Multisided arrays of control points for multisided Bézier patches Original Research Article

A survey is presented examining a variety of different techniques, some old and some new, for extending the construction of standard three sided and four sided Bézier patches to n-sided surface patches. Standard triangular and rectangular Bézier patches can be defined either explicitly using Bernstein blending functions or recursively using de Casteljau pyramid algorithms based on barycentric coordinate functions. Underpinning both of these constructions are control points organized into triangular and rectangular arrays. Thus to extend the notion of Bézier patches to multisided schemes, we need to construct multisided arrays of control points and we need to generalize either the Bernstein basis functions or the barycentric coordinate functions to polygonal domains. But what exactly are multisided arrays of control points, and how precisely do we construct Bernstein basis functions or barycentric coordinates for polygonal domains? There is no single answer to any of these questions: different answers lead to different types of multisided Bézier schemes. Here we focus on different techniques for indexing multisided arrays, including Greek gnomons, spider webs, planar tessellations, fractal gaskets, Minkowski sums, and lattice polygons. For each of these indexing sets, we provide, whenever possible, either the associated bivariate Bernstein blending functions or the barycentric coordinate functions and the associated de Casteljau pyramid algorithm for the corresponding polygonal domains.