发表机构
National Taiwan Normal University; University of Tokyo（台湾师范大学; 东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对紧致二维黎曼流形间的保面积参数化问题，通过拉伸能建立变分表征，提出authalic流，开发离散计算方法并验证其面积保留性能。
AI 中文摘要
保面积参数化应用于必须保留相对表面积的场景中。我们通过拉伸能研究该问题。对于总面积相等的紧致二维黎曼流形之间的保定向微分同胚，我们证明拉伸能由面积比的方差表征，且其临界点是保面积的。该变分表征自然导出了我们称之为authalic流的L²梯度流。随后，我们基于离散拉伸能开发了其单纯形对应形式，得到了适用于多种拓扑类型的开曲面和闭曲面的计算方法。为连接离散形式与光滑理论，我们证明了拉伸能相对于网格细化的一阶一致性，并在给定几何近似假设下，建立了离散全局极小值的一阶L²面积失真界。在基准网格上的数值实验在所有测试中均生成无折叠映射，且与现有方法相比表现出具有竞争力的面积保留效果。
英文摘要
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.
