Polygon Meshes

A graph bases 3D Represenation . Nodes of the graph are called vertices which are connected by edges and form faces. They offer a piecewise linear surface representation. They can easily be textured via texture mapping.

Definition #

1. A polygon mesh is a finite set of closed, simple polygons .

   

2. Intersection of two polygons in is either empty, a vertex, or an edge.

3. Every edge belongs to at least one polygon

Degree of a Vertex #

The degree or valence of a vertex is equalt to the number of edges that connect to it.

Boundary of a Polygon #

The boundary of a mesh is the set of all edges that belong only to only one face.

• Either empty or forms closed loops
• If empty, then the polygon mesh is closed

Properties #

• Can represent arbitrary topologies
• Easy to manipulate
• Efficient to render
• The number of vertices, edged and faces correspond to Eulers Formula (Graphs)

Triangle meshes #

A polygon mesh where every polygon is a triangle. Representation and rendering are simplified. Each face is planar and convex. Any polygon can be triangulated, so a triangle mesh can always be constructed from a poygon mesh.

Possible implementations #

• Triangle List
• Indexed Face Set
• Face-Based Connectivity
• Winged Edge Data Structure

Conversion #

Can be converted to a Point Cloud :

Option 1

1. For each face calculate the area
2. normalize to have the the sum of all alreas as 1
3. use this as probability density to sample the corresponding face
4. genreate radom Barycentric Coordinates coordinates via:
   • 2 random variables
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   • 
   • 

Option 2

“Farthest Point Sampling” – Start with initial point and then iteratively find the new farest-away point. Notion of distance can vary. Distance on the Mesh: path along edges. >>I guess you just use the original vertices as “points”<<