Journal for Geometry and Graphics 13 (2009), No. 2, 145--162
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
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