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Abstract:
Bernstein polynomials are a classical tool in Computer Aided Design to
create smooth maps
with a high degree of local control.
They are used for the construction of B\'ezier surfaces, free-form
deformations, and many other applications.
However, classical Bernstein polynomials are only defined for simplices
and parallelepipeds.
These can in general not directly capture the shape of arbitrary
objects. Instead,
a tessellation of the desired domain has to be done first.
We construct smooth maps on arbitrary sets of polytopes
such that the restriction to each of the polytopes is a Bernstein
polynomial in mean value coordinates
(or any other generalized barycentric coordinates).
In particular, we show how smooth transitions between different
domain polytopes can be ensured.
