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.
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