For this definition, it is assumed that
. Let ,
, , , be linearly
independent6.5 points in
. A -simplex,
, is formed from these points as
is the scalar multiplication of by each
of the point coordinates. Another way to view (6.3) is
as the convex hull of the points (i.e., all ways to linearly
interpolate between them). If , a triangular region is obtained.
For , a tetrahedron is produced.
For any -simplex and any such that
. This yields a -dimensional simplex that is
called a face of the original simplex. A 2-simplex has three
faces, each of which is a 1-simplex that may be called an edge. Each
1-simplex (or edge) has two faces, which are 0-simplexes called vertices.
To form a complex, the simplexes must fit together in a nice way.
This yields a high-dimensional notion of a triangulation, which
is a tiling composed of triangular regions. A
simplicial complex, , is a finite set of simplexes that
satisfies the following:
Figure 6.15 illustrates these requirements. For , a
-cell of is defined to be interior,
, of any -simplex. For , every
0-simplex is a 0-cell. The union of all of the cells forms a
partition of the point set covered by . This therefore
provides a cell decomposition in a sense that is consistent with
- Any face of a simplex in is also in .
- The intersection of any two simplexes in is either a
common face of both of them or the intersection is empty.
To become a simplicial complex, the simplex
faces must fit together nicely.
Steven M LaValle