Page 12 - Geometric Modeling and Algebraic Geometry

6 T. Dokken
• To establish the theoretical foundation for a new generation of methods for inter-
section and self-intersection calculation of 3D CAD-type sculptured surfaces by
introducing approximate algebraic methods and qualitative geometric descrip-
tions.
• To demonstrate through software prototypes the feasibility of the approach.
• To investigate other uses of the approximate algebraic methods developed for
extending functionality in modeling and interrogation of 3D geometries.
• To demonstrate that cooperation between mathematical domains such as approxi-
mation theory, classical algebraic geometry and computer aided geometric design
is an important part of improving mathematical based technology on computers.
• To interact with CAD systems developers to improve both friendly use and ro-
bustness of future CAD systems.

To address these objectives the project activities have been structured in four
main work areas, where each partner has had one or two work areas as their main
focus:
• Classification, where we have used the tools and knowledge of classical alge-
braic geometry to better understand singularities, see Section 1.5.
• Implicitization, where we have looked into resultants and approximate impliciti-
zation to better find exact and approximate implicit representations of parametric
surfaces, see Section 1.6.
• Intersection, where we have looked into algebraic intersection methods, com-
bined numeric and algebraic intersection algorithms, and combined recursive and
approximate implicit intersection methods, see Section 1.7.
• Applications, where we have searched for other problem domains where the ap-
proach of approximate implicitization can be used for better solving challenging
problems related to systems of polynomial equations, see Section 1.8.
In addition to the topics above we will in this paper also address:

• Project background and partners in Section 1.2.
• Why CAD-type intersection is still a challenge in industry in Section 1.3.
• The algorithmic challenges of CAD type intersections in Section 1.4.
• The potential impact of the GAIA project in section 1.9.
The list of references at the end of this paper is a bibliography of papers related to the
GAIA-project published by the project partners during and after the GAIA-project.

1.2 Project background and facts

The Ph.D. dissertation Aspects of Intersection Algorithms [16] from 1997 established
close dialogue between the Department of Applied Mathematics at SINTEF ICT in
Oslo, and the algebraic geometry group in the Department of Mathematics, Uni-
versity of Oslo. Gradually the idea of establishing a closer cooperation with other
European groups matured, and the algebraic geometry group at the University of Nice