Details

Given a set of data points , and , this Demonstration constructs a nonrational -degree B-spline surface interpolating these points, namely:

(1)

Again, the first order of business is to compute reasonable values for the and the knot vectors and . We show how to compute the ; the are analogous.

A common method is to use equations (2) or (3) to compute parameters , , ⋯, for each and then to obtain each by averaging across all for , that is,

, ,

where for each fixed , was computed by equation (2) or (3).

(a) Let , then

, for , and . (2)

(b) Let , then

for , and . (3)

Once the are computed, the knot vectors and can be obtained by equation (4):

, , , . (4)

Now to the computation of the control points. Clearly, equation (1) represents linear equations in the unknown . However, since is a tensor product surface, the can be obtained more simply and efficiently as a sequence of curve interpolations.

For fixed , write equation (1) as

, (5)

where . (6)

Notice that equation (5) is just curve interpolation through the points . The are the control points of the isoparametric curve on at fixed . Now fixing and letting vary, equation (6) is curve interpolation through the points , with as the computed control points. Thus, the algorithm to obtain all the is as follows:

1. Using and the , do curve interpolations through , which yields the .

2. Using and the , do curve interpolations through , which yields the .

Reference

[1] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer-Verlag, 1997 pp. 376–382.