area-preserving piecewise affine mappings
For any two polygons that have the same area, we show how to construct isomorphic triangulations that have additional area-correspondence properties. We use those isomorphic triangulations to prove that there always exist everywhere-area-preserving piecewise-affine homeomorphisms between any two simple polygons of the same area. We prove that, moreover, every piecewise-affine homeomorphism between the boundaries of the two same-area simple polygons extends to a piecewise-affine area-preserving homeomorphism of the interiors as well.
authalic transformations piecewise affine homemorphisms