# QUALITY Quality Measures for Pointsets in M dimensions

QUALITY, a C++ code which computes some measures of the quality of dispersion of a set of N points in the unit hypercube in M dimensional space.

The region to be analyzed is defined by a single routine which the user may specify. We will refer to that routine as sample_routine. This routine is simply required to return sample points from the region.

The library includes routines for two simple regions, namely the unit hypercube and the unit hypersphere. If either of these regions is of interest, then the user simply has to pass the appropriate name, either sample_hypercube_uniform or sample_sphere_uniform. For other regions, the user should write the appropriate version of a sampling routine.

The unit hypercube in M dimensional space is simply the set of all vectors whose entries are between 0 and 1. It is a natural abstraction of the unit line segment, the unit square, and the unit cube, and represents a very simple chunk of M dimensional space.

The unit hypersphere in M dimensional space is simply the set of all points whose distance from the origin is 1 or less.

The quality measures computed include:

• Alpha, the minimum angle divided by the maximum possible minimum angle (for triangulated 2D datasets only);
• Area, the ratio of the minimum and maximum triangle areas (for triangulated 2D datasets only);
• Beta, the standard deviation of the minimum spacing between nodes, normalized by the average spacing between nodes;
• Chi, the regularity measure;
• D, the second moment determinant measure;
• E, the Voronoi energy measure;
• Gamma, the mesh ratio;
• H, the point distribution norm;
• Lambda, the COV measure;
• Mu, the point distribution ratio;
• Nu, the cell volume deviation;
• Q, two times the radius of the inscribed circle divided by the radius of the circumscribed circle (for triangulated 2D datasets only);
• R0, the Riesz S=0 energy;
• S, the sphere volume measure (unit hypercube datasets only).
• Tau, the second moment trace measure;

### Languages:

QUALITY is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Programs:

STAR_DISCREPANCY, a C++ program which reads a TABLE file of N points in M dimensions (presumed to lie in the unit hypercube) and computes bounds on the star discrepancy, a measure of dispersion, by Eric Thiemard.

TABLE_QUALITY, a C++ program which analyzes the point spacing quality of data stored in a file.

TET_MESH, a C++ library which includes some quality measures for tetrahedral meshes.

TET_MESH_QUALITY, a C++ program which computes quality measures of a tetrahedral mesh.

TRIANGULATION_QUALITY, a C++ program which computes quality measures of a triangulation.

### Reference:

1. David Field,
Qualitative Measures for Initial Meshes,
International Journal of Numerical Methods in Engineering,
Volume 47, 2000, pages 887-906.
2. DP Hardin, Edward Saff,
Notices of the AMS,
Volume 51, Number 10, November 2004, pages 1186-1194.