surfacetopology 0.2.0

Triangulated compact manifolds

The connectivity of a triangulated surface composed of $n_v$ vertices and $n_t$ triangles can be specified by a 3-by-$n_t$ matrix of vertex indices ranging from 0 to $n_v-1$. Each row contains the index of the three vertices of a triangle. This representation is less general than a simplicial complex, but is more convenient and widespread for large meshes. Note that the vertex positions are not needed since only topological properties are considered.

This triangulation describes a compact manifold surface if and only if:

1. every triangle has 3 distinct vertices;
2. every edge is adjacent to 1 or 2 triangles;
3. the set of triangles adjacent to any vertex forms a closed or open fan.

A fan is a collection of triangles sharing a common vertex and chained together by a sequence of shared edges.

Objective

The purpose of this package is to check if a triangulation satisfies the above conditions and to calculate the basic topological properties such as connectedness, orientability, genus and boundaries that are used for the classification of compact 2-manifolds.

Examples

The function surface_topology separates the connected components and provides for each one some information about the mesh. The attribute manifold indicates if the conditions are satisfied. The attribute genus is calculated from the Euler characteristic (euler).

>>> import surfacetopology as topo
>>> S1 = topo.small_mesh('tetrahedron')
>>> S1
[[0 1 2]
 [0 2 3]
 [1 0 3]
 [2 1 3]]
>>> topo.surface_topology(S1)
[SurfaceTopology(n_vertices=4, n_edges=6, n_triangles=4, n_boundaries=0,
euler=2, genus=0, manifold=True, oriented=True, closed=True)]

If a surface has multiple connected components (here artificially generated using disjoint_sum), a list of SurfaceTopology objects are returned. When there are isolated vertices, a large number of connected components of size 1 may be generated. In that case, it is useful to call the function renumber_vertices first.

>>> S2 = topo.small_mesh('torus')
>>> S = topo.disjoint_sum(S1, S2, 10+S1)
>>> len(topo.surface_topology(S))
13
>>> S = topo.renumber_vertices(S)
>>> len(topo.surface_topology(S))
3

If a surface is not oriented, the orientability will be checked by attempting to fix the orientation (stored in the attribute triangles if the surface turns out to be orientable). Here, the Moebius strip is not orientable and has genus 1.

>>> S3 = topo.small_mesh('moebius')
>>> topo.surface_topology(S3)
[SurfaceTopology(n_vertices=6, n_edges=12, n_triangles=6, n_boundaries=1,
euler=0, genus=1, manifold=True, oriented=False, orientable=False, closed=False)]

Surfaces with higher genus can be generated using connected_sum

>>> S = topo.connected_sum(S2, S2, S2)
>>> topo.surface_topology(S)
[SurfaceTopology(n_vertices=15, n_edges=57, n_triangles=38, n_boundaries=0,
euler=-4, genus=3, manifold=True, oriented=True, closed=True)]

In case manifold is false, the SurfaceTopology object contains attributes for trouble-shooting (nonmanifold_vertices, nonmanifold_edges, collapsed_triangles).

Implementation

The code is implemented in C++, interfaced and compiled using cython, and with a wrapper for python (surfacetopology/wrapper.py). It was designed for meshes with 100k vertices, but should be appropriate for 1M vertices.

Installation

Can be installed using the command pip install surfacetopology (on Windows, a compiler such as Microsoft Visual C++ is required).

Tested using Anaconda 2023.09 (python 3.11) on Linux and Windows.