A Discrete-to-Continuum Variational Theory for Area-Preserving Surface Parameterization
Abstract
We establish a discrete-to-continuum variational theory for area-preserving parameterization of surfaces based on the stretch energy. For orientation-preserving diffeomorphisms between compact, connected, oriented Riemannian surfaces of equal total area, we prove that area-preserving maps are precisely the critical points of the stretch energy under boundary-fixing variations. We then establish discrete-to-continuum convergence of the optimal energy values: within a uniformly geometrically controlled admissible class, the discrete infimum converges to the smooth minimum with second-order accuracy, while discrete almost minimizers have first-order decay of their L2 area distortion. Thus, the discrete problem not only approximates the smooth stretch energy but also recovers area preservation in the refinement limit. We further derive the L2-gradient flow of the stretch energy and its simplicial counterpart, whose projected quasi-implicit iteration is shown to be globally convergent. The resulting method applies to open and closed surfaces of several topological types, and numerical experiments on benchmark surfaces illustrate its improvement over existing methods.
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BibTeX
@misc{liu2026discretetocontinuum,
title = {{A Discrete-to-Continuum Variational Theory for Area-Preserving Surface Parameterization}},
author = {Shu-Yung Liu and Kento Sakai and Mei-Heng Yueh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.17873},
eprint = {2608.17873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17873},
}
RIS
TY - UNPB
TI - A Discrete-to-Continuum Variational Theory for Area-Preserving Surface Parameterization
AU - Shu-Yung Liu
AU - Kento Sakai
AU - Mei-Heng Yueh
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.17873
UR - https://arxiv.org/abs/2608.17873
AB - We establish a discrete-to-continuum variational theory for area-preserving parameterization of surfaces based on the stretch energy. For orientation-preserving diffeomorphisms between compact, connected, oriented Riemannian surfaces of equal total area, we prove that area-preserving maps are precisely the critical points of the stretch energy under boundary-fixing variations. We then establish discrete-to-continuum convergence of the optimal energy values: within a uniformly geometrically controlled admissible class, the discrete infimum converges to the smooth minimum with second-order accuracy, while discrete almost minimizers have first-order decay of their L^2 area distortion. Thus, the discrete problem not only approximates the smooth stretch energy but also recovers area preservation in the refinement limit. We further derive the L^2-gradient flow of the stretch energy and its simplicial counterpart, whose projected quasi-implicit iteration is shown to be globally convergent. The resulting method applies to open and closed surfaces of several topological types, and numerical experiments on benchmark surfaces illustrate its improvement over existing methods.
ER -