trimeshMeanBreadth

TRIMESHMEANBREADTH Mean breadth of a triangular mesh.

   MB = trimeshMeanBreadth(VERTICES, FACES)
   Computes the mean breadth (proporitonal to the integral of mean
   curvature) of a triangular mesh.

   Example
     [V, F] = createCube;
     F2 = triangulateFaces(F);
     MB = trimeshMeanBreadth(V, F2)
     MB = 
         1.5000

   See also
   meshes3d, trimeshSurfaceArea, trimeshEdgeFaces, polyhedronMeanBreadth

   References
   Stoyan D., Kendall W.S., Mecke J. (1995) "Stochastic Geometry and its
       Applications", John Wiley and Sons, p. 26
   Ohser, J., Muescklich, F. (2000) "Statistical Analysis of
       Microstructures in Materials Sciences", John Wiley and Sons, p.352

0001 function mb = trimeshMeanBreadth(vertices, faces)
0002 %TRIMESHMEANBREADTH Mean breadth of a triangular mesh.
0003 %
0004 %   MB = trimeshMeanBreadth(VERTICES, FACES)
0005 %   Computes the mean breadth (proporitonal to the integral of mean
0006 %   curvature) of a triangular mesh.
0007 %
0008 %   Example
0009 %     [V, F] = createCube;
0010 %     F2 = triangulateFaces(F);
0011 %     MB = trimeshMeanBreadth(V, F2)
0012 %     MB =
0013 %         1.5000
0014 %
0015 %   See also
0016 %   meshes3d, trimeshSurfaceArea, trimeshEdgeFaces, polyhedronMeanBreadth
0017 %
0018 %   References
0019 %   Stoyan D., Kendall W.S., Mecke J. (1995) "Stochastic Geometry and its
0020 %       Applications", John Wiley and Sons, p. 26
0021 %   Ohser, J., Muescklich, F. (2000) "Statistical Analysis of
0022 %       Microstructures in Materials Sciences", John Wiley and Sons, p.352
0023 
0024 % ------
0025 % Author: David Legland
0026 % E-mail: david.legland@inrae.fr
0027 % Created: 2015-08-19, using Matlab 8.5.0.197613 (R2015a)
0028 % Copyright 2015-2024 INRA - Cepia Software Platform
0029 
0030 %% Check input validity
0031 
0032 if size(faces, 2) ~= 3
0033     error('meshes3d:trimeshMeanBreadth:NonTriangularMesh', ...
0034         'Requires a triangular mesh as input');
0035 end
0036     
0037 %% Compute edge and edgeFaces arrays
0038 % Uses the same code as in trimeshEdgeFaces
0039 
0040 % compute vertex indices of each edge (in increasing index order)
0041 edges = sort([faces(:,[1 2]) ; faces(:,[2 3]) ; faces(:,[3 1])], 2);
0042 
0043 % create an array to keep indices of faces "creating" each edge
0044 nFaces = size(faces, 1);
0045 edgeFaceInds = repmat( (1:nFaces)', 3, 1);
0046 
0047 % sort edges, keeping indices
0048 [edges, ia, ib] = unique(edges, 'rows'); %#ok<ASGLU>
0049 nEdges = size(edges, 1);
0050 
0051 % allocate memory for result
0052 edgeFaces = zeros(nEdges, 2);
0053 
0054 % iterate over edges, to identify incident faces
0055 for iEdge = 1:nEdges
0056     inds = find(ib == iEdge);
0057     edgeFaces(iEdge, 1:length(inds)) = edgeFaceInds(inds);
0058 end
0059 
0060 
0061 %% Compute dihedral angle for each edge
0062 
0063 % compute normal of each face
0064 normals = meshFaceNormals(vertices, faces);
0065 
0066 % allocate memory for resulting angles
0067 alpha = zeros(nEdges, 1);
0068 
0069 % iterate over edges
0070 for iEdge = 1:nEdges
0071     % indices of adjacent faces
0072     indFace1 = edgeFaces(iEdge, 1);
0073     indFace2 = edgeFaces(iEdge, 2);
0074     
0075     % normal vector of adjacent faces
0076     normal1 = normals(indFace1, :);
0077     normal2 = normals(indFace2, :);
0078     
0079     % compute dihedral angle of two vectors
0080     alpha(iEdge) = vectorAngle3d(normal1, normal2);
0081 end
0082 
0083 
0084 %% Compute mean breadth
0085 % integrate the dihedral angles weighted by the length of each edge
0086 
0087 % compute length of each edge
0088 lengths = meshEdgeLength(vertices, edges);
0089 
0090 % compute product of length by angles
0091 mb = sum(alpha .* lengths) / (4*pi);