**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",
https://www.cma.uio.no/,
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, http://www.math.sintef.no/gaiatwo/,
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.

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.

Introductory 33% Intermediate 34% Advanced 33%

Researchers and practitioners in the field of Computer Aided Design

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.)

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')