Overview

The aim of this chapter is to review some methods for surface definition. A surface can be defined using various categories of methods, and, in particular, we encounter parametric, implicit or explicit surfaces. A parametric surface, given u and v two parameters living in some interval, is the data of a function a whose values ?( u, v) describe the surface. Implicit surfaces are given by a relation like f( x, y, z)=0 while explicit surfaces take the form z= f( x, y).

In the first sections of this chapter we discuss the methods extensively used in practice (for instance, in CAD systems), i.e., parametric surfaces. Most methods developed for surface definition are derived from methods used for curve definition. Thus, the material discussed in Chapter 12 will be reviewed here and extended to the case of surfaces.

First, we introduce the basic notions related to surface definitions (as we did for curves in Chapter 12) to give a rough idea of the various methods that can be used to define (construct) a surface. It is not our intention to be exhaustive, and we refer the reader to the ad-hoc literature. In this respect, references listed in Chapter 12 for curve topics remain relevant.

Surface definitions can be classified into several categories depending on what the surface looks like or what the geometry of the surface must be. Particular surfaces are briefly discussed including surfaces of revolution, ruled