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BernsteinB'ezier Polynomials on Spheres and SphereLike Surfaces

Summary: Bernstein­B'ezier Polynomials on Spheres
and Sphere­Like Surfaces
Peter Alfeld 1) , Marian Neamtu 2) , and Larry L. Schumaker 3)
Abstract. In this paper we discuss a natural way to define barycentric coordi­
nates on general sphere­like surfaces. This leads to a theory of Bernstein­B'ezier
polynomials which parallels the familiar planar case. Our constructions are
based on a study of homogeneous polynomials on trihedra in IR 3 . The special
case of Bernstein­B'ezier polynomials on a sphere is considered in detail.
1. Introduction
Bernstein­B'ezier (BB­) polynomials defined on triangles are useful tools for con­
structing piecewise functional and parametric surfaces defined over triangulated
planar domains. They play an extremely important role in CAGD (computer­aided
geometric design), data fitting and interpolation, computer vision, and elsewhere
(see e.g. the books [Farin '88, Hoschek & Lasser '93]).
In many applications we need to work on the sphere, or on sphere­like surfaces.
Researchers have been searching for a number of years for an appropriate analog
of the BB­polynomials in the spherical setting, but have been hampered by the
perceived lack of a reasonable way to define barycentric coordinates on spherical
triangles. In fact, recently, [Brown & Worsey '92] showed that such coordinates


Source: Alfeld, Peter - Department of Mathematics, University of Utah


Collections: Mathematics