meshVolume

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, meshCentroid

0001 function vol = meshVolume(varargin)
0002 %MESHVOLUME (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, meshCentroid
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 2012-2024 INRA - Cepia Software Platform
0030 
0031 % parse input
0032 [vertices, faces] = parseMeshData(varargin{:});
0033 
0034 % ensure mesh has triangle faces
0035 faces = triangulateFaces(faces);
0036 
0037 % initialize an array of volume
0038 nFaces = size(faces, 1);
0039 vols = zeros(nFaces, 1);
0040 
0041 % Shift all vertices to the mesh centroid
0042 vertices = bsxfun(@minus, vertices, mean(vertices,1));
0043 
0044 % compute volume of each tetraedron
0045 for i = 1:nFaces
0046     % consider the tetrahedron formed by face and mesh centroid
0047     tetra = vertices(faces(i, :), :);
0048     
0049     % volume of current tetrahedron
0050     vols(i) = det(tetra) / 6;
0051 end
0052 
0053 vol = sum(vols);