Smooth Two-Dimensional Interpolations: A Recipe for All Polygons

This paper presents a method for constructing smooth and bounded interpolations on any polygon, whether convex or concave in shape, or even one containing holes or isolated points in its interior. The resulting two-dimensional function distributes the value at any given vertex or internal node over the remaining portion of the domain. The representation depends only on simple geometrical properties such as lengths and areas. Accordingly, it is invariant with respect to any chosen coordinate system. The resulting set of interpolations is smooth within the domain. Within a triangle, the behavior is akin to a linear color gradient. If necessary, linear boundary behavior can also be assured. A Java implementation is available online.