# Simplex Mesh

## Topological Properties

I have introduced an original surface representation called Simplex Meshes that have the following properties:

• Each vertex is adjacent to a fixed number of neighboring vertices : 2 for a contour (1-simplex mesh), 3 for a surface (2-simplex mesh)
• The topology of a simplex mesh is dual of that of a triangulation

Below are some examples of Simplex Meshes :

1-Simplex Mesh
2-Simplex Mesh
3-Simplex Mesh

The duality between k-simplex meshes and k-triangulations is shown below: dashed lines correspond to k-simplex meshes and solid lines to k-triangulations . Thus a vertex of a 2-simplex mesh is associated by duality to a triangle and an edge of a 2-simplex mesh to a edge of the triangulation. Note that a 1-simplex mesh and 1-triangulation are basically the same thing : a polygonal line.

Duality of a 1-Simplex Mesh
Duality of a 2-Simplex Mesh
Duality of a 3-Simplex Mesh

## Geometrical Properties

We consider here only 2-simplex meshes representing three-dimensional surfaces, but the notion can be extended to simplex meshes of higher dimensions.

Each vertex of a 2-simplex mesh is surrounded by three neighbors. On the figure below, vertex P is adjacent to vertices A, B and C. In a tridimensional space, one can fit a sphere through 4 points. Therefore we can use the osculating sphere to define:

• A normal vector n at P as the normal at the osculating sphere.
• A mean curvature H as the curvature of the osculating sphere.

Those properties are especially useful when characterizing (shape recognition) and smoothing (shape reconstruction) tridimensional shapes.

## Selected Publication

 Hervé Delingette International Journal of Computer Vision, 32(2):111-146, September 1999 Summarizes topological and geometrical properties of simplex meshes Defines also contours embedded on simplex meshes. Largely describes use of simplex meshes for shape reconstruction The topological definition of simplex meshes presented in this paper is a bit outdated though