Group Measure Voronoi
The Voronoi tesselation produces a polygonal representation, plus a report of topological and geometrical properties.

For each atom seed there is a Voronoi polyhedron. However, users may choose (TODO) to represent and analyse just a subset of the total number of polyhedrons. Unselected polyhedrons are used in the tesselation but not in the representation and analysis.

Throughout the report, Minimum is the minimum value, Maximum is the maximum value, Average is the mean value and Deviation is the standard deviation, in the universe formed by the selected polyhedrons only.

### Polyhedrons

Total number of polyhedrons. Polyhedrons with outer faces are considered External, all the others are Internal.

### Faces

Total number of faces. Outer faces are considered External, all the others are Internal.

### Edges

Total number of edges. Edges in outer faces are considered External, all the others are Internal.

### Vertices

Total number of vertices. Vertices in outer faces are considered External, all the others are Internal.

### Polyhedron Faces

Number of faces per polyhedron. The minimum value is 4 (tetrahedron).

### Polyhedron Edges

Number of edges per polyhedron. The minimum value is 6 (tetrahedron).

### Polyhedron Vertices

Number of vertices per polyhedron. The minimum value is 4 (tetrahedron).

### Face Vertices

Number of vertices per face. The minimum value is 3 (triangle).

### Edge polyhedrons

Number of polyhedrons sharing an edge. The minimum value is 1. (system edge).

### Edge faces

Number of faces sharing an edge. The minimum value is 2. (system edge).

### Vertex Polyhedrons

Number of polyhedrons sharing a vertex. The minimum value is 1 (system corner).

### Vertex Faces

Number of faces sharing a vertex. The minimum value is 3 (system corner).

### Vertex Edges

Number of edges sharing a vertex. The minimum value is 3 (system corner).

### System Volume

Volume of the global system (including polyhedrons non-selected for analysis), obtained by: 1) limiting the volume with six outer planes (the Real value); 2) summing the volumes of all the polyhedrons (the Calculated value).

Comparing the two values provides a quantitative assertion of the global geometrical accuracy achieved with the Voronoi tesselation.

### Polyhedron Volumes

The volume for each polyhedron, obtained by summing the volumes of all its triangular piramids.

### Polyhedron Areas

The area for each polyhedron, obtained by summing the areas of all its faces.

### Polyhedron Lengths

The length for each polyhedron, obtained by summing the lengths of all its edges.

### System Anisotropy

For each polyhedron face, determine the following matrix, where n is the unit vector normal to the face:

    q11 = n(1) * n(1) - 1/3
    q12 = n(1) * n(2)
    q13 = n(1) * n(3)
    q21 = n(2) * n(1)
    q22 = n(2) * n(2) - 1/3
    q23 = n(2) * n(3)
    q31 = n(3) * n(1)
    q32 = n(3) * n(2)
    q33 = n(3) * n(3) - 1/3

Then scale the matrix q by A/V**2/3, where A is the face area and V is the polyhedron volume. Finally sum for all polyhedron faces. The q matrix thus obtained describes the polyhedron matrix anisotropy.

To obtain the System Anisotropy, sum these matrices for all polyhedrons, multiplied by the corresponding polyhedron volume fractions. The trace of the resulting matrix must be zero.

The System Anisotropy is independent of the orientation of the system.

### Polyhedron Anisotropy

To obtain a scalar Polyhedron Anisotropy, the following equation is applied:

    q = sqrt { [q11**2 + q12**2 + q13**2 + q21**2 +
    q22**2 + q23**2 + q31**2 + q32**2 + q33**2] / 9 }

For symmetrical solids such as the tetrahedron, the octahedron, the cube, the dodecahedron, or even the Kelvin 8+6 polyhedron, the scalar Polyhedron Anisotropy is always zero.

The scalar Polyhedron Anisotropy is independent of the orientation of the polyhedron.

### Polyhedron Aberration

The ratio between the area of a polyhedron and the area of a sphere with the same volume. As a sphere is the 3D object with a smaller area/volume ratio, Spherical Aberration is always above 1.0.

### Face Areas

The area for each face, obtained by summing the areas of all its triangles. Faces with an area smaller than a given tolerance (the Area tolerance in the Group->Measure dialog) are removed.

### Face Lengths

The perimeter for each face, obtained by summing the lengths of all its edges.

### Face Angles

The angle between adjacent faces (between 0 and 180 degrees).

### Face Aberration

The ratio between the perimeter of a face and the perimeter of a circumpherence with the same area. As a circumpherence is the 2D object with a smaller perimeter/area ratio, Circular Aberration is always above 1.0.

### Edge Lengths

The length for each edge. Edges with a length smaller than a given tolerance (the Vertex tolerance in the Group->Measure dialog) are removed.

### Edge Angles

The angle between adjacent edges (between 0 and 180 degrees).

### Seed Lengths

The distance between atom seeds. The minimum distance must be larger than a given tolerance (GAMGI_MATH_TOLERANCE_LENGTH, currently 1.0E-4), otherwise an error is flagged and the tesselation is canceled.
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