
Journal for Geometry and Graphics 13 (2009), No. 2, 145162 Copyright Heldermann Verlag 2009 On the Representation of Dupin Cyclides in Lie Sphere Geometry with Applications Miroslav Lávicka Dept. of Mathematics, Faculty of Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzen, Czech Republic lavicka@kma.zcu.cz Jan Vrsek Dept. of Mathematics, Faculty of Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzen, Czech Republic vrsekjan@kma.zcu.cz Dupin cyclides are canal surfaces defined as envelopes of a family of oriented spheres which touch three given oriented spheres. With respect to their attractive geometric properties they are often used in Computer Aided Geometric Design and in many engineering applications. In this paper, we study these surfaces from the point of view of Lie sphere geometry. This representation enables to solve many complicated problems through simple and well known methods of linear algebra. As for applications, we present an algorithm for computing their rational parametrizations and demonstrate a construction of blends between two canal surfaces using methods of Lie geometry. Keywords: Quadratic spaces, Lie sphere geometry, Dupin cyclides, rational parametrizations, PN surfaces, blending surfaces. MSC: 51N25; 68U05 [ Fulltextpdf (705 KB)] for subscribers only. 