High genus surface parameterization using the Euclidean Ricci flow method

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.