| Multivariate splines are smooth piecewise polynomial functions defined
on a triangulation. A long standing open question in the field is that
of an explicit formula for the dimension of differentiable piecewise cubic
splines. The fundamental problem is that the dimension of the spline space
depends not just on the topology of the triangulation, but also on the
precise location of the vertices. This talk describes an approach to answering
that question by using the same techniques that were used to solve the
four color map problem. Thus we construct an unavoidable set of sub-triangulations
using a discharging technique. The ideas are illustrated by proving a dimension
formula for differentiable piecewise quartic splines (a much simpler result)
by the four color map techniques. That result, however, can also be obtained
by simpler means. The work described here is very tentative but it does
illustrate a perhaps unexpected connection between multivariate splines
and the four color map problem. Central to the talk is the Bernstein Bezier
form of a bivariate polynomial which will be introduced and explained in
the talk.
You can view annotated
slides of this talk. |