 
Summary: Dimension and Local Bases of
Homogeneous Spline Spaces
by
Peter Alfeld 1) , Marian Neamtu 2) , and Larry L. Schumaker 3)
Abstract. Recently, we have introduced spaces of splines defined on trian
gulations lying on the sphere or on spherelike surfaces. These spaces arose
out of a new kind of BernsteinB'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 wellknown theory of polynomial splines
on planar triangulations. Rather than working with splines on spherelike
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.
1. Introduction
Let \Delta := fT [i] g N
1 be a planar triangulation of a
set\Omega\Gamma and let 0 Ÿ r Ÿ d be
integers. The classical space of splines of degree d and smoothness r is defined
by
S r
