ABSTRACT
When generalized barycentric coordinates are gaining popularity in an ever-growing set of application contexts, there is a need for a clear understanding of the relationship between the interpolation properties of various coordinate types and the geometries on which they are defined. The authors present both practical computational evidence and theoretical mathematical analyses for a variety of circumstances in which generalized barycentric interpolation will predictably succeed. They then compute interpolants using the following coordinates: Wachspress, mean value, discrete harmonic, moving least squares with linear, quadratic, and cubic weights, and maximum entropy with a "triangle inequality" prior and the uniform prior. The ideas have been generalized to tetrahedra and higher-dimensional simplices. On tetrahedra, the most widely known and used shape quality metric is aspect ratio, which on polyhedra is the ratio of the diameter of the element to the radius of the smallest inscribed sphere.
