# About | ECITTT | Conference Programme | Abstracts (15.)

# APPROXIMATE IMPLICITIZATION FOR

SURFACE INTERSECTION AND SELF-INTERSECTION

Dr Tor Dokken
SINTEF Applied Mathematics

P O Box 124

Blindern

0314 Oslo

Norway

Within CAGD parametric and algebraic descriptions are combined to achieve good computational efficiency for conic sections and quadric surfaces. For more complex smooth shapes piecewise rational parametric descriptions (NURBS) dominate. The reason is that the process of finding an algebraic description is computationally expensive, results in algebraic surfaces of high degree and requires exact arithmetic. A method for the approximation of piecewise rational parametric manifolds of a chosen degree with an algebraic hypersurface is presented. This method has good convergence rate, uses singular value decomposition to find a set of alternative approximations and enables the selection of an approximation to be based on required behavior of the gradient of the resulting approximating algebraic surface. The FET assessment project GAIA IST-1999-29010 "Application of approximate algebraic geometry in industrial computer aided geometry" (October 1st 2000 - September 30th 2001) addresses the use of approximate implicitization within CAD. The potential of this approach for improving intersection and self-intersection of CAD-type surfaces is analyzed. By the end of March this 1-year project will be half way through. The talk will give the theoretical background of this approach and report on the progress of the GAIA project (www.math.sintef.no/gaia/).