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Common Tangents to Spheres in R^3

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(1)INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Common Tangents to Spheres in R3 Ciprian Borcea — Xavier Goaoc — Sylvain Lazard — Sylvain Petitjean. N° 5325 Septembre 2004. N 0249-6399. ISRN INRIA/RR--5325--FR+ENG. Thème COG. apport de recherche.

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(143) 

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(147) ™

(148) >6W)x v tLgix NtLš˜nkW3twx:` !6gTe*g(J™>:W)x ¦W

(149) 5”itŽbkh˜ewgiap )ƒ-\tLgix p IA “ewn p IE   * ew”Tš–x

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(151)  tvb ¦|*•W•™igixgTe\uDe5acach–‚Tš˜W•’Žtvš–”TWl“eLn ¦v”TgTš–W3aka ž sFW#bc”incgTh–gTfbcRTh¨a\gTW)d#W3akaktvnk`dVeLgixThjbkvh˜ewg=h–gw(a+b) bke+'€vƒ- h–UXuTš–h˜W)a v = (a ± b) dVeLg5bcnzvtLxTh–d#bch–gTfœbkRTαW=lNtLβdÅb bkRitŽb +€vƒ tLgix '€: % RTeLš¨xTal“ewnFtvš–š v ∈ P ž  ‘ WygTe* dVeLgidVš˜”Dx:W\eLgbcRTWd3tLacWJeLl acuTRTW3ncW)Ca *hjbkR>d#eLuiš–tLgitvnd#W3gwbkWVnzaVž  Œ ¤¤)(¤ )  Œ ¤ , ¤

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(154) twa—bcRTW›‚-eL”TgDxTtvnk` evl•bcRiW›uTnceLopW3d#bch–eLg evlW3tLdzR

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(168) 5”itLš1bce

(169) α1 − α2

(170)

(171) = (α1 − α2 ) (α1 − α3 ) (α2 − α3 ). α1 − α 3

(172).

(173) 2

(174) α1 − α22

(175) 2

(176) α1 − α23. WVgidVWtvbJš–W3twapb—3b * ed#W3gwbkWVnza tLncWW

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(179) oT. A= B 2 − 4AC =. `q

(180) \. dij = αj − αi  1 (r2 d23 + r22 d31 + r32 d12 ), D 1. −1 (r1 d23 + r2 d31 + r3 d12 ) (r1 d23 + r2 d31 − r3 d12 ) d23 d31 d12 D (r1 d23 − r2 d31 + r3 d12 ) (−r1 d23 + r2 d31 + r3 d12 ),. D = d23 d31 d12 + r12 d23 + r22 d31 + r32 d12. :. ‘ WdVtLggTe* uTnkeŽ’LW\QRTWVewncW3U vž set  6¤ Œ   uD Y ewgiach–x:W3n\l“eL”Tnx:h–aPbch–gid#baPuiRTWVnkW3a *h˜bcR

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(182) h˜bka\uTš¨tvgTWwž›Q;e‚-WbztvgTfwWVg5b\bceh–g:™igTh˜bcW3š˜` UXtLg6` evl bcRTW)aPWXš–h˜gTW)aV¦;bcRTWl“ew”TnPbkR aPuTRiWVnkW  Œ ¤¤)¤). Γ½©ÎµÑ.

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(209) 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 Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY 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 - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).   

(210).   . ISSN 0249-6399.

(211)

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