Differential Geometry of Surfaces

Abstract

Surfaces define the boundaries of a solid. They themselves are bounded by curves (Figure 6.1). Surface design may be regarded as an extension of curve design in two parametric dimensions. Developments in previous chapters, therefore, can all be applied in surface modeling. In curve design, emphasis is laid on the generic (non-analytical) parametric representation of low degree polynomial segments that can be composed together to model a curve. Reasons are to encompass a variety of shapes that analytical curves fail to provide, to prevent undue oscillations that may be observed in higher degree polynomial segments, and to make the representation free from singularities like vertical slopes. Also, parametric representation makes easier to compute the intersection points or plot curves easier that is not so with explicit and implicit representations. Following the above, we can treat surface modeling in a manner similar to curve design, that is, we represent surface patches