On The Topology of Polygonal Meshes

TL;DR

This paper provides an informal overview of polygonal mesh topology, covering fundamental concepts, formulas, and methods to manipulate meshes for topological simplification.

Contribution

It offers a clear, accessible exposition on mesh topology, including proofs of key formulas and practical approaches to modify mesh topology.

Findings

Derived Euler and Euler-Poincaré formulas for meshes

Defined Betti numbers using mesh statistics

Outlined methods to cut meshes into topological discs

Abstract

This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the space in which the mesh is immersed. A distinction is also made between quantitative topological properties and qualitative properties. Next, we outline proofs of the Euler and the Euler-Poincar\'e formulas. The Betti numbers are then defined in terms of the Euler-Poincar\'e formula and other mesh statistics rather than as cardinalities of the homology groups which allows us to avoid abstract algebra. Finally, we discuss how it is possible to cut a polygonal mesh such that it becomes a topological disc.