Differential Properties at Singular Points of Parametric Surfaces

Abstract

Differential properties, e.g., a normal vector, are important characteristics of a surface. Most of the applications such as surface rendering, surface-surface intersection, and surface fairing, require the differential properties. These properties are well defined in differential geometry which has some assumptions in order to guarantee the cal­ culation of differential properties. For instance, a normal vector of a parametric surface can be calculated as a cross product of two partial derivatives under the assumptions. However, in geometric modeling, SUI­ faces may have singular points where the assumptions are not satisfied. This chapter aims at computing differential properties at singular points. A normal vector can be calculated by taking a limit of cross product nor­ mal vectors. Higher order differential properties are deduced from the normal vector.