Size analysis of nearly regular Delaunay triangulations
Consider a two-dimensional spatial point pattern in which the points are nearly regular. A pattern is nearly regular if the Delaunay triangles formed by points of the pattern are nearly equilateral, and of roughly the same size. Consider \(k\) distinct, labeled points in the complex plane \(\mathbb C\) denoted by \(z = \{z_1,\dots z_k\}^T\). Typically, these \(k\) points to be a subset of \(n\geq k\) points in \(\mathbb C\) are considered. The centroid size is defined by \(S(z) = \{\sum\|z_i -\overline{z}\|^2\}^2\), where \(\overline{z}\) is the center of gravity. The size \(S\) of a triangle with edge lengths \(a\), \(b\) and \(c\) is determined by \(3 S^2 = a^2 + b^2 + c^2\), and it can be seen that the average squared size of a triangle is closely related to the average of three neighboring squared distances (in the case of a nearly regular pattern). The authors propose to model such patterns with Gaussian perturbations about a regular mean configuration.
scientific article; zbMATH DE number 1367827