The plan is to briefly describe the backend fmesher C++ library.
Introduction
The fmesher C++ library was initially written by Finn
Lindgren in April/May 2010, as an implementation of the data structures
and refined constrained Delaunay triangulation (RCDT) methods from Hjelle and Dæhlen (2006), extended to RCDTs on
spheres.
Data structures
The key to the algorithm speed is the use of traversal operators for the mesh graph, made efficient by support in the data structure for key operations.
Triangle centric data model
The mesh storage is composed of a 3-column matrix of vertex
coordinates, S, and three triangle graph topology 3-column
integer matrices:
-
TV: 3 vertex indices, in CCW order -
TT: For each of the 3 corners, the index of the opposing edge neighbouring triangle, if any, otherwise -1. -
TTi: For each of the 3 corners, the in-triangle index of the opposing edge’s neighbouring triangle’s opposing in-triangle index. This structure can be turned on/off to allow certain operations to be more efficient.
This is similar to a winged-edge or half-edge structure, but treats triangles as primary objects, allowing compact storage as well as efficient graph traversal.
Note: A further structure, VT, can be enabled, that for
each vertex holds the set of triangle indices that it is used by. Before
2025, only a single triangle index was stored, so that there was no
cheap way to access all triangles that connect to a given
vertex, unless the mesh was a proper edge-connected manifold. For
example, some operation would not handle meshes where the forward and
inverse orbit0 operations could not reach all the connected
triangles.
Graph traversal algebra
-
Dartlocation objects; originator vertex, edge direction, and triangle - Each
alphaoperator alters only one simplex quantity:-
alpha0: swap the originating vertex for the edge, keep the triangle -
alpha1: swap to the edge pointing to the remaining 3rd vertex, keep the originator vertex and triangle -
alpha2: swap to the adjacent triangle, stay along the same edge and keep the originator vertex
-
- Each
orbitkeeps one simplex quantity fixed while altering the other two, keeping orientation intact; If one starts with aDartpointing in CCW direction in a triangle, each successful orbit operation will keep that property.-
orbit0 = alpha1,alpha2: move to the next CCW adjacent triangle, except on boundary -
orbit1 = alpha2,alpha0: move to the adjacent triangle and swap edge direction, except on boundary -
orbit2 = alpha0,alpha1: move to the next CCW vertex in the triangle
-
These operators, and their inverses, allow arbitrary graph traversal of 2-manifold meshes with triangles connected via edges.