Circular Bernstein-b Ezier Polynomials

We discuss a natural way to deene barycentric coordinates associated with circular arcs. This leads to a theory of Bernstein-B ezier polynomials which parallels the familiar interval case, and which has close connections to trigonometric polynomials. x1. Introduction Bernstein-B ezier (BB-) polynomials deened on an interval are useful tools for constructing piecewise functional and parametric curves. They play an important role in CAGD, data tting and interpolation, and elsewhere. The purpose of this paper is to develop an analogous theory where the domain of the polynomials is a circular arc rather than an interval. In addition to their intrinsic interest, the circular BB-polynomials studied here are also useful for describing the behavior of spherical BB-polynomials 1, 2, 3] on the circular arcs making up the edges of spherical triangles. The paper is organized as follows. In Section 2 we introduce circular barycentric coordinates as the basis for our developments. These are used in Section 3 to deene circular BB-polynomials. Several basic properties of BB-polynomials are developed in this section, including a de Casteljau algorithm , subdivision, smoothness conditions for joining BB-polynomials, and degree raising. In Section 4 we discuss certain curves naturally associated with circular BB-polynomials. We introduce control curves, and describe various geometric properties of the them. We conclude with a collection of remarks and references. x2. Barycentric Coordinates on Circular Arcs Deenition 1. Let C be the unit circle in IR 2 with center at the origin, and let A be a circular arc on C of length less than with vertices v 1 6 = v 2. Let v be a point on C. Then the (circular) barycentric coordinates of v relative to A are the unique pair of real numbers b 1 ; b 2 such that v = b 1 v 1 + b 2 v 2 : (1) All rights of reproduction in any form reserved.