Computational Methods for Algebraic Spline Surfaces

Computational Methods for Algebraic Spline Surfaces

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This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces (a€œCOMPASSa€), which was held from September 29 to October 3, 2003, at SchloAŸ Weinberg, Kefermarkt (A- tria). The workshop was mainly devoted to approximate algebraic geometry and its - plications. The organizers wanted to emphasize the novel idea of approximate implici- zation, that has strengthened the existing link between CAD / CAGD (Computer Aided Geometric Design) and classical algebraic geometry. The existing methods for exact implicitization (i. e. , for conversion from the parametric to an implicit representation of a curve or surface) require exact arithmetic and are too slow and too expensive for industrial use. Thus the duality of an implicit representation and a parametric repres- tation is only used for low degree algebraic surfaces such as planes, spheres, cylinders, cones and toroidal surfaces. On the other hand, this duality is a very useful tool for - veloping ef?cient algorithms. Approximate implicitization makes this duality available for general curves and surfaces. The traditional exact implicitization of parametric surfaces produce global rep- sentations, which are exact everywhere. The surface patches used in CAD, however, are always de?ned within a small box only; they are obtained for a bounded parameter domain (typically a rectangle, or a€“ in the case of a€œtrimmeda€ surface patches a€“ a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice.p 1 p 2 p 3 p 4 Fig. 5. Left: Planar picture of a Del Pezzo surface with 2 components. ... In order to see the two connected components, we project the surface onto the first three projective coordinates. The complex image is the conic C. The real image is the subset of points on the conic for which the form x1x2 is positive or zero. This subset of ... 5 (left). Example 33. Let F(x0 , x 1 , x2) be the quartic equation F = 17(x41 + x42) + 30x21x22 aˆ’ 160(x21 + x22)x20 + 380x40, and let S be the Delanbsp;...

Title:Computational Methods for Algebraic Spline Surfaces
Author: Tor Dokken, Bert Jüttler
Publisher:Springer Science & Business Media - 2006-05-24

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