Metrization in space families of points in $\mathbb{R}^n$ and adjoining questions

A. Yu. Igumnov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 40-54

Résumé

In the work we introduced the concept of a family of points in $\mathbb{R}^n$ and metrization of space of points families. Under the family understood the points numbered set of points in $\mathbb{R}^n$. In the interpretation of points in space as the nodes of a grid (lattice) introduced in this paper, the concept of distance can be used as a kind of measure of the differences between the test grid of reference or in some critical sense. Moreover, this measure of the difference can be determined through measure differences corresponding to the grid elements—for example, in the case of tetrahedral mesh—for its individual tetrahedrons adjacent, for couples tetrahedra. A family of $k$ points ($k$-point family) is a function $F:\{1,\ldots,k\} \to \mathbb{R}^n$. We define the distance $\rho(F,G)$ between the families $F$ and $G$ as the logarithm of some expression that contains the Euclidean distance $|F(i)F(j)|$, $|G(i)G(j)|$. Distance $\rho$ is invariant relatively orthogonal mapping: $\rho(O\circ F,G)=\rho(F,G)$ for any orthogonal mapping $O:\mathbb{R}^n\to \mathbb{R}^n$. We give an estimate of the distance that moves the family $F$ under the action of quasi-isometric mapping $f$: $\displaystyle\rho(F,f \circ F) \leq \log \frac{L}{l}$, where $l$ is minimum distortion mapping $f$, $L$ is maximum distortion mapping $f$. Next, we prove the following sufficient sign of preservation any properties of families of points at quasi-isometric mapping: Тheorem 2. Let $F$ is $k$-point family in $\mathbb{R}^n$, $f:F(I)\to \mathbb{R}^n$ is quasi-isometric mapping; ${\mathcal Z}$—a set of $k$-point families. If $F\not\in{\mathcal Z}$ and $$ log \frac{L}{l}\rho(F,{\mathcal Z}), $$ then $$ f\circ F \not\in{\mathcal Z}. $$ (${\mathcal Z}$ is set of families considered the property of not having). Also we provide a general scheme of finding the value $\rho(F,{\mathcal Z})$. For example, we explore the three-point families. We calculated distance from the arbitrary triangle to set of degenerate triangles. Also we prove that the most remote from set of degenerate triangles is an equilateral triangle, and calculated the corresponding distance. It is equal to $\log 2$.

Détail

Keywords: Delaunay's condition of empty ball, quasiisometrique mapping, triangle nondegeneracy, meshes.