SIAM Short Course on Approximate Algebraic Methods for Computer Aided Geometric Design

Sunday November 9

Organizers: Tor Dokkenm, SINTEF, Norway
Laureano Gonzalez-Vega, Universidad de Cantabria, Spain
Bert Juettler, J. Kepler University, Austria


So far, the use of algebraic geometry in Geometric Design has been limited by the difficulties associated with the conversion processes to and from implicit form (implicitization and parameterization). Recently, new methods for approximate implicitization have emerged. These methods will help to exploit the potential of algebraic techniques. The course will present these techniques and their possible applications (e.g., to surface--surface intersection) to a wider audience. It is expected that this will stimulate further research in this field.


Tor Dokken has been employed by the industrial research institute SINTEF since he graduated from the University of Oslo in 1978. He is affiliated to the "Centre of Mathematics for Application",, which opened at the University of Oslo in March 2003. His work has been focused on mathematical challenges in CAD-type applications with special attention to intersection algorithms. In his Dr. thesis presented in 1997 he introduced the concept of approximate implicitization for the use in intersection algorithms. Currently this work is continued in the project GAIA II,, sponsored by the European Union through the IST-program

Laureano Gonzalez-Vega is professor of mathematics at the University of Cantabria of Santander, Spain (since November, 2002) where he is the Dean of the Faculty of Sciences since December 2000. He gained his Ph. D. in 1989 at Rennes (France) under the supervision of Professors Marie-Francoise Roy and Tomas Recio. His research interests include Computer Aided Geometric Design, Computer Algebra, Real Algebraic Geometry and Scientific Computing.

Bert Juettler is professor of mathematics at the Johannes Kepler University of Linz, Austria (since October 2000). He gained his PhD in 1994 at Darmstadt (Germany) under the supervision of Professor Josef Hoschek. His research interests include Computer Aided Geometric Design, Kinematics and Robotics, Differential Geometry and Computer Graphics.


The course will cover in detail methods for implicitization and approximate implicitization (resultants and related methods, surface fitting, numerical implicitization via composition of functions). In addition, it will include an outline of related applications (surface-surface intersection, detection of self-intersections, reverse engineering). The content of the course is closely related to the on-going project GAIA II entitled 'Intersection algorithms for geometry--based IT-applications using approximate algebraic methods', which is funded by the European Commission (Fifth Framework Programme, IST 2002-35512). This project aims to explore the potential of approximate algebraic methods for applications in Computer Aided Design.

Course Objectives

The attendees will gain a clear understanding of methods for approximate implicitization of curves and surfaces. In addition, they will get an impression of where these new methods may be useful. It is expected that the attendees will be able to use these methods in their future work.

Level of Material

Introductory 33% Intermediate 34% Advanced 33%

Who Should Attend

Researchers and practitioners in the field of Computer Aided Design

Recommended Background

Understanding of Bezier and B-spline techniques, and basic knowledge about implicit representations, basic Linear Algebra

Recommended textbooks: J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, AK Peters, Wellesley Mass.; G. Farin, Curves and Surfaces for CAGD - a practical guide, Academic Press.)

Course Outline

1. Introduction (10')

2. Approximate Implicitization (110')

2.1 Resultants and related techniques (37')
2.2 Surface fitting (37')
2.3 Composition of function (36')

3. Applications (50')

3.1 surface-surface-intersection (17')
3.2 self intersections (17')
3.3 others (16')