# Bijective mappings with generalized barycentric coordinates: a counterexample

## Alec Jacobson

## December 11, 2012

After attending the NFS Workshop on Barycentric Coordinates this summer I wrote up a quick counterexample showing that generalized barycentric coordinates cannot be designed to always produce bijective mappings. To be clear, no existing coordinates have claimed to obtain this property, but as far as I know no one has written explicitly that it's out of reach.
**Abstract**
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.
**Sketch**
Consider the four corners of square above. As we move **c**_{3} towards **c**_{1} the point **x** originally at the origin will either:
- be
*overrun* by **c**_{3} before **c**_{3}=**c**_{1} (not bijective),
- reach (and thus overrun)
**c**_{1} before **c**_{3}=**c**_{1} (not bijective)
- or the coordinate function of
*w*_{3}(**x**)=0.5 (bijective)

But by rotational symmetry we may apply the same logic for all 4 corners implying that to get a bijection we have all four coordinates *=0.5* contradicting the partition of unity assumption.