7.1 Bicubic Interpolation

This is one of the simplest surface construction methods. Its uses are limited, but it prepares us for more general approaches. Suppose you are given 16 points p _i,j arranged as in Sketch 56 and also a pair of parameter values ( u _i, v _j) with each of them.

Sketch 56: A 4 4 grid of points.

We seek an interpolating bicubic patch x( u, v), such that

Following the matrix notation of Section 6.3, we may write each x( u _i, v _j) as a matrix product. We list an example:

x( u ₁ v ₂) =

We may combine all individual equations for the x( u _i, v _j) into one matrix equation:

where P is the given matrix of data points as in (7.1) and B is the matrix containing the unknown control points b _i,j. The matrix M ^T contains the values of the Bernstein polynomials at the given parameters:

and N is given by

Equation (7.3) is most conveniently decomposed into a sequence of linear systems. First, we define

which reduces (7.3) to

This constitutes four "vector" systems each with four linear equations, all having the...