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Intersection algorithms for geometry based IT-applications using approximate algebraic methods

Objective

Dual parametric and implicit representations are central in algorithms for low degree (typically 1 or 2) algebraic surfaces. Although available in theory, exact implicit representations of sculptured CAD-type surfaces are not useful in practice, as they are computationally too expensive. This is due to the high degrees and the exploding number of coefficients. Since the advent of approximate implicitisation, new methods are within reach. A previous FET assessment project verified that self-intersection algorithms work with approximate implicit of total degree as low as 4. Intersection algorithms are the most complex part of 3D CAD kernels, and the source of many problems in advanced CAD model exchange and use. Approximate implicitisation opens a bridge to algebraic geometry. By combining results from different branches of mathematics, we plan to investigate the use of approximate algebraic geometry in surface intersection algorithms.

DESCRIPTION OF WORK
1. Singularities: Classification, detection and localisation. The partners believe that Computer Aided Geometric Design will benefit from algebraic geometry. The project will address both real and classical algebraic geometry to achieve a better understanding of intersection problems and identify results that can improve intersection algorithms.
2. Representation: Different methods for finding algebraic representations exist. We will examine in more detail the resultant types, further investigate approximate methods and address piecewise algebraic methods. Topics of special interest are approximation with parametrical implicit curves/ surfaces and classes of algebraic surfaces governed by a small numbers of parameters.
3. Intersection: As the main objective is intersection algorithms, we will examine how the approximate algebraic approach complements state-of-the-art methods, and develop prototype integration. More accurate methods and compact representations for intersection tracks will also be in focus. The results will be demonstrated in an industrial prototype.
4. Applications: The assessment project resulted in a number of open issues that will be addressed. Some applications have already been identified, such as detection of loops on curves and ray tracing for graphics; others are foreseen. When a complex geometric constellation arises, a local approximate algebraic geometry is a tool to gain more insight and sort out the solution. As the project will benefit from industrial feedback from a larger industrial audience, the GAIA users club will invite industrial companies interested in influencing new CAD-type technology.

Funding Scheme

CSC - Cost-sharing contracts

Coordinator

SINTEF - STIFTELSEN FOR INDUSTRIELL OG TEKNISK FORSKNING VED NORGES TEKNISKE HOEGSKOLE
Address
Strindveien 4
7034 Trondheim
Norway

Participants (6)

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE
France
Address
3, Rue Michel-ange
75794 Paris Cedex 16
JOHANNES KEPLER UNIVERSITAET LINZ - INSTITUT FUER ANALYSIS
Austria
Address
Altenbergerstrasse 69
4040 Linz
THINK3
France
Address
55 Avenue Galline
69100 Villeurbanne
UNIVERSIDAD DE CANTABRIA
Spain
Address
Avenida De Los Castros S/n
39005 Cantabria
UNIVERSITE DE NICE - SOPHIA ANTIPOLIS
France
Address
28 Av. De Valrose - Parc Valrose
06108 Nice - Cedex 2
UNIVERSITETET I OSLO
Norway
Address
Problemveien 5-7
0316 Oslo