Relaxation and $\mathrm{\Gamma }$-Convergence of Stretch Energy on Surface Homeomorphisms

Area-preserving parameterization of triangular meshes is an essential task in geometry processing, yet an exactly area-preserving simplicial mapping may not exist in general. In this paper, we study area preservation of simplicial mappings in the limit of mesh refinement by establishing the $\mathrm{\Gamma }$-convergence of the discrete stretch energy, a functional that measures their area distortion. We first introduce its limit functional, the relaxed stretch energy on surface homeomorphisms, defined through the pushforward of the area measure, and show that it is the relaxation of the stretch energy of diffeomorphisms with respect to uniform convergence. Then, to compare simplicial mappings with homeomorphisms, we construct a geometric lift that requires no shape-regularity assumption on the image mesh. Finally, we establish the $\mathrm{\Gamma }$-convergence and show that the minimal discrete energies converge to the continuum minimum. Moreover, any homeomorphism arising as a limit of lifted discrete local minimizers in fixed-radius balls is area-preserving.