Classical Polygonal Interpolation: A Unified Geometric Approach

finite elements, parameterization, interpolation, morphing, and shading.
The classical barycentric coordinates were only defined for triangles
and higher dimensional simplices.
Three commonly used generalizations are the Wachspress coordinates that
are affine invariant and positive inside of
convex polygons, mean value coordinates that are defined in the whole
plane and guaranteed to be positive
inside of arbitrary polygons, and discrete harmonic coordinates that
serve as a discrete version
of the Laplacian operator.
We present a unified, geometric, and intuitive construction that
explains the
'linear precision' property of all these coordinates. Using it, we
obtain the discrete harmonic, mean value,
and Wachspress coordinates for arbitrary dimensions. It is also
perceivable that this construction simplifies
many proofs in the theory of barycentric coordinates, and leads to new
results.