Computing Conformal Structure of Surfaces

TL;DR

This paper presents a comprehensive method for computing conformal structures of 2-manifolds represented as triangle meshes, enabling applications in geometry classification, surface parametrization, and computer graphics.

Contribution

It introduces a novel pipeline for deriving conformal structures from mesh topology to harmonic forms and period matrices, facilitating surface analysis and mapping.

Findings

Successfully computes conformal structures for complex meshes.

Enables accurate global conformal mappings between surfaces and planes.

Provides tools for geometry classification and surface parametrization.

Abstract

This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures, then construct dual cohomology bases and diffuse them to harmonic 1-forms. Next, we construct bases of holomorphic differentials. We then obtain period matrices by integrating holomorphic differentials along homology bases. We also study the global conformal mapping between genus zero surfaces and spheres, and between general meshes and planes. Our method of computing conformal structures can be applied to tackle fundamental problems in computer aid design and computer graphics, such as geometry classification and identification, and surface global parametrization.