I recently participated in a minisymposium (SIAM Conference ),
where geometric modeling, graphics, and finite elements were the focus.
Over the past 4 to 5 years, there has been a lot of interest in the
construction of barycentric coordinates on polygons/polyhedra, and the
minisymposium brought together many of us with common interests. There
are many open-questions, interesting connections to be established, and
potential applications (in FEM/solution of PDEs) that can be explored.
As a follow-up to the minisymposium, a web page has been created that
provides links to many online resources. If you're inclined, you can
take a look at Barycentric Coordinates and Transfinite Interpolation for more information.