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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 breadth = polyhedronMeanBreadth(vertices, edges, faces)
%POLYHEDRONMEANBREADTH Mean breadth of a convex polyhedron.
%
% BREADTH = polyhedronMeanBreadth(V, E, F)
% Return the mean breadth (average of polyhedron caliper diameter over
% all direction) of a convex polyhedron.
%
% The mean breadth is computed using the sum, over the edges of the
% polyhedron, of the edge dihedral angles multiplied by the edge length,
% the final sum being divided by (4*PI).
%
% Note: the function assumes that the faces are correctly oriented. The
% face vertices should be indexed counter-clockwise when considering the
% supporting plane of the plane, with the outer normal oriented outwards
% of the polyhedron.
%
% Typical values for classical polyhedra are:
% cube side a breadth = (3/2)*a
% cuboid sides a, b, c breadth = (a+b+c)/2
% tetrahedron side a breadth = 0.9123*a
% octaedron side a beradth = 1.175*a
% dodecahedron, side a breadth = 15*arctan(2)*a/(2*pi)
% icosaehdron, side a breadth = 15*arcsin(2/3)*a/(2*pi)
%
% Example
% [v e f] = createCube;
% polyhedronMeanBreadth(v, e, f)
% ans =
% 1.5
%
% See also
% meshes3d, meshEdgeFaces, meshDihedralAngles, checkMeshAdjacentFaces
% trimeshMeanBreadth
%
% 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: 2010-10-04, using Matlab 7.9.0.529 (R2009b)
% Copyright 2010-2023 INRA - Cepia Software Platform
% compute dihedral angle of each edge
alpha = meshDihedralAngles(vertices, edges, faces);
% compute length of each edge
lengths = meshEdgeLength(vertices, edges);
% compute product of length by angles
breadth = sum(alpha.*lengths)/(4*pi);
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