Geometric algorithms for the intersection of curves and surfaces

Abstract

The problem of finding the intersection of curves and surfaces arises in numerous computer aided design applications. The methods generally used rely on iterative numerical techniques based on the solution of a set of non-linear equations. These systems of equations are generally local and need adequate starting points in order to yield convergent solutions. This article presents two general algorithms based on geometric considerations to find the intersections of C 0 curves and surfaces. The first method can be applied when one object is defined by a parametric equation and the other by an implicit equation. The second method is based on a succession of orthogonal projections from one object to the other. The same algorithm can be applied to curves and surfaces. These methods are implemented in the general framework provided by dual kriging for parametric curve and surface modelling. Finally, the conjugate tangent approach can speed up considerably the algorithm by considering alternatively tangent lines or planes in the iterative process together with orthogonal projections.