Approximation of Normal Cycles
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Approximation of Normal Cycles David Cohen-Steiner — Jean-Marie Morvan. N° 4723 Février 2003. ISSN 0249-6399. ISRN INRIA/RR--4723--FR+ENG. THÈME 2. apport de recherche.
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(131) ® 7³ ±Ð³.®.
(132). 7. ¼ ¼ é¼
(133) 4°RHZ ¯. . b k lf 7G7 d e 7G7K9 + 7^@bfGi`( g]i4f.g h i!h [ b k lf 7G7 z de 787:9 + ^ bf.i` g]i4f.g h i!h [ 3 H "+ ^@bfGi`pfR+ @^ bfGi`a
(134) 7
(135) ]
(136)
(137) B 7
(138) FE
(139) bP
(140) 4 i1 _LGE
(141) B + ^@bf.i4`!8 + ^ bf.i`. y° ° ° ¬{® g¯¥± g± _° ° d³ ® g z® I¯ ±Ð³e¯ ³ JLKNQbwNns\*\_DYehC\7k>*Y]\_fghoc_Xgxt[]QaCap\asKNQbnsQGc_xyQPnPJLKNQvQNc7`¥aCcd[]fNQPhC\d+ cdXIx+ cdnpQ + ^@bfGi` 9 + e fR+ ^@bfGi` 9 ^Ni2jl`p^ b[k l ki`i]^@b'k i4` `+ e W Z S Z S JLKNQthjapcdXIxNcdn{xé\_npO"h¡_¦ dr7f W8WGW,f d ¨` c_XkDQ¡fghsQPxas\(xyQgXgQ¡`fgnscapfNnpQOtQPc7hjfNnpQPhb\_X©<4c7hr hs\y`YecapQPx as\"cdX*i^7Q \_OtQ asnpY]`W`\7O2wIc_`¥Q avS hsfNkghsQna V
(142) ¸ é¼ - ° ± T As° >°{ ® "°± ³.¬t¬{ ® 7¬± J¯ I#A{¯ ° ±#® L 6 - ° ± Ap° ® ° <J¯ I#A¥¯ °±L® L# 6 8 H*R° iN² ± H¬ I d ± I %° ".° J¯ I °¡® L!¿³e¯ ± H*° G°
(143) #IQ Ap° e ^2` 9P \m^_T ` F f de S W u QG` c_[Z[4apKgca KNQP'X Z S Y]hLcthjOt\*\dapK K*i*wIQPnphsfNns.c_` Q_ e ^2`oc_nsQWXN\_asKNY]XN^tkNfyaLasKgQz` []c7hshsYe` cd[ -,YZwIhs`{KNYZpa + /zY][][ZY]XN^t`fNnpcasfgnsQz\dI [ Z S ] ( 9 *Ð -,Qa xNQ XN\_asQAc fNXNYZaXN\7 nsO"cd[_QG`¥as\7n"gQP[]x xN8 QgXNQGx 8 \_X Q S cdXgx +HxyQ Xg\dasQ¶asKNQ hjQG`\7Xg x é¦.fgYZXXgxNapKNcdOtQxyQPX7Y]nsa{QGcd`¥[asY]é\_\7X nsO¿_¨¥\_ Z S tµ QxyQ XN\_asQtk*i . f WGWGW8f0. Z S asKNQ"wgnsY]Xg`Y]wgc_[C`fNnpcasfgnsQGh\_ Z S
(144) ¸ é¼ 8 HyO° i S
(145) j ° Z° t°
(146) g0± EJ¯ E t° ± ³.¬ LGIQ
(147) D¬ ±Ð³. ®
(148) aeZ^7 Z` 9 M . 8A f WGW8W,f . 8A V ® Lt ± H*° # ³
(149) ¬ ³ ¬ I N± I °¥¯® Ln Q S ³
(150) A@± Hy° ³ °p¬ ±Ð³. ®
(151) ³e¯G¬ ´].° ± H*° i S 0tG° 7
(152) AK¬ I N± I °® Ln ³
(153) ¶ ± H*° d³ °{¬±Ð³. ®
(154) 6 u QPO"cdn asKgcda e Z S xNQa ^G44.k 8 ` 9 e e ^7Z` . e f ¦Ð7R >7¨. Z Y s a K aeZ^ Z` 9 ^7Z`pfy KNQPnsQ ^ Z`Y]h asKgQzhsfNO \_9cd[][ i>ÐOtY]XN\_n{h \d<apKNQzOtcdasnpY 8d¤¦ e xNQ XN\_asQPh#apKNQ`[ec_hphL\dhjfgkghjQ aphL\_ M lf WGWGW8f ^ \ k l`V YapK i QP[ZQPOtQ X>aph{¨¥ JLKNQzé\7[Z[]\ Y]XN^¡npQPhsfN[ZavY]hL#Q [][ XN\ X<<¦§hjQP Q (ZJ *,é\_n Y]Xghra{cdXg` QGK¨ ¼ ¼ é¼ e ^ ` 9 e ^ Z` z f 3 Hy° ° z 7K°
(155) 4®±²°¥¯ ± H*° ®dF IQ"° L® ® LU Z S 6 D.F
(156) hjY]XN^JLKNQP\_npQ O«tcdXgx(monp\_wD\7hsYZasY]\_X}gy#QW^7Qa Y]O2OtQGxyY]cdasQP[Zi asKNQzé\7[Z[]\ YZXg^ ¼ ¼ .y ¹ + - °± Ap2° ¬{ ® 7¬±vG¯ I#A¥¯ °±v® Lb 3 H*®¯ c° Ap ® IQ
(157) " E Z S ³e ¯ G¯ "®G® ± H ²¬ Z®¯G° ® ³.K°
(158) I±òG° ° 2AsG° " _G° 0YHBE ° J¯ I LK _¬{°® LW 6 - °± T Ap!° 2 7°{7® t° ± ³.¬¬{ ® 7¬±J¯ I#A{¯ ° ±® LW 3 Hy®=¯ ° Þ9Þ ã sNì í{è¥î. 6.
(159) =R. C®;Hy°K
(160) I±²°³
(161) 4° ®
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(163) #E ° Iy8 ® ° ¯JI#A{¯ ° ±® LRV 698 H*°
(164) &1 L® .
(165) B FE A 7IQ
(166) E V E i4f.g=h i!h ! b k l1. °°. 7. e ^ ` k e. ^ ^2` ` 7 h. ;>= ? ^plf 7G7 + 787 ` ^@bf.i4` ` ^7$ f `.^ ^ l k $ 7G7 + 7G7 ` ZS ^N ^N\^_T ` F ` ^ r^ \^@Tr` F ` `p`pf 3 Hy° °'$ 9 $Z^c f # ^ `p`b³e¯± H*° 7I*¯K 7® 3 ³e¯¥±0
(167) 4¬p° Ap°±3©°{°K
(168) 7
(169) ^ `R
(170) 787 + 787 ³e¯W± H*° 7³ IQ ® L±@Hy°%
(171) 4® ® L± H*°¯ °p¬{®7
(172) ULGIQ
(173) 4 7t°
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(175) ® 7³ ±Ð³.®.
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(177) I³ L®8 ¯{bQ Z Q nG]l g ¾ Q L
(178) C\_n ( R * <g KgQ Q ^7Q nGNµ0 N[Z[]Q nGIu
(179) y`{KNn{c_xNQ nG
(180) A@± Hy°tK¬ I N± I °2® L ³.°p¬{°^3o³e¯ ° ± ¯ _¬{°¯¥ \7O2O( casK,gmoK*iyh ¦ l>¨ Bgk I U ( 7 * <g KNQ QP^_Q nGNµ0 N[][ZQPnP4u
(181) y`{KNn{c_xyQPnP ¡ ³
(182) 4K° ±Ð³. ¯ _¬{°¯{# &²XgxyYecdXgctbXNYZ4 casK,) <NgB 9 ¦j ?">:7¨ ~d>~Gj7~ ¬ 9d}g
(183) 4 N± I#Ap° L® Iy8 =X¯ L® ³.°p¬{°^3o³e¯ °t̳
(184) D°G. ( } *%DgNQGxyQ npQ nG" I d ± I % ° ".° J¯ I ° ± H*°{® EgJ,n{cdXIh O2QPnP casK,
(185) *\y ` ^plQ` )l! k5l_ ( 9 *%DgNQGxyQ npQ nG" v°{ ® "°± ³.¬ G° G¯ I ° 8 H*°{® E
(186) *wNnpY]XN^_QPnj$ #oQ np[]c_^g ¾ QP% C\7n 4 l& N ( :7* <gdNf, I d ± I °b® L I ³
(187) Z 7Iy, _¬{°¯ @³ ± H*° '® = d% rE_¬ Z° O2QPnP casK,
(188) y\*`_ + +)( ^ l)` )l k*&gg (Ä;~ * <gyNf<" I N± I R ° "G° =J¯ I °R¯
(189) 4 >K°
(190) 4° dÌ,³ +GG° ® ¯ ° ± H*°{® E) <N) Y£gTWQ \7Oy7 ¦r6 ?>"?7a¨ :g ?d :_}>RN ( >7* <gggf< ®
(191) Z 7° .- ° LGIQ
(192) 4¬±Ð³. ®
(193) g¯ ) &²XgxNY]c_XgctvXNY]4 capK<) <gg >¦r6 ?>?>¨¥I~B} 9=j~_~y_ ( ?7* <gggf<u C®
(194) ° >K°
(195) 4¬p°2® L¡K¬ I N± I °¯W ³
(196) ¯ °{G¬
(197) I± "# ® 7 ³ = d±§³.7®
(198) N¯¥4 <g bYZ£DQPnsQPX>asYecd[,TWQ \7OtQasnp/ i ^pl& Q` l &nk l!gg (]P7 *%<gggf<u I d ± I % ° ".° J¯ I °¥¯® LG¯ I#A. 7
(199) M E±§³.¬¯ °±§¯¥ OtQ nG) <g capK<] ll!0N] ^
(200) 5l k*&"gQ`s ¾ ly (]_ *z# \_np^7cdX,yTWQP\_OtQapnsYe`WOtQPc_hsfNnpQWasKNQP\_npi_ ` c7x¤NmonpQPhph Q &
(201) (]G R *zm4µ Y]X7ap^_QPX<1 '¡® oK¬ E_¬ Z2 ° 7
(202) ³
(203) g±²° oK¬ I N± I ° L® n ®dF E;H*G° ( ³
(204) 32b³.° = 7
(205) )
(206) g@³ 7
(207) P
(208) ³ L®d8 =¯¥4 YZ£DQPnsQPX7apY]c_[TWQP\_OtQapnsi¦§TWi7
(209) *\*\>hbcdXIWx <N
(210) Z+ QPX>asKNQ7DQGxNhP ¨ ¾ \_nsasKy0 Dv\7[Z[ecdXgx< O"hjasQPnj xgcdO( lN (]P7 *! 143_KN[]Q_N I d ± I ° "G° =J¯ I °¯
(211) 4 52n
(212) 4 _7® 0¯ °±§¯¥>7 RN# casK, ¾ c_`{KNnGd_6 ?N*¦r6 ?>d}*¨¥=7R>~G²_ ?g nk60y (] } *! 143dKN[]Q_
(213) g±²N° d 7
(214) ¬ I J °
(215) g± ° # °¯ K°
(216) I± d±§³.7®
(217) N¯"® L 7,G° 7° ° .8 ¯t¬ I N± I = ° ".° J¯ I °¥¯¥ n{`{K< casK,,@¦ Lc_hsQ [´¨ "} :g<¦r ?">:>¨¥5 99>~G 97:>~y (]6 9 *!51]3dKN[]Q_) 9 ®F E;HyG° 7®
(218) @± Hy°{® ° ¯ L®
(219) D7®
(220) J¯ "®P®d@± H¬{° ´<¬p7® Z.° >°¯¥ capK< XIc_`{KNnGgGN¦r6 ?>*~d¨¥ R ??dògPN (] :7*!a1]3dKN[]Q_ I d ± I °¥!¯ 7
(221) K¬ I J K°
(222) I±.n¯ L® IQ
(223) g³.7®
(224) ® L¯ ° ±.¯ 3o³´ ± H ®¯¥³´±§³ ° G° _K¬ H7#TWQ \_OtQ asnpY]c_Q QPxNY]`PcapcR_¦j ?">>~d¨; 99={=~y_. Þ9Þ ã sNì í{è¥î.
(225) P}. C®;Hy°K
(226) I±²°³
(227) 4° ®
(228). ¡$v% + P ¼ ½ 9d é¼ 2N g ¼ ½ ¼ ,= ¸ G RNZ TW QPXNQ n{cd[<` fNnpnsQPX7a{h zzzWzzzzWzzzzW RNRN R uvfgQP`¥nsapnpYQ DX>cda kNns[]QPQwNcdnpXgQPhsxQ X>Y]X7apapc_Q kN^7[ZnpQWc_k*[4i` fNY]X7npnsapQPQ X7^7a{nph#cdasWY]\_X zzzzzzzWWzzzzzzzzWW RN } c_hph cdXgxXN\_npO"hL\do`fNnpnpQ X>aph WzzzzWzzzzW RNRN :9 JL¾ Kg\_npQO"`cd\_[<XIhr`a{iycd` Xg[ZQ` itc_hpashsKN\*QP` \_Y]npcdQ asOQGx"éas\7\n ctYZX>` asQP\_^_Otn{wgcd[<c_`` avfNnshjnpfNQ kIX>hja{Q haLz\_9 zWWzzzzzzzzWW 9 ¼ y é¼ ¸ ´ gZ JLgZKg7QZ hjQG`JL\_XIKNxQéTzfNcdXIfgxNhpcdhLOtO"Q X>cda{wcd[4c_hpé\_hj\ynpO«` Y]cd\_asQPxc2asK*2\ i*wIQPnphsfN ns.c_z`QW\dz©W zzzzzzzzWW Z S g Z 7 R L J N K z Q N w s n _ \ r q P Q ` s a ] Y _ \ X _ \ ( X 2 c s h t O * \ d \ s a ( K * K * i I w P Q g R \7OtwgcdnpYZXg^apKNQXN\_npO"cd[eh zWzznphsfNns.zc_` QzWWzzzzzzzzWW g Kg\_Ot\dap\_w*i"kIQ ar QPQ X(XN\7nsO"cd[¤` i*` [ZQGhzzzzWzzzzW g } wNwNnp\=yYZO"capYZ\7X\_` fNnscdasfNnpQzO2QGc_hsfNnpQPh zzzWzzzzW. } }} 9 9 9 (. :. ~ ~ >. >. G. ÔéçÞ<Ô.
(229) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(230). . ISSN 0249-6399.
(231)
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