查找四面体网格边的算法

Question

I have a few thousands of tetrahedra meshes that is generated for modelling. I have the vertices and surfaces information which are indexed into two matrices which some are related. The nodes of surfaces are linked to the vertices' matrix. I want to find which vertices are connected to each other to form a tetrahedra. Is there an algorithm to detect the connectivity of the vertices that formed edges of a tetrahedra?

Answer 1

If your node matrix is an adjacency matrix, ie nearest neighbors, then choosing any vertex, the other three remaining vertices of some tetrahedron it forms must satisfy:

(i) They are immediate neighbors of the chosen vertex,

(ii) They are are immediate neighbors of each other . If this is not true then you'll end up with tetrahedra crossing each other.

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If I misunderstood your statement, ie it's not an adjacency matrix, then you'll have to do the 3D-version of Delaunay Triangulation , ie calculating the 3D Voronoi cell graph for your point cloud. https://en.wikipedia.org/wiki/Voronoi_diagram

Answer 2

Starting from a list of possibly duplicated edges:

Suppose you got an ASCII file with possibly duplicate edges. On a Unix system, here is a simple way of extracting from it the list of unique edges:

cat edges.txt | awk '{printf "%d %d\n",  ($1 < $2 ? $1 : $2),  ($1 > $2 ? $1 : $2)}' | sort | uniq > unique_edges.txt

What it does:

1. The 'awk' command ensures that in each edge, the vertex of lower id comes first (thus, there will be a single way of representing the same edge in the file)
2. the 'sort' command sorts all the lines. This will ensure that all duplicated edges are together
3. the 'uniq' command suppresses the duplicated lines

If you are under Windows, you can install cygwin (to have awk,sort and uniq). It is also quite easy to do something similar in other scripting languages (eg perl).

Starting from a list of tetrahedra:

Now if you start with a file with the tetrahedra, you can generate a (duplicated) list of edges as follows (then you can run the command above to remove duplicates):

cat tetrahedra.txt | awk '{printf "%d %d\n%d %d\n%d %d\n%d %d\n%d %d\n%d %d\n", $1, $2, $1, $3, $1, $4, $2, $3, $2, $4, $3, $4}' > edges.txt

What it does: it generates for each tetrahedron its 6 edges, one edge per line.

Clearly, you can combine both commands, to directly extract a unique list of edges from the tetrahedra file.