A triangle mesh is a two-dimensional mesh whose maximal cells are triangles. It is
commonly used to represent a piecewise-linear
surface, but a triangle mesh need not be a manifold.
It can be described combinatorially by a triple 
, where 
is a finite set of vertices with
geometric coordinates, 
is a set of pairs of vertices called edges, and 
is a set of triples of vertices called
faces. The set 
is the union of the three vertex pairs determined by every
face in 
(Botsch et al. 2010).
The 1-skeleton of the triangle mesh is the graph 
. In general this graph does not
determine 
,
since a 3-cycle of edges need not bound a face.
Under the additional condition that every 3-cycle is filled, the faces
can be recovered as
 |
Thus 
is the set of 2-simplices of the clique
complex of the 1-skeleton.
In a manifold triangle mesh, an edge belonging to exactly one triangle is a boundary edge, while an interior edge belongs to exactly two triangles. An edge belonging to more than two triangles is nonmanifold. A vertex at which the incident faces form disconnected fans is also nonmanifold. If the represented surface is an orientable surface, the mesh can be oriented by consistently ordering the three vertices of every face. Reversing this order reverses the corresponding normal vector.
