On pseudo-harmonic barycentric coordinates

Harmonic coordinates are widely considered to be perfect barycentric coordinates of a polygonal domain due to their attractive mathematical properties. Alas, they have no closed form in general, so must be numerically approximated by solving a large linear equation on a discretization of the domain. The alternatives are a number of other simpler schemes which have closed forms, many designed as a (computationally) cheap approximation to harmonic coordinates. One test of the quality of the approximation is whether the coordinates coincide with the harmonic coordinates for the special case where the polygon is close to a circle (where the harmonic coordinates have a closed form - the celebrated Poisson kernel). Coordinates which pass this test are called "pseudo-harmonic". Another test is how small the differences between the coordinates and the harmonic coordinates are for "real-world" polygons using some natural distance measures. We provide a qualitative and quantitative comparison of a number of popular barycentric coordinate methods. In particular, we study how good an approximation they are to harmonic coordinates. We pay special attention to the Moving-Least-Squares coordinates, provide a closed form for them and their transfinite counterpart (i.e. when the polygon converges to a smooth continuous curve), prove that they are pseudo-harmonic and demonstrate experimentally that they provide a superior approximation to harmonic coordinates.