Mean value coordinates for arbitrary planar polygons

Abstract

Barycentric coordinates for triangles are commonly used in computer graphics, geometric modeling, and other computational sciences because they provide a convenient way to linearly interpolate the data that is given at the corners of a triangle. The concept of barycentric coordinates can also be extended in several ways to convex polygons with more than three vertices, but most of these constructions break down when used in the nonconvex setting. Mean value coordinates offer a choice that is not limited to convex configurations, and we show that they are in fact well-defined for arbitrary planar polygons without self-intersections. Besides their many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation. They are particularly useful for interpolating data that is given at the vertices of the polygons and we present several examples of their application to common problems in computer graphics and geometric modeling.