Recently, we have introduced spaces of splines defined on triangulatio
ns lying on the sphere or on sphere-like surfaces. These spaces arose
out of a new kind of Bernstein-Bezier theory on such surfaces. The pur
pose of this paper is to contribute to the development of a constructi
ve theory for such spline spaces analogous to the well-known theory of
polynomial splines on planar triangulations. Rather than working with
splines on sphere-like surfaces directly, we instead investigate more
general spaces of homogeneous splines in R(3). In particular, we pres
ent formulas for the dimensions of such spline spaces, and construct l
ocally supported bases for them.