The dual parametric and implicit representations are central in algorithms for low degree (typically 1 or 2) algebraic surfaces. Although available in the theory, exact implicit representations of sculptured CAD-type surfaces are not useful, as they are computationally too expensive. This is due to the high degrees and the exploding number of coefficients. Since the advent of approximate implicitization, new methods are within reach. The FET assessment project verified that self-intersection algorithms work with approximate implicits 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 implicitization opened 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.
The project combines knowledge from Computer Aided Geometric Design (CAGD) and classical algebraic geometry to improve intersection algorithms for Computer Aided Design (CAD) types system.
The focus within the project is on:
· Exact and approximate implicitization
· Classification and identification of singularities
· Recursive subdivision based intersection algorithms
· Industrial testing
We want to improve intersection algorithms in CAD by integrating approximate algebraic geometry and state-of-the art approaches, and further, to understand better approximate implicitization with respect to limitations and possibilities of different polynomial degrees, choice of polynomial bases and other aspects. To improve the use the algebraic surfaces our aim is to integrate knowledge from real & classical algebraic geometry into the CAD-domain. We intend to find better methods for approximating intersection tracks, investigate other uses for algebraic geometry, and combine knowledge from CAD, approximation theory and classical algebraic geometry. Our project will span the whole chain from basic research to industrial prototyping.
Description of work
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.
Milestones and expected results
GAIA II - Project Information and Partners
Full title: Intersection algorithms for geometry based IT-applications using approximate algebraic methods.
Project Acronym: GAIA II
Contract number: IST-2001-35512
Key Action/Action line: FET-Open, IST VI.1.1, Fifth framework program
Coordinator: Tor Dokken, SINTEF ICT, Department of Applied Mathematics SINTEF. e-mail tor.dokken "at" sintef.no
About the project