We illustrate the use of the applet on this page with several examples. Brief descriptions are given here, for details click on the relevant head line.

The greatest obstacle in multivariate spline research is the
fact that the dimension of spline spaces (and also for
example the solvebility of interpolation problems and the
composition of minimal determining sets) depends on not just
the *topolgy* of the
triangulation
but also its >em>geometry. An arbitrarily small
change in the location of the vertices (without changing the
way triangles are connected) can change the dimension of a
spline space! This is in complete contrast to to the
univariate case (of splines in one variable defined on a
partition of an interval).

The case of *r=1, d=3* is of particular interest for
applications, and it leads to some famous and difficult open
problems.

One of the fundamental aspects of
multivariate splines
is that things get easier as the polynomial degree
increases. In particular, if *d > 4r+1* a minimal
determining set can be found by considering vertices, edges,
and interiors of triangles individually.

Finite elements are special cases of multivariate splines. Their
key attribute is that on each triangle they can be defined completely
in terms of data or parameters given on that triangle. They are
usual used for the solution of differential equations in which case
the coefficients are unknowns that are found by assembling and
solving a larger linear system. Finite elements being local causes
this system to be (*sparse*

When the function and derivative values are given one can use finite element for interpolation, and that is how they will be presented on this page.

[15-Mar-1999]