Approximate methods for change of representation and their applications in CAD

The project relates to methods for conversion between the two main representations of curves and surfaces, the implicit and the parametric form. Parametric representations are very useful for designers when creating geometry. However, implicit representat ions encompass a larger class of shapes and are more powerful for geometric queries. The exact conversion procedures, implicitization and parameterization, have been studied in classical algebraic geometry and in symbolic computation, but their practical application in CAD is rather limited due to high polynomial degrees of exact implicitization and the use of symbolic computations. In his Doctor Philos dissertation in 1997, the coordinator of SAGA, Tor Dokken introduced a class of approximate implicitiza tion techniques focused on parametric spline curves and surfaces. The ESR fellow will work on extending the knowledge on approximate implicitization by addressing a selection of open issues from the list below. The fellow will start from the first item as it will be a very good introduction to the topic of approximate implicitization. Topics will then be chosen according the achieved results, and the natural next steps in the research training. Tentative list of topics are: - Approximate implicitization of triangular Bezier surfaces - Extending the work on "Weak approximate implicitization" where numerical integration is used rather than directly combining the unknown algebraic surface and the parametric surface. - Approximate implicitization and algebra ic spline surfaces. Rather than approximate by a single algebraic surface, the approximate implicitization is by a piecewise polynomial algebraic surface. - Approximate implicitization of procedural surfaces, surface where no closed form expression of the surface exists. - Applications of approximation implicitization for simplification of shape descriptions (detection of planes, spheres, cylinders, cones) in composite shape descriptions.