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I have a polyhedron which is defined by a series of vertices, which are vectors in R^3, and triangular faces, which are defined by a mapping of the three vertices which define the face.

As an example, here is V and F

V=[-0.8379    0.1526   -0.0429;
   -0.6595   -0.3555    0.0664;
   -0.6066    0.3035    0.2454;
   -0.1323   -0.3591    0.1816;
    0.1148   -0.5169    0.0972;
    0.2875   -0.2619   -0.3980;
    0.2995    0.4483    0.2802;
    0.5233    0.2003   -0.3184;
    0.5382   -0.3219    0.2870;
    0.7498    0.1377    0.1593]

F=[2     3     1;
   7     3     4;
   3     2     4;
   7     9    10;
  10     8     7;
   9     5     6;
   9     8    10;
   1     6     2;
   7     8     1;
   2     6     5;
   8     9     6;
   5     9     4;
   9     7     4;
   4     2     5;
   7     1     3;
   6     1     8]

Euler's formula gives the relationship between face, edges, and vertices

V-E+F = 2

I'm trying to find the unique set of edges of the polyhedron from the vertices. I can already find all edges for each face (3 edges per face and every edge is a member of two adjacent faces) by doing the following

Fa = F(:,1);
Fb = F(:,2);
Fc = F(:,3);


e1=V(Fb,:)-V(Fa,:);
e2=V(Fc,:)-V(Fb,:);
e3=V(Fa,:)-V(Fc,:);

However, this finds all the edges for each face and includes duplicates. An edge, e_i on Face A, is also -e_i on Face B.

Anyone have a good method of finding the unique set of edges (positive and negative directions), or determining a mapping within e1,e2,e3 that links the positive edge to it's negative?

shanksk
  • 115
  • 4

1 Answers1

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It's better to work with the edges encoded in the same way as the faces are encoded: e.g., [3 7] is an edge between the vertices 3 and 7. From this representation, you can get the coordinates of the vector just by subtracting those vertices.

Here is a one-line command to get the unique set of edges from F: 

E = unique(sort([F(:,[1,2]); F(:,[1,3]); F(:,[2,3])], 2), 'rows');

The first argument of sort is a matrix with two columns holding all edges, with duplication. Then it's sorted so that row 7 3 becomes 3 7. Finally, unique returns the unique rows only. Output: 

 1     2
 1     3
 1     6
 1     7
 1     8
 2     3
 2     4
 2     5
 2     6
 3     4
 3     7
 4     5
 4     7
 4     9
 5     6
 5     9
 6     8
 6     9
 7     8
 7     9
 7    10
 8     9
 8    10
 9    10

You can then get coordinate form of the edges with coords = V(E(:,1))-V(E(:,2))

  • Just want to finally thank you for the great suggestion. It's only now, over 1.5 years later that I've come to see the efficiency of your approach. I appreciate the help. – shanksk Nov 28 '17 at 18:42