Degeneration estimation of a tetrahedral in a tetrahedral partition of the three-dimensional space

Based on the geometric characteristics of the tetrahedron, quantitative estimates of its degeneracy are proposed and their relationship with the condition number of local bases generated by the edges emerging from the same vertex is established. The concept of the tetrahedron degeneracy index is introduced in several versions and their practical equivalence to each other is established. To assess the quality of a particular tetrahedral partition, it is proposed to calculate the empirical distribution function of the degeneracy index on its tetrahedral elements. A model irregular triangulation (tetrahedralization or tetrahedral partition) of three-dimensional space is proposed, depending on the control parameter that determines the quality of its elements. The coordinates of the tetrahedra vertices of the model triangulation tetrahedrons are the sums of the corresponding coordinates of the nodes of some given regular grid and random increments to them. For various values of the control parameter, the empirical distribution function of the tetrahedron degeneration index is calculated, which is considered as a quantitative characteristic of the quality of tetrahedra in the triangulation of a three-dimensional region.