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Abstract

We develop spherical barycentric coordinates. Analogous to classical,
planar barycentric coordinates that describe the positions of points in a plane
with respect to
the vertices of a given planar polygon, spherical barycentric coordinates
describe the positions
of points on a sphere with respect to the vertices of a given spherical
polygon.
In particular, we introduce spherical mean value coordinates that inherit many
good properties of their planar counterparts.
Furthermore, we present a construction that gives a simple and intuitive
geometric interpretation for
classical barycentric coordinates, like Wachspress coordinates, mean value
coordinates, and discrete
harmonic coordinates.

One of the most interesting consequences is the possibility to
construct mean value coordinates for arbitrary polygonal meshes.
So far, this was only possible for triangular meshes. Furthermore, spherical
barycentric coordinates
can be used for all applications where only planar barycentric coordinates were
available up to now.
They include B\'ezier surfaces, parameterization, free-form deformations, and
interpolation of rotations.