meshVolume

 (Signed) volume of the space enclosed by a polygonal mesh.

   V = meshVolume(VERTS, FACES)
   Computes the volume of the space enclosed by the polygonal mesh
   represented by vertices VERTS (as a Nv-by-3 array of cooridnates) and
   the array of faces FACES (either as a Nf-by-3 array of vertex indices,
   or as a cell array of arrays of vertex indices).

   The volume is computed as the sum of the signed volumes of tetrahedra
   formed by triangular faces and the centroid of the mesh. Faces need to
   be oriented such that normal points outwards the mesh. See:
   http://stackoverflow.com/questions/1838401/general-formula-to-calculate-polyhedron-volume

   Example
     % computes the volume of a unit cube (should be equal to 1...)
     [v f] = createCube;
     meshVolume(v, f)
     ans = 
         1

   See also
   meshes3d, meshSurfaceArea, tetrahedronVolume, meshComplement

0001 function vol = meshVolume(varargin)
0002 % (Signed) volume of the space enclosed by a polygonal mesh.
0003 %
0004 %   V = meshVolume(VERTS, FACES)
0005 %   Computes the volume of the space enclosed by the polygonal mesh
0006 %   represented by vertices VERTS (as a Nv-by-3 array of cooridnates) and
0007 %   the array of faces FACES (either as a Nf-by-3 array of vertex indices,
0008 %   or as a cell array of arrays of vertex indices).
0009 %
0010 %   The volume is computed as the sum of the signed volumes of tetrahedra
0011 %   formed by triangular faces and the centroid of the mesh. Faces need to
0012 %   be oriented such that normal points outwards the mesh. See:
0013 %   http://stackoverflow.com/questions/1838401/general-formula-to-calculate-polyhedron-volume
0014 %
0015 %   Example
0016 %     % computes the volume of a unit cube (should be equal to 1...)
0017 %     [v f] = createCube;
0018 %     meshVolume(v, f)
0019 %     ans =
0020 %         1
0021 %
0022 %   See also
0023 %   meshes3d, meshSurfaceArea, tetrahedronVolume, meshComplement
0024 
0025 % ------
0026 % Author: David Legland
0027 % e-mail: david.legland@inrae.fr
0028 % Created: 2012-10-01,    using Matlab 7.9.0.529 (R2009b)
0029 % Copyright 2012 INRA - Cepia Software Platform.
0030 
0031 % HISTORY
0032 % 2013-08-16 speed improvement by Sven Holcombe
0033 
0034 % parse input
0035 [vertices, faces] = parseMeshData(varargin{:});
0036 
0037 % ensure mesh has triangle faces
0038 faces = triangulateFaces(faces);
0039 
0040 % initialize an array of volume
0041 nFaces = size(faces, 1);
0042 vols = zeros(nFaces, 1);
0043 
0044 % Shift all vertices to the mesh centroid
0045 vertices = bsxfun(@minus, vertices, mean(vertices,1));
0046 
0047 % compute volume of each tetraedron
0048 for i = 1:nFaces
0049     % consider the tetrahedron formed by face and mesh centroid
0050     tetra = vertices(faces(i, :), :);
0051     
0052     % volume of current tetrahedron
0053     vols(i) = det(tetra) / 6;
0054 end
0055 
0056 vol = sum(vols);