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Open AccessArticle10.3929/ethz-a-007600337

Bijective mappings with generalized barycentric coordinates: A counterexample

Alec Jacobson-2012-01-01-Repository for Publications and Research Data (ETH Zurich)
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TL;DRAbstract

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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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.

Keywords

Barycentric coordinate systemMathematicsBijectionBipolar coordinatesCounterexamplePartition (number theory)GeneralizationLog-polar coordinates

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