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@inra.fr
0027 % Created: 2015-08-19,    using Matlab 8.5.0.197613 (R2015a)
0028 % Copyright 2015 INRA - Cepia Software Platform.
0029 
0030 
0031 %% Check input validity
0032 
0033 if size(faces, 2) ~= 3
0034     error('meshes3d:trimeshMeanBreadth:NonTriangularMesh', ...
0035         'Requires a triangular mesh as input');
0036 end
0037     
0038 %% Compute edge and edgeFaces arrays
0039 % Uses the same code as in trimeshEdgeFaces
0040 
0041 % compute vertex indices of each edge (in increasing index order)
0042 edges = sort([faces(:,[1 2]) ; faces(:,[2 3]) ; faces(:,[3 1])], 2);
0043 
0044 % create an array to keep indices of faces "creating" each edge
0045 nFaces = size(faces, 1);
0046 edgeFaceInds = repmat( (1:nFaces)', 3, 1);
0047 
0048 % sort edges, keeping indices
0049 [edges, ia, ib] = unique(edges, 'rows'); %#ok<ASGLU>
0050 nEdges = size(edges, 1);
0051 
0052 % allocate memory for result
0053 edgeFaces = zeros(nEdges, 2);
0054 
0055 % iterate over edges, to identify incident faces
0056 for iEdge = 1:nEdges
0057     inds = find(ib == iEdge);
0058     edgeFaces(iEdge, 1:length(inds)) = edgeFaceInds(inds);
0059 end
0060 
0061 
0062 %% Compute dihedral angle for each edge
0063 
0064 % compute normal of each face
0065 normals = meshFaceNormals(vertices, faces);
0066 
0067 % allocate memory for resulting angles
0068 alpha = zeros(nEdges, 1);
0069 
0070 % iterate over edges
0071 for iEdge = 1:nEdges
0072     % indices of adjacent faces
0073     indFace1 = edgeFaces(iEdge, 1);
0074     indFace2 = edgeFaces(iEdge, 2);
0075     
0076     % normal vector of adjacent faces
0077     normal1 = normals(indFace1, :);
0078     normal2 = normals(indFace2, :);
0079     
0080     % compute dihedral angle of two vectors
0081     alpha(iEdge) = vectorAngle3d(normal1, normal2);
0082 end
0083 
0084 
0085 %% Compute mean breadth
0086 % integrate the dihedral angles weighted by the length of each edge
0087 
0088 % compute length of each edge
0089 lengths = meshEdgeLength(vertices, edges);
0090 
0091 % compute product of length by angles
0092 mb = sum(alpha .* lengths) / (4*pi);