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A Novel Algorithm for Periodic Conformal Flattening of Genus-one and Multiply Connected Genus-zero Surfaces

Zhong-Heng Tan, Tiexiang Li, Wen-Wei Lin, Shing-Tung Yau

TL;DR

The paper introduces periodic conformal flattening to map genus-one and multiply connected genus-zero surfaces onto fundamental periodic domains by minimizing a discrete conformal energy, producing a sparse quadratic problem. It yields two algorithms: DPCF for genus-one and SPCF for multiply connected surfaces, both solving via small linear systems and independent of cut-path choices. The authors prove cut-path independence and bijectivity under intrinsic Delaunay conditions, with an IDT preprocessing strategy to guarantee bijectivity when needed. Numerical experiments show 4–5× efficiency gains over prior methods and robust texture-mapping results, illustrating practical impact in mesh processing and graphics. The work highlights limitations to higher-genus generalization and points to ongoing extension toward broader applicability.

Abstract

In this paper, we propose a novel parameterization method for genus-one and multiply connected genus-zero surfaces, called periodic conformal flattening. The conformal energy minimization technique is utilized to compute the desired conformal map, which is characterised as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The advantages of the proposed algorithms DPCF and SPCF are a) independence from the cut path selection, which introduces no additional conformal distortion near the cut seams; b) bijectivity guaranteeing for intrinsic Delaunay triangulations. The numerical experiments illustrate that DPCF and SPCF express high accuracy and a 4-5 times improvement in terms of efficiency compared with state-of-the-art algorithms.Based on the theoretical proof of the bijectivity guaranteeing, a simple strategy is applied for to guarantee the bijectivity of the resulting maps for non-Delaunay triangulations. The application on texture mapping illustrates the practicality of our developed algorithms.

A Novel Algorithm for Periodic Conformal Flattening of Genus-one and Multiply Connected Genus-zero Surfaces

TL;DR

The paper introduces periodic conformal flattening to map genus-one and multiply connected genus-zero surfaces onto fundamental periodic domains by minimizing a discrete conformal energy, producing a sparse quadratic problem. It yields two algorithms: DPCF for genus-one and SPCF for multiply connected surfaces, both solving via small linear systems and independent of cut-path choices. The authors prove cut-path independence and bijectivity under intrinsic Delaunay conditions, with an IDT preprocessing strategy to guarantee bijectivity when needed. Numerical experiments show 4–5× efficiency gains over prior methods and robust texture-mapping results, illustrating practical impact in mesh processing and graphics. The work highlights limitations to higher-genus generalization and points to ongoing extension toward broader applicability.

Abstract

In this paper, we propose a novel parameterization method for genus-one and multiply connected genus-zero surfaces, called periodic conformal flattening. The conformal energy minimization technique is utilized to compute the desired conformal map, which is characterised as an easily solvable quadratic functional minimization problem, yielding a sparse linear system. The advantages of the proposed algorithms DPCF and SPCF are a) independence from the cut path selection, which introduces no additional conformal distortion near the cut seams; b) bijectivity guaranteeing for intrinsic Delaunay triangulations. The numerical experiments illustrate that DPCF and SPCF express high accuracy and a 4-5 times improvement in terms of efficiency compared with state-of-the-art algorithms.Based on the theoretical proof of the bijectivity guaranteeing, a simple strategy is applied for to guarantee the bijectivity of the resulting maps for non-Delaunay triangulations. The application on texture mapping illustrates the practicality of our developed algorithms.

Paper Structure

This paper contains 15 sections, 10 theorems, 63 equations, 13 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Discrete Surface and Conformal Energy
  3. Double Periodic Conformal Flattening for Genus-One Surfaces
  4. Single Periodic Conformal Flattening for Multiply Connected Surfaces
  5. Annulus Conformal Parameterizations
  6. Poly-annulus conformal parameterizations
  7. Main Analysis
  8. Cut Path Independence
  9. Bijectivity Guaranteeing
  10. Numerical Experiments and Applications
  11. Double Periodic Conformal Flattening
  12. Single Periodic Conformal Flattening
  13. Bijectivity Guaranteeing Strategy
  14. Application on Texture Mapping
  15. Conclusion

Key Result

Theorem 1

A simply connected Riemann surface is conformally equivalent to one of the following three canonical Riemann surface:

Figures (13)

  • Figure 1: A 2D lattice and the unit cell domain $\mathcal{T}$ with doubly periodic boundary.
  • Figure 2: An illustrative example of double periodic conformal flattening.
  • Figure 3: A 1D lattice $R$ with two boundaries and the unit cell domain with single periodic boundary.
  • Figure 4: An illustrative example for conformal maps from a doubly connected surface to a quadrilateral band and then to an annulus.
  • Figure 5: An illustrative example for conformal map from a multiply connected surface to a disk with multiple circular holes.
  • ...and 8 more figures

Theorems & Definitions (19)

  • Theorem 1: Poincaré-Klein-Koebe Uniformization XGST07book
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Theorem 3
  • ...and 9 more