Differential properties at singular points of parametric surfaces

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 calculation 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, surfaces 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 normal vectors. Higher order differential properties are deduced from the normal vector.