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 :
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1-Simplex Mesh
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2-Simplex Mesh
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3-Simplex Mesh
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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.
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Duality of a 1-Simplex Mesh
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Duality of a 2-Simplex Mesh
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Duality of a 3-Simplex Mesh
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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
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Hervé Delingette
International Journal of Computer Vision, 32(2):111-146, September 1999
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