Many recent works attempt to generalize barycentric coordinates to arbitrary polygons. I construct a counterexample proving that no such generalization will produce purely bijective mappings in the plane, provided the coordinates meet the Lagrange, reproduction, and partition of unity properties. The proof concerns generalized barycentric coordinates in a square, but trivially generalizes to arbitrary polygons with degree greater than three.
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Alec Jacobson (2013) studied this question.
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