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The Conjugate Function Method for Surfaces with Elaborate Topological Types

H. Hakula, A. Rasila, Y. Zheng

TL;DR

The paper tackles the challenge of computing conformal mappings for multiply connected domains on planar surfaces and Riemann surfaces with high accuracy. It extends the conjugate function method to these complex topologies by leveraging hp-FEM, a fast linear-algebra-based construction for the conjugate problem, and a generalized reciprocal identity for error control. A key contribution is reducing the Dirichlet data for the conjugate problem to a small n×n system via a reduction matrix, enabling direct, efficient computation without reliance on costly optimization. The authors demonstrate broad applicability through torus and higher-genus examples, extensive numerical experiments (including Nasser’s challenge and random slit configurations), and practical texture-mapping and polyhedral-surface applications, highlighting the method’s accuracy and scalability for complex topologies.

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is generalized and refined to achieve the same level of accuracy on multiply connected planar domains and Riemann surfaces. The main challenge is the accurate and efficient construction of boundary values for the conjugate problem on multiply connected domains. The method relies on high-order finite element methods which allow for highly accurate computations of mappings on surfaces, including domains of complex boundary geometry containing strong singularities and cusps. We also derive the reciprocal error estimate for the multiply connected case. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments with error estimates.

The Conjugate Function Method for Surfaces with Elaborate Topological Types

TL;DR

The paper tackles the challenge of computing conformal mappings for multiply connected domains on planar surfaces and Riemann surfaces with high accuracy. It extends the conjugate function method to these complex topologies by leveraging hp-FEM, a fast linear-algebra-based construction for the conjugate problem, and a generalized reciprocal identity for error control. A key contribution is reducing the Dirichlet data for the conjugate problem to a small n×n system via a reduction matrix, enabling direct, efficient computation without reliance on costly optimization. The authors demonstrate broad applicability through torus and higher-genus examples, extensive numerical experiments (including Nasser’s challenge and random slit configurations), and practical texture-mapping and polyhedral-surface applications, highlighting the method’s accuracy and scalability for complex topologies.

Abstract

The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is generalized and refined to achieve the same level of accuracy on multiply connected planar domains and Riemann surfaces. The main challenge is the accurate and efficient construction of boundary values for the conjugate problem on multiply connected domains. The method relies on high-order finite element methods which allow for highly accurate computations of mappings on surfaces, including domains of complex boundary geometry containing strong singularities and cusps. We also derive the reciprocal error estimate for the multiply connected case. The efficacy of the proposed method is illustrated via an extensive set of numerical experiments with error estimates.

Paper Structure

This paper contains 27 sections, 3 theorems, 33 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Illustrative Example: 3D Face Recognition
  3. Organization
  4. Preliminaries
  5. Riemann surfaces
  6. High-Order Finite Element Method
  7. Error Estimation
  8. Special Types of Boundaries
  9. The Conjugate Function Method On Planar Domains and Riemann Surfaces
  10. Q-type canonical domains
  11. Variational Formulation for Finite Elements
  12. Fast Construction of the Conjugate Problem
  13. Reciprocal Identity for Multiply Connected Domains
  14. Linear Algebra Based Construction of the Conjugate Problem
  15. Details of the Reduction Step
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^2$ be a polygon, $v$ the FEM-solution of the Dirichlet-Neumann problem on some polygonal domain, and let the weak solution $u_0$ be in a suitable countably normed space where the derivatives of arbitrarily high order are controlled. Then where $C$ and $b$ are independent of $N$, the number of degrees of freedom. Here $v$ is computed on a proper geometric mesh, where

Figures (15)

  • Figure 1: Map of a face. Left: The map on the parameter space. Right: Checkerboard illustration of the map on the surface. Original data source: snap2025facemesh.
  • Figure 1: Two Holes Revisited. Convergence of the reciprocal error as a function of degrees of freedom using optimization and the new construction. The current version is eventually more accurate on exactly the same mesh, and dips below the threshold limit of $10^{-7}$ that indicates the best possible accuracy when the black-box optimization routine was used (loglog-plot) 24M1656840.
  • Figure 1: Torus with checkerboard colouring of the conformal map.
  • Figure 1: Nasser's Configuration: Map
  • Figure 1: Opera Mask. Conformal map on the domain.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem 2.1
  • Lemma 3.1
  • Proposition 4.1
  • Proof 1
  • Remark 4.2
  • Remark 4.3