ABSTRACT
In this chapter, we will encounter surfaces for the first time. We will cover the basic definitions and go on to extend the concept of Be´zier curves to surfaces. A famous object which is composed of Be´zier patches is the “Utah teapot,” shown in Figure 6.1.1
6.1 Parametric Surfaces A parametric curve is the result of a mapping of the real line into 2-or 3-space. A parametric surface is defined in a similar way: It is the result of a map of the real plane into 3-space. This “real plane” is called the domain of the surface. It is simply a plane with a
has (u, v). corresponding 3D surface point is then a point:
x(u, v) =
⎡ ⎣ f(u, v)g(u, v)
h(u, v)
⎤ ⎦ . (6.1)
EXAMPLE 6.1 The parametric surface given by
x(u, v) =
⎡ ⎣ uv
u2 + v2
⎤ ⎦
is illustrated in Sketch 39. Of course, only a portion of the surface is illustrated; the surface extends infinitely from each edge. This parametric surface happens to be a functional surface because two of the coordinate functions in (6.1) are simply u and v.
