On The Topology of Polygonal Meshes

TL;DR

This paper presents an accessible treatment of polygonal-mesh topology tailored for visual computing. It distinguishes intrinsic versus extrinsic topology, and derives key quantitative invariants (s, g, b) and Betti numbers from the Euler's formula without invoking full homology. It outlines practical procedures for identifying cycles, verifying manifoldness, computing the Euler characteristic, and constructing a cut graph to flatten genusier meshes. The work emphasizes algorithmic approaches and provides connections to algebraic topology while remaining computationally tangible.

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 characterize topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the embedding space. A distinction is also made between quantitative topological properties and qualitative properties. We outline a proof of the Euler and the Euler-Poincaré formulas. The Betti numbers are defined in terms of the Euler-Poincaré 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.

Figures (13)

• Figure 1: The Fertility model is shown on the left. It may not be immediately obvious how many holes (or handles) it possesses, but on the right we see a sphere with four handles which is topologically equivalent, i.e. there is a homeomorphism between these two shapes, and on the right it is obvious that the model has four handles.
• Figure 2: Top left: A homeomorphism, $f: A \rightarrow B$, is a continuous, bijective map which has a continuous inverse. Middle bottom: A homotopy is a continuous deformation. In this example an annulus is homotopic to a ring since the former can be continuously shrunk into the latter. Top right: The red curves are homologous because they differ only by the boundary of a region. However, the red curves are not homologous to the green or blue curves. The blue curve, on the other hand, is homologous to zero because it is the boundary of a region.
• Figure 3: A polygonal mesh, $\mathcal{M} = \langle \mathcal{V}, \mathcal{E}, \mathcal{F} \rangle$, where $\mathcal{V} = \{v_0, v_1, v_2, v_3, v_4, v_5\}$, $\mathcal{E}=\{e_0, e_1, e_2, e_3, e_4, e_5\}$, and $\mathcal{F} = \{f_0, f_1\}$. $V=5$, $E=6$. and $F=2$. The boundaries of the two faces are $\partial f_0=\{e_0,e_1,e_2,e_3\}$ and $\partial f_1=\{e_1,e_4,e_5\}$. The edge boundaries are $\partial e_0=\{v_0,v_1\}$, $\partial e_1=\{v_1,v_2\}$, $\partial e_2=\{v_2,v_3\}$, $\partial e_3=\{v_0,v_3\}$, $\partial e_4=\{v_1,v_4\}$, and $\partial e_5=\{v_4,v_2\}$.
• Figure 4: Left: two sets of edges where the left one is not a cycle, but the right one is a cycle. The cycle instigator is shown as a red line segment. Right: a cube which is not a cycle since it is missing a face is shown on the left. On the right a closed cube which constitutes a cycle. Both of the shown cycles are also simple.
• Figure 5: Two non-manifold configurations: on the left the two tetrahedra share a vertex, and on the right, two boxes share an edge. From the images we cannot tell whether the meshes are intrinsically or extrinsically non-manifold.