000 Society for Industrial and Applied Mathematics Dimension and Local Bases of Homogeneous Spline Spaces*

Recently, we have introduced spaces of splines deened on triangulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-B ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangula-tions. Rather than working with splines on sphere-like surfaces directly, we instead investigate more general spaces of homogeneous splines in IR 3. In particular, we present formulae for the dimensions of such spline spaces, and construct locally supported bases for them.