ABSTRACT
This chapter discusses properties of transfinite barycentric interpolation schemes, which can be considered as continuous counterparts of generalized barycentric coordinates and are currently a subject of intensive research due to their fascinating mathematical properties and numerous applications in computational mechanics, computer graphics, and geometric modeling. While traditional generalized barycentric coordinates interpolate function values given at the vertices of a polygon or polyhedron, transfinite barycentric interpolation is based on smooth domains. Transfinite barycentric interpolation schemes establish bridges between barycentric coordinates and methods used to interpolate continuous data. Discrete harmonic and Wachspress coordinates are the same for a circumscribable polygon. The continuous or transfinite version of this remarkable result is established. Harmonic coordinates and harmonic mappings deliver extremely useful tools for high quality interpolation and shape deformation. Harmonic mappings and coordinates do not allow for closed-form solutions in general and sophisticated numerical schemes are required for computing accurate approximations.
