Curves and Surfaces for CAGD, 5th Edition

19

W. Boehm

Differential Geometry II

19.1 Parametric Surfaces and Arc Element

A surface may be given by an implicit form f (x, y, z) = 0 or, more useful for CAGD, by its parametric form

(19.1)

where the cartesian coordinates x, y, z of a surface point are differentiable functions of the parameters u and v and [a, b] denotes a rectangle in the u, v-plane; see Figure 19.1 (sometimes other domains are used, for example, triangles). To avoid potential problems with undefined normal vectors, we will assume

that is, that both families of isoparametric ...