Computational Methods for Algebraic Spline Surfaces: ESF Exploratory Workshop

This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces (“COMPASS”), which was held from September 29 to October 3, 2003, at Schloß 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 – in the case of “trimmed” surface patches – a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice.

Table of Contents:
Approximate Parametrisation of Confidence Sets.- Challenges in Surface-Surface Intersections.- Computing the Topology of Three-Dimensional Algebraic Curves.- Distance Properties of ?-Points on Algebraic Curves.- Distance Separation Measures Between Parametric Curves and Surfaces Toward Intersection and Collision Detection Applications.- Elementary Theory of Del Pezzo Surfaces.- The Geometry of the Tangent Developable.- Numerical and Algebraic Properties of Bernstein Basis Resultant Matrices.- Polynomial C2 Spline Surfaces Guided by Rational Multisided Patches.- A Recursive Taylor Method for Algebraic Curves and Surfaces.- Self-Intersection Problems and Approximate Implicitization.- Singularities of Some Projective Rational Surfaces.- On the Shape Effect of a Control Point: Experimenting with NURBS Surfaces.- Third Order Invariants of Surfaces.- Universal Rational Parametrizations and Spline Curves on Toric Surfaces.- Panel Discussion.