Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow

TL;DR

This work addresses the challenge of numerically computing quasi-conformal maps on general Riemann surfaces by solving the Beltrami equation $\frac{\partial f}{\partial \bar{z}} = \mu(z) \frac{\partial f}{\partial z}$ for a prescribed Beltrami differential $\mu$. The key idea is to construct an auxiliary metric $\tilde{g_1} = e^{2\lambda_1(z)} |dz + \mu d\overline{z}|^2$ so that the desired map becomes conformal between $(S_1,\tilde{g_1})$ and $(S_2,g_2)$, enabling a discretized conformal map via discrete Yamabe flow. The method combines discrete Euclidean and hyperbolic curvature flow (discrete Yamabe flow), discrete Beltrami differentials, and Yamabe energy to guarantee convergence to constant-curvature metrics on general meshes. Experimental results on genus-0, genus-1, and genus-2 surfaces demonstrate accuracy, compositional consistency, and applicability to complex topologies, showing the framework's broad utility for geometric processing tasks involving quasi-conformal maps.

Abstract

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichmüller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

Figures (14)

• Figure 1: Conformal and Quasi-conformal maps for a topological disk. (A) is the original face. (B) is the conformal mapping of (A). (C) is the circle packing induced by (B). (D) is the checkerboard texture induced by (B). (E) is the quasi-conformal mapping. (F) is the circle packing induced by (E).
• Figure 2: Illustration of how the Beltrami coefficient $\mu$ measures the distortion by a quasi-conformal mapping that maps a small circle to an ellipse with dilation $K$.
• Figure 3: The figure illustrates the definition of quasi-conformal maps between Riemann surfaces
• Figure 4: (A) shows the Euclidean and Hyperbolic triangle. (B) shows the discrete surface Yamabe flow
• Figure 5: The figure illustrates the idea of discrete curvature flow. A planar domain with non-zero curvature on the boundary is being deformed iteratively into a flat metric. Under the deformed metric, curvature is zero everywhere in the interior and is constant on the boundary (i.e. the boundary becomes a circle).

Theorems & Definitions (7)

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