Applications of a Fruitful Relation between a Dupin Cyclide and a Right Circular Cone
A powerful way of handling a Dupin cyclide is presented. It is based on the concept of inversion, which yields a fruitful relation between the symmetric Dupin horn cyclide, from which all other Dupin cyclides may be obtained by offsetting, and a right cir
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Applications of a Fruitful Relation between a Dupin Cyclide and a Right Circular Cone G. Albrecht, M iinchen Abstract
A powerful way of handling a Dupin cyc1ide is presented. It is based on the concept of inversion, which yields a fruitful relation between the symmetric Dupin horn cyc1ide, from which all other Dupin cyc1ides may be obtained by offsetting, and a right circular cone. This relation has two important applications. First, it is used for constructing rational rectangular and triangular Bezier patches on the cyc1ide. Second, it allows to establish an approximative isometry between cyc1ide and cone patches, a useful result e.g. for scattered data interpolation techniques on Dupin cyc1ides. Key words: Dupin cyc1ide, cone, inversion, rational Bezier rectangles and triangles, scattered data interpolation.
1. Introduction
Dupin cyclides have proven to be a useful class of surfaces for CAGD. They offer a simple means for augmenting the flexibility of solid modellers since they encompass planes, tori, and the so-called natural quadrics (spheres, right circular cylinders/cones) as special cases (see e.g. [20]), i.e., the surfaces traditionally used in solid modelling systems. In order to be able to fully utilize Dupin cyclides in this context, intersection algorithms-among other things-must be provided. This problem has been addressed by several authors, see e.g. [11, 16], where Johnstone [11] makes use of the concept of inversion. In engineering applications Dupin cyclides have been found to be particularly important as blending surfaces for planes, cones, cylinders, and spheres. References [3, 9, 10, 16-18] are some contributions concerning this area. Dupin cyclides are of practical use also for 3D motion planning [7] and for creating and manipulating geometrically more complex objects [8]. For using Dupin cyclides in standard CAD systems based on free form surfaces, Bezier or B-spline representations are required for them. Pratt [17] and Kaps [12] present biquadratic rational Bezier representations of a so-called principal eyclide patch, a rectangular surface patch on a Dupin cyclide bounded by lines of curvature, which was first introduced by Martin [15]. Pratt then uses his previous results in the application of Dupin cyclides as blending surfaces [18], i.e., G. Farin et al. (eds.), Geometric Modelling © Springer-Verlag/Wien 1998
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he determines the position of the Bezier control points for certain cyclide blends. Normalized principal cyclide patches, i.e., rational Bezier patches on a Dupin cyclide having weights equal to 1 at three of four corner points, are studied by Ueda [22], and a NURBS representation of a principal cyclide patch is given by Zhou and Strasser [24]. Also, so-called generalized cyclides [5] or double Blutel surfaces [4], which are projective images of Dupin cyclides, as well as an even more general class of surfaces [6] having-among other things-plane rational curves as parameter lines and containing the generalized cyclides as a speci
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