TABLE OF CONTENTS The B zier surface is a direct extension of the B zier curve. Many topics applicable to these surfaces, but discussed under B zier curves, are not repeated here. This chapter does discuss the bicubic B zier patch, conversion between bicubic B zier and Hermite forms, degree elevation, the composite B zier surface, and, briefly, the rational B zier surface.
Points on a B zier patch are given by the following tensor product, a simple extension of the general equation for points on a B zier curve:
where u, w ?[0, 1].
The p ij comprise a ( m+1) ( n+1) rectangular array of control points defining the vertices of the characteristic polyhedron of the B zier patch, which lies entirely within its convex hull. B im (u) and B jn (w) are the basis functions, defined in the same way as for B zier curves.
The general matrix equation for a B zier patch is
or
where the dimensions of the matrices depend on the dimensions of the control point array. The subscripts indicating the matrix sizes are not repeated in later expressions, and they are used here merely to illustrate their relationship to the control points. The B zier patch need not be described by a square array of control points, although we will begin by exploring a 4 4 array of points, which defines the bicubic B zier patch.
We can express a bicubic B zier patch in a form similar to the bicubic Hermite patch.
