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 SchemeCSC - Cost-sharing contracts
75794 Paris Cedex 16
06108 Nice - Cedex 2