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Fast Conformal Parameterization of Disks and Sphere Sectors

Tom Gilat, Ben Gilat

TL;DR

Addresses fast, conformal-like parameterization of 3-fold rotationally symmetric sphere-type meshes onto plane tiles under a translations tiling constraint. Introduces a novel torus-based branched covering constructed from 63 copies of the mesh, and solves two full-rank linear systems with cotangent weights to obtain a Dirichlet-energy-minimizing embedding onto the plane torus. The method extends to mappings onto equilateral triangles and rectangles via symmetry arguments and Tutte embeddings, with explicit constructions and energy-optimality proofs. Applications include rapid, minimally distorted representations of 3-fold symmetric biomolecule surfaces such as PIEZO1/PIEZO2, enabling ML and structural analysis.

Abstract

We prove a novel method for the embedding of a 3-fold rotationally symmetric sphere-type mesh onto a subset of the plane with 3-fold rotational symmetry. The embedding is free-boundary with the only additional constraint on the image set is that its translations tile the plane, in turn this forces the angles at the embedding of the branch points in the construction. These parameterizations are optimal with respect to the Dirichlet energy functional defined on simplicial complexes. Since the parameterization is over a fixed area domain, it is conformal (i.e. a minimizer of the LSCM energy). The embedding is done by a novel construction of a torus from 63 copies of the original sphere. As a foundation for this result we first prove the optimality of the embedding of disk-type meshes onto special types of triangles in the plane, and rectangles. The embedding of the 3-fold symmetric torus is full rank and so cannot be reduced by simpler constructions. 3-fold symmetric surfaces appear in nature, for example the surface of the 3-fold symmetric proteins PIEZO1 and PIEZO2 which are an important target of current studies.

Fast Conformal Parameterization of Disks and Sphere Sectors

TL;DR

Addresses fast, conformal-like parameterization of 3-fold rotationally symmetric sphere-type meshes onto plane tiles under a translations tiling constraint. Introduces a novel torus-based branched covering constructed from 63 copies of the mesh, and solves two full-rank linear systems with cotangent weights to obtain a Dirichlet-energy-minimizing embedding onto the plane torus. The method extends to mappings onto equilateral triangles and rectangles via symmetry arguments and Tutte embeddings, with explicit constructions and energy-optimality proofs. Applications include rapid, minimally distorted representations of 3-fold symmetric biomolecule surfaces such as PIEZO1/PIEZO2, enabling ML and structural analysis.

Abstract

We prove a novel method for the embedding of a 3-fold rotationally symmetric sphere-type mesh onto a subset of the plane with 3-fold rotational symmetry. The embedding is free-boundary with the only additional constraint on the image set is that its translations tile the plane, in turn this forces the angles at the embedding of the branch points in the construction. These parameterizations are optimal with respect to the Dirichlet energy functional defined on simplicial complexes. Since the parameterization is over a fixed area domain, it is conformal (i.e. a minimizer of the LSCM energy). The embedding is done by a novel construction of a torus from 63 copies of the original sphere. As a foundation for this result we first prove the optimality of the embedding of disk-type meshes onto special types of triangles in the plane, and rectangles. The embedding of the 3-fold symmetric torus is full rank and so cannot be reduced by simpler constructions. 3-fold symmetric surfaces appear in nature, for example the surface of the 3-fold symmetric proteins PIEZO1 and PIEZO2 which are an important target of current studies.

Paper Structure

This paper contains 9 sections, 2 theorems, 5 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Constructing the torus
  3. Optimal embedding of the torus
  4. Properties of the mapped torus
  5. Mapping to an equilateral triangle
  6. The rectangular case
  7. Complexity of the mappings
  8. Embedding of a 3-fold symmetric sphere
  9. Conclusion

Key Result

Corollary 1

$\Phi (\mathcal{M}_1)$ is precisely one of the octants of the fundamental domain of $\mathbb{R}^2/\widetilde{\Lambda}=\mathbb{R}^2/\mathbb{Z}^2$, and is a right angle isosceles triangle with leg length $\frac{1}{2}$ (see Figure fig:torus1).

Figures (8)

  • Figure 1: Embedding of the torus constructed out of 63 sphere copies (using equal weights for simplicity).
  • Figure 2: 3-fold symmetry in symbolism and in nature.
  • Figure 3: Embedding of the 8 disks construction onto $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$.
  • Figure 4: Gluing instructions for the copies of the disk mesh comprising the torus.
  • Figure 5: Hexagonal tiling of the plane with marked fundamental domain.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Claim 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof