L. Gonzalez-Vega and I. Necula
Dpto. Matematicas, Estadistica y Computacion
Universidad de Cantabria, Santander, Spain
{gvega,ioana}@matesco.unican.es
One of the main problems in the manipulation of parametric surfaces in Computer Aided Design is the finding of efficient algorithms for computing the implicit equation of surfaces parametrized by rational functions. This is due, for example, to the fact that, if for tracing the considered surface the parametric representation is the most convenient, to decide in an efficient way the position of a point with respect to the surface considered, the implicit equation is desired.
There are many algorithms coming from Computer Algebra solving this problem but most of them depends heavily on the fact that the coefficients of the parametrization are known exactly which is not the usual case when dealing with real problems: usually such coefficients are floating-point real numbers. To overcome this drawback we propose the using of symbolic methods to compute the implicit equation of different classes of parametric surfaces (ie through the consideration of the coefficients as parameters) and then to compute the implicit equation of the considered surface via a well-controlled evaluation.
The advantages and disadvantages of this approach will be presented through the study of a concrete set of problems and examples coming from a company devoted to car design together with the impact that this new approach has produced over some useful tecniques in CAD/CAM such as sectioning, off-setting and blending.