The Intersection Problems of Parametric Curves and Surfaces by Means of Matrix-Based Implicit Representations

Abstract

In this paper, we introduce and study an implicit representation of parametric curves and parametric surfaces. We show how these representations, which we will call the matrix implied, establish a bridge between geometry and linear algebra, thus opening the possibility of a more robust digital processing. The contribution of this approach is discussed and illustrated on important issues of geometric modeling and Computer Aided Geometric Design (CAGD): The curve/curve, curve/surface and surface/surface intersection problems, the point-on-curve and inversion problems and the computation of singular points.