Our intuitive notion of a surface is a continuous set of points approximating a plane in the neighborhood of each of the points. Mathematically more succinct, we also may think of a surface as a two-parameter family of points. Yet another conception, analogous to our notion of a curve, is that of a surface as the locus of a point moving with two degrees of freedom. In addition, we can describe surfaces with special properties: for example, as the locus of points of a moving line or curve. These notions become useful to geometric modeling when we express them analytically. The intrinsic equations and characteristics of a surface are a subject of differential geometry and are far more subtle and complex than those of a curve. Where the theory of curves centers on curvature and torsion as functions of arc length, nothing quite so straightforward applies to surfaces. Although, in the chapters that follow, we will discuss many surface properties arising from their differential geometry, we will not pursue the fundamental forms or more rigorous derivations. In this chapter, we review the explicit and implicit equations of surfaces, and introduce the bivariate parametric equations. These topics are the basis for later discussions of quadric surfaces expressed implicitly, and the Hermite, B zier, and B-spline surfaces expressed parametrically. The implicit and parametric surfaces are the most commonly used surfaces in geometric modeling, and most of our studies here are particularly about the tensor product parametric surfaces.

We construct,...