On the Representation of Dupin Cyclides in Lie Sphere Geometry with Applications
نویسندگان
چکیده
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.
منابع مشابه
Conversion of biquadratic rational Bézier surfaces into patches of particular Dupin cyclides: the torus and the double sphere
Toruses and double spheres are particular cases of Dupin cyclides. In this paper, we study the conversion of rational biquadratic Bézier surfaces into Dupin cyclide patches. We give the conditions that the Bézier surface should satisfy to be convertible, and present a new conversion algorithm to construct the torus or double sphere patch corresponding to a given Bézier surface, some conversion ...
متن کاملRepresentation of Dupin cyclides using quaternions
Dupin cyclides are surfaces characterized by the property that all their curvature lines are circles or lines. Spheres, circular cylinders, cones and tori are particular examples. We introduce a bilinear rational Bézier-like formula with quaternion weights for parametrizing principal patches of Dupin cyclides. The proposed construction is not affine invariant but it is Möbius invariant, has low...
متن کاملDo Blending and O setting Commute for Dupin Cyclides?
A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r > 0 involves three steps: (1) computing the (?r)-oosets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the ooset cones, and (3) computing the r-ooset of the cyclide. Unfortunately , this process does not always work properly. Worse, for so...
متن کاملGluing Dupin cyclides along circles, finding a cyclide given three contact conditions
Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tan...
متن کاملDo blending and offsetting commute for Dupin cyclides?
A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r > 0 involves three steps: (1) computing the (−r)-offsets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the offset cones, and (3) computing the r-offset of the cyclide. Unfortunately, this process does not always work properly. Worse, for ...
متن کامل