Chapter 8 - Pyramid Algorithms for Multisided Bezier Patches

Abstract

The purpose of this chapter is to investigate pyramid algorithms for multisided Bezier patches. This chapter builds dynamic programming procedures—pyramid algorithms—based upon special recurrence relations. Such recurrences arise quite naturally in three settings: Lagrange interpolation, discrete convolution, and blossoming. Standard Lagrange interpolation procedures do not lend themselves to generating multisided patches. This chapter highlights discrete convolution and blossoming, which are unavoidably interrelated. Construction of multisided Bezier patches requires multisided arrays of control points and barycentric coordinate functions for multisided polygonal domains. Multisided Bezier patches have many different formulations: S-patches, C-patches, and toric Bezier patches are the three most important paradigms. Three common threads tie these three schemes together: discrete convolution, Minkowski sum, and the general pyramid algorithm. Each of these schemes, S-patches, C-patches, and toric Bezier, is an example of a pyramid patch with a polygonal domain, but with a different type of indexing set and a different collection of barycentric coordinate functions. This chapter reviews the properties and algorithms that multisided schemes share with the standard three-sided and four-sided Bezier patches.