|Wh{vx^kf #". .. ) (. ". IP-03 249 / 2KJ 781E=+24/ L?9;3 2<7= >?/ 9 / . -1 7D - F -
(606) 781E= B >R-T + 9. b;]mn{^cW
(607) vah{SmW V|Whxv{^kfj».^c]µSU^cfSºhxSUW plU]U^_h fg^_vf`cW |^cnij]¿iK´^knz¶ij`c^cde]UW5"nxt4lmijvxWe²+W
(608) deW
(609) ]UWgviKhxW
(610) f ien{W
(611) n
(612) »e^c`_`clmnph{viKhxW
(613) ^c] ^cdelUvxWX»qbfgfglUv°SmWg]UW
(614) eWgvV|bevxWh{SIij] hxSUv{W
(615) W~sbe^c]4hxn`_^cWb;] h{SUWÄnxijV|Wnxt;lIijvxW b;v$SUW
(616) ]UWg;WgvÅhzb~sbe^c];hn$`c^_W°be]Äh{SmW°n{ieV|W°np^kW;² hO^knAn{lÁÂf^cWg]4h$hxbrsWijrm`_W.hxbh{W5nzh^ÈÃhzb~Ib;^_]4hn$Smi·;W hxSUWnxijV|W
(617) Âfbqbev^c]miKhxW
(618) n.b;v fbqbev^c]miKhxW
(619) n
(620) »jbevw^_Ãz»;ÃÆbev.ÃÆbelmv?~Ib;^_]4hng»KhxSUW^ GWgvxWg]mfgW^_]
(621) Âfbqb;vx^c]mijh{W
(622) n beÃ.h{SUWmvnzh~Iij^cvW
(623) t4lmie`cnh{SUW ^ sW
(624) v{W
(625) ]mfW^_] +fbqb;vx^c]mijh{W
(626) nbjÃwh{SmWn{W
(627) fb;]m+~mie^_v5² n{~sW
(628) f^kij`fgi;npWbeà hxhxSUSU^kWgn^cv`ci;rUnz^khnpW5Uf£WghxdebeW
(629) v]UrsWgW
(630) vfieb;fgV|o W
(631) bnfgfi lmv{vxW
(632) nde^cbeSU]W
(633) ]+vxijhzh{SUW
(634) b|v~sh{beSm^cie]4]+hxni|iefgv{W\lUv{h{;SmWeWa² eWgv{h{^kfW5nbjÃwij]ÇiK´^kn~mijvij`c`cWg`$nxt;lIijvxWe»Uij]I n{^_V|~U`cW+i·oºhxbº~IW
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