Barycentric Coordinates for Arbitrary Planar Polygons

$v$ inside a triangle $[v_1,v_2,v_3]$ as a \emph{convex
combination} of the three vertices $v_i$. This concept has
recently been generalized in various ways to points inside a
convex polygon and one possible generalization are the mean
value coordinates that are also known to work for points inside
the kernel of a star-shaped polygon, which is important in the
context of triangle mesh parameterization.

We show that mean value coordinates are even more general
and allow to write \emph{any} point $v$ in the plane as an
\emph{affine combination} of the vertices of an \emph{arbitrary}
polygon (non-intersecting and with multiple, possibly nested
components). The weights of the affine combination depend smoothly
on $v$ and can be used to interpolate data that is given at the
$v_i$. The interpolating surface is $C^\infty$, except at the
$v_i$, where it is only $C^0$, and very similar to the
interpolating thin plate spline, albeit much faster to compute.