30 - Approximation of Surfaces

Summary

Computer-aided geometric design focuses on the representation and design of surfaces in a computer graphics environment. A popular way to represent surfaces in such an environment is with Bézier surface patches. In this chapter we extend most of the results about Bernstein polynomials and Bézier curves to surfaces.

We have seen in Fig. 27.2 how complicated curved shapes can be defined by joining together several Bézier curves. Each is really a segment of a curve, restricted to parameter values 0 ≤ t ≤ 1. Similarly, Bézier surface patches are portions of surfaces that can be pieced together to form complicated shapes. The two most common Bézier patches are called “three-sided” (or, triangular) and “four-sided” (or, rectangular). Of course, these are not triangles and rectangles in the traditional sense, but regions of surfaces with three- or four-sided boundaries.

The conceptually simplest way of extending the univariate Bernstein polynomials to several variables is to use rectangular patches. (See [Lorentz '531.) We want to approximate a function f(x, y) in the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, using values of f on the grid points x = i/m, y = j/n where m, n are positive integers and i = 0, 1, …, m, j = 0, 1, …, n. We replace, for each fixed y, f(x, y) as a function of x by its mth degree Bernstein approximant.