Estimation of Tetrahedron Degeneration in a Tetrahedral Partition of Three-Dimensional Space

Based on the geometric characteristics of a tetrahedron, quantitative estimates of its degeneracy are proposed and their relationship with the condition number of local bases generated by the edges emerging from a single 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, we propose calculating the empirical distribution function of the degeneracy index on its tetrahedral elements. An irregular model triangulation (tetrahedralization or tetrahedral partition) of three-dimensional space is proposed, depending on a control parameter that determines the quality of its elements. The coordinates of the tetrahedra vertices of the model triangulation tetrahedra are the sums of the corresponding coordinates of the nodes of some given regular mesh and random increments to them. For various values of the control parameter, the empirical distribution function of the tetrahedron degeneracy index is calculated, which is considered as a quantitative characteristic of the quality of tetrahedra in the triangulation of a three-dimensional region.