Barycentric Interpolation

Interpolating given discrete data with continuous functions in one or more variables is a fundamental problem in diverse fields of science and engineering. This project focusses on barycentric data interpolation, which is characterized by the fact that the interpolant can be expressed as a linear combination of the data with certain cardinal basis functions as well as by the reproduction of linear functions, which in turn justifies to call the basis functions "barycentric". While many classical interpolation methods, including interpolation with splines, radial basis functions, and subdivision schemes, just to name a few, could be called barycentric, we prefer to use the term barycentric interpolation whenever simple closed-form expressions for the barycentric basis functions exist, so that evaluating the barycentric interpolant is efficient and can be done without needing to first solve a linear or even non-linear interpolation problem.In one variable, the idea of non-polynomial barycentric interpolation was pioneered by Berrut in 1988, who proposed a simple barycentric rational interpolant with guaranteed absence of real poles. This basic idea has since been generalized and has risen in popularity, especially in the case of equidistant data points. These interpolants are similar in spirit to Shepard's method, which remains a common choice for interpolating data given at scattered two- and three-dimensional sample points, but they have better approximation properties.For the special case of data prescribed at the vertices of a simplex, barycentric coordinates, which were introduced by Möbius in 1827, provide a very convenient way to express the unique linear interpolant. This kind of barycentric interpolation is widely used in computer graphics for interpolating vertex attributes such as colour, normals, or texture coordinates over the individual triangles of a triangle mesh, but it also plays a key role in other disciplines. Starting with the seminal work published by Wachspress in 1975, the ideas of barycentric coordinates and barycentric interpolation have been extended in recent years to arbitrary polygons in the plane and to general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, mesh deformation, as well as finite element and boundary element methods.