TABLE OF CONTENTS The equation of a B-spline surface follows directly from the equation of a B-spline curve. This relationship is analogous to that between B zier curves and surfaces. Furthermore, we define the B-spline surface, like the B zier surface, in terms of a characteristic polyhedron. The shape of the surface approximates the polyhedron. The approximation is weaker the higher the degree.
The tensor product equation of the B-spline surface is
The p ij are control points and are the vertices of the characteristic polyhedron. The N i, K (u) and N j, L (w) are the basis functions, and they are the same as those for B-spline curves. The degree of each of the basis-function polynomials is controlled by K and L, respectively.
For a nonperiodic B-spline surface, we select the values of K, L, m, and n, and compute the knot values just as for curves. We compute N i, K (u) and N j, L (w) recursively, using Equations 5.2 and 5.3. Notice that two sets of knot values are required and that the control points form an ( m+1) ( n+1) array.
The matrix form of the B-spline surface is similar to the form of the B-spline curve. Recall that the B-spline curve is computed in segments of a unit interval on the parametric variable u (see Equations 5.14, 5.15, and 5.23). A unit square on the parametric variables u
