Computational Methods for Algebraic Spline Surfaces

1. Front Matter

   Pages I-VII

   PDF

2. Approximate Parametrisation of Confidence Sets

   • Zbyněk Šír

   Pages 1-10

3. Challenges in Surface-Surface Intersections

   • Vibeke Skytt

   Pages 11-26

4. Computing the Topology of Three-Dimensional Algebraic Curves

   • G. Gatellier, A. Labrouzy, B. Mourrain, J. P. Técourt

   Pages 27-43

5. Distance Properties of ∈-Points on Algebraic Curves

   • Sonia Pérez-Díaz, Juana Sendra, J.Rafael Sendra

   Pages 45-61

6. Distance Separation Measures Between Parametric Curves and Surfaces Toward Intersection and Collision Detection Applications

   • Gershon Elber

   Pages 63-75

7. Elementary Theory of Del Pezzo Surfaces

   • Josef Schicho

   Pages 77-94

8. The Geometry of the Tangent Developable

   • Pal Hermunn Johansen

   Pages 95-106

9. Numerical and Algebraic Properties of Bernstein Basis Resultant Matrices

   • Winkler Joab R.

   Pages 107-118

10. Polynomial C² Spline Surfaces Guided by Rational Multisided Patches

    • Kȩstutis Karčciauskas, Jörg Peters

    Pages 119-134

11. A Recursive Taylor Method for Algebraic Curves and Surfaces

    • Huahao Shou, Ralph Martin, Guojin Wang, Adrian Bowyer, Irina Voiculescu

    Pages 135-154

12. Self-Intersection Problems and Approximate Implicitization

    • Jan B. Thomassen

    Pages 155-170

13. Singularities of Some Projective Rational Surfaces

    • Ragni Piene

    Pages 171-182

14. On the Shape Effect of a Control Point: Experimenting with NURBS Surfaces

    • Panagiotis Kaklis, Spyridon Dellas

    Pages 183-192

15. Third Order Invariants of Surfaces

    • Jens Gravesen

    Pages 193-211

16. Universal Rational Parametrizations and Spline Curves on Toric Surfaces

    • Rimvydas Krasauskas, Margarita Kazakevičiūté

    Pages 213-231

17. Panel Discussion

    Pages 232-237

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.

• Algebra
• Algebraic Geometry
• Approximation Theory
• Computer Aided Geometric Design
• Invariant
• Numerical Methods
• Topology
• geometry

• SINTEF Applied Mathematics, Oslo, Norway

  Tor Dokken

• Institute of Applied Geometry, Johannes Kepler University, Linz, Austria

  Bert Jüttler

Policies and ethics