A new differential approach for parametric-implicit surface intersection

Abstract

In this paper, we focus on the parametric-implicit surface intersection problem. In our approach, this problem is formulated in terms of an initial value problem of first-order ordinary differential equations (ODEs). To this end, we take advantage of the orthogonality at any point on the intersection curve between the tangent vector to that curve and the normal vector to the implicit surface. This yields an initial value system of ODEs that is numerically integrated through an adaptive Runge-Kutta method. In order to determine the initial value for this system, a simple procedure based on the scalar and vector fields associated with the function defining the implicit surface and its gradient is described. Such a procedure yields a starting point on the nearest branch of the intersection curve. The performance of the presented method is analyzed by means of some illustrative examples. © Springer-Verlag Berlin Heidelberg 2003.