Differential Geometry of Intersection Curves

Abstract

In Chap. 5 we have studied the classification, detection, and solution of intersection problems. In this chapter we focus on the differential geometry properties of intersection curves of two surfaces. To compute the intersection curves more accurately and efficiently, higher order approximation of intersection curves may be needed. This requires the computation of not only the tangents of the intersection curves, but also curvature vectors and higher order derivative vectors, i.e. higher order differential properties of the curves. The two types of surfaces commonly used in geometric modeling systems are parametric and implicit surfaces that lead to three types of surface-sufrace intersection problems: parametric-parametric, implicit-implicit and parametric-implicit. While differential geometry of a parametric curve can be found in textbooks such as in [412, 444, 76], there is little literature on differential geometry of intersection curves. Faux and Pratt [116] give a for mula for the curvature of an intersection curve of two parametric surfaces. Willmore [444] describes how to obtain the unit tangent t, the unit principal normal n, and the unit binormal b, as well as the curvature k and the torsion τ of the intersection curve of two implicit surfaces. Hartmann [154] provides formulae for computing the curvature k of intersection curves for all three types of intersection problems. They all assume transversal intersections where the tangential direction at an intersection point can be computed simply by the cross product of the normal vectors of the both surfaces.