In this paper we discuss a natural way to define barycentric coordinat
es on general sphere-like surfaces. This leads to a theory of Bernstei
n-Bezier polynomials which parallels the familiar planar case. Our con
structions are based on a study of homogeneous polynomials on trihedra
in R(3). The special case of Bernstein-Bezier polynomials on a sphere
is considered in detail.