The Voronoi tesselation produces a polygonal representation,
plus a report of topological and geometrical properties.
For each atom there is a Voronoi polyhedron.
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 all the polyhedrons available.
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
Cell->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
Cell->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.