Detection and Computation of Degenerate Normal Vectors on Tensor Product Polynomial Surfaces

Abstract

One of the essential properties of a surface is its normal vector. Surface rendering, surface-surface intersection, and offset surface generation all require normal vectors. A normal vector at a point on a tensor product surface is usually obtained by taking a cross product of the two partial derivatives. However a normal vector can sometimes degenerate so the cross product yields a zero vector. Application programs might collapse by the degenerate normal vector. This paper is aimed at detecting any degenerate normal vectors of a tensor product Bézier surface as well as computing normal vectors at those points where they can be uniquely defined. Both the detection and the computation of degenerate normal vectors are carried out in a uniform manner based on a Bézier normal vector surface.