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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function mb = trimeshMeanBreadth(vertices, faces)
%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
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2015-08-19, using Matlab 8.5.0.197613 (R2015a)
% Copyright 2015-2023 INRA - Cepia Software Platform
%% Check input validity
if size(faces, 2) ~= 3
error('meshes3d:trimeshMeanBreadth:NonTriangularMesh', ...
'Requires a triangular mesh as input');
end
%% Compute edge and edgeFaces arrays
% Uses the same code as in trimeshEdgeFaces
% compute vertex indices of each edge (in increasing index order)
edges = sort([faces(:,[1 2]) ; faces(:,[2 3]) ; faces(:,[3 1])], 2);
% create an array to keep indices of faces "creating" each edge
nFaces = size(faces, 1);
edgeFaceInds = repmat( (1:nFaces)', 3, 1);
% sort edges, keeping indices
[edges, ia, ib] = unique(edges, 'rows'); %#ok<ASGLU>
nEdges = size(edges, 1);
% allocate memory for result
edgeFaces = zeros(nEdges, 2);
% iterate over edges, to identify incident faces
for iEdge = 1:nEdges
inds = find(ib == iEdge);
edgeFaces(iEdge, 1:length(inds)) = edgeFaceInds(inds);
end
%% Compute dihedral angle for each edge
% compute normal of each face
normals = meshFaceNormals(vertices, faces);
% allocate memory for resulting angles
alpha = zeros(nEdges, 1);
% iterate over edges
for iEdge = 1:nEdges
% indices of adjacent faces
indFace1 = edgeFaces(iEdge, 1);
indFace2 = edgeFaces(iEdge, 2);
% normal vector of adjacent faces
normal1 = normals(indFace1, :);
normal2 = normals(indFace2, :);
% compute dihedral angle of two vectors
alpha(iEdge) = vectorAngle3d(normal1, normal2);
end
%% Compute mean breadth
% integrate the dihedral angles weighted by the length of each edge
% compute length of each edge
lengths = meshEdgeLength(vertices, edges);
% compute product of length by angles
mb = sum(alpha .* lengths) / (4*pi);
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