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.