Circular Bernstein-b Ezier Polynomials X2. Barycentric Coordinates on Circular Arcs

In this paper we discuss a natural way to deene barycen-tric 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] on the circular arcs making up the edges of spherical triangles. The paper is organized as follows. In Sect. 2 we introduce a kind of circular barycentric coordinate which is the basis for our developments. These are used in Sect. 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 Sect. 4 we discuss certain curves naturally associated with our circular BB-polynomials. There we introduce control curves, and describe various geometric properties of the curves. We conclude with a collection of remarks and references. In this section we introduce 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 with vertices v 1 6 = v 2 which are not antipodal. Let v