V _{B} is the number of boundary vertices of the
triangulations, and V _{I} the number of interior
vertices.

The simplest example of a non-generic triangulation is formed by a singular vertex.

It's an open problem whether every non-degenerate triangulation is generic. Not every degenerate triangulation is non-generic.

The major feature that distinguishes bivariate splines from
univariate splines (which are functions of just one
independent variable) is that the dimension depends not just
on the number of triangles and how they are connected, but
also on the precise location of the vertices. By contrast,
in the univariate case it suffices to specify the number
*N* of subintervals, the degree *r* of
smoothness, and the polynomial degree *d* to
determine the dimension of *Srd* (in fact, it's *
d+1+(N-1)(d-r)* ). In the bivariate case an arbitrarily
small change in the location of one vertex can cause the
dimension of the spline space to change!

However, for every *topology* of the triangulation
(which describes how triangles are connected) there is a
*generic dimension.* If the spline space does not
have that dimension then there is an arbitrarily small
perturbation of the location of the vertices that will cause
the dimension to assume that generic value. Moreover, the
generic dimension is always *the smallest possible*
dimension.

This is actually easy to see by a beautiful and simple
argument: Express the polynomials on each triangle as a
linear combination of basis functions, e.g., in the power
form. Then write down the differentiability conditions.
These form a set of *homogeneous linear equations*.
The dimension of the spline space equals the number of the
available coefficients minus the rank of the system of
smoothness conditions. Now consider a set of vertices that
causes the matrix of that linear system to have the largest
rank (and the spline space the smallest dimension) possible.
Pick a largest possible non-singular square submatrix of
that linear system. Its determinant is non-zero, and it is
a *rational function* of the locations of the
vertices. Think of those *n* locations as one point
in *2n* dimensional space. The determinant can
vanish only on a set of vertex locations of measure zero,
and if it does vanish the vertex vector can be moved out of
that set by an arbitrarily small movement.

Billera investigated the spline problem using methods from
algebraic homology that led to a particular linear system
based on *cofactors*. Whiteley analyzed that system
using methods from rigidity theory.

These ideas are summarized, and then applied to *
trivariate* splines, in a paper by
Larry Schumaker,
Walter Whiteley,
and myself:

The generic dimension of the space of C1 splines of degree
* d >= 8* on tetrahedral decompositions, SIAM JNA,
v. 30, pp. 889--920, 1993.

Click here to view a
dvi file
or a
postscript file
of the paper.

[21-Apr-1997]