Square-Domain Area-Preserving Parameterization for Genus-Zero and Genus-One Closed Surfaces

The parameterization of closed surfaces typically requires either multiple charts or a non-planar domain to achieve a seamless global mapping. In this paper, we propose a numerical framework for the seamless parameterization of genus-zero and genus-one closed simplicial surfaces onto a unit square domain. The process begins by slicing the surface with either the shortest-path or the Reeb graph method. The sliced surface is then mapped onto the unit square using a globally convergent algorithm that minimizes the weighted variance of per-triangle area ratios to achieve area preservation. Numerical experiments on benchmark models demonstrate that our method achieves high accuracy and efficiency. Furthermore, the proposed method enables applications such as geometry images, producing accurate and high-quality surface reconstructions.