Wang Yuan-Guang
Beijing Electro-Mechanical Engineering Institute, Beijing, China.
Sci Rep. 2025 May 22;15(1):17784. doi: 10.1038/s41598-025-97421-5.
The parameterization of surfaces, which is related to many frontier problems in mathematics, has long been a challenge for scientists and engineers. On the other hand, Ricci flow is a powerful tool in geometric analysis for studying low-dimensional topology. Owing to the natural cooperative impetus, Ricci flow has been increasingly employed to parameterize closed surfaces. However, due to the lack of choices when addressing high genus surfaces, engineers must still rely on the mainstream tool of hyperbolic Ricci flow, which is inconsistent with human intuition. Therefore, this disadvantage is a potential barrier for humans in designing textures in the parameter domain. By making a small modification to traditional Euclidean Ricci flow to sacrifice its tessellation capability, we develop a new Euclidean Ricci flow method with special features characterized by its ability to embed the fundamental domain of high genus surfaces in 2-dimensional Euclidean space. Based on this method, the parameter domain is more suitable for exploring the nature of singularity points on high genus surfaces and more suitable for designing the checkerboard textures. Four illustrative examples demonstrated the robust, rigorous features of our method, abandoning dogma and challenging the traditional views that only the hyperbolic Ricci flow can be used to parameterize high genus surfaces.
曲面的参数化与数学中的许多前沿问题相关，长期以来一直是科学家和工程师面临的挑战。另一方面，里奇流是几何分析中研究低维拓扑的有力工具。由于自然的协同推动，里奇流越来越多地被用于对封闭曲面进行参数化。然而，由于在处理高亏格曲面时缺乏选择，工程师们仍然必须依赖双曲里奇流这一主流工具，而这与人类直觉不符。因此，这一缺点是人类在参数域中设计纹理时的潜在障碍。通过对传统欧几里得里奇流进行微小修改以牺牲其细分能力，我们开发了一种具有特殊功能的新欧几里得里奇流方法，其特点是能够将高亏格曲面的基本域嵌入二维欧几里得空间。基于此方法，参数域更适合探索高亏格曲面上奇点的性质，也更适合设计棋盘纹理。四个示例展示了我们方法的稳健、严谨的特点，摒弃了教条，挑战了只有双曲里奇流可用于对高亏格曲面进行参数化的传统观点。
