Summary: Dimensions of Multivariate Spline Spaces 1
Introduction. This is a sequel to the Case Study, Alfeld, 1987. That paper was updated a few times and
distributed privately. In this paper dimension results are collected more systematically. Contact the author
for the last available edition of the Case Study.
Note on Tables and Figures. Due to the peculiarities of the T E X typesetting system Figures and Tables
may not appear in their natural order. They may also appear far from the point they are referenced in the
text, and they may separate paragraphs and even formulas. Since the assembly of these notes is an ongoing
process no effort has been made to smooth out the arrangement of the paper's ingredients.
Note on Software. All dimensions were computed using the software package Goliath, see Alfeld and Eyre,
1991a and 1991b.
Note on Notation. To be consistent with a larger part of the literature, we use:
d = polynomial degree
r = degree of smoothness
k = dimension of underlying domain.
Boxed Entries. Boxed entries in the tables are particularly interesting and are commented on in the text.
The ThreeDimensional CloughTocher Split, 4 Tetrahedra. The trivariate CloughTocher split is
a straightforward generalization of the bivariate CloughTocher split. Thus we consider a tetrahedron with
vertices V 1 , V 2 , V 3 , and V 4 , that has been split about its centroid
V 5 = (V 1 + V 2 + V 3 + V 4 ) =4: (2)