Abstract:
In the work we introduced the concept of a family of points in Rn
and metrization of space of points families.
Under the family understood the points numbered set of points in Rn. 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,…,k}→Rn.
We define the distance ρ(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 ρ is invariant relatively orthogonal mapping:
ρ(O∘F,G)=ρ(F,G)
for any orthogonal mapping O:Rn→Rn.
We give an estimate of the distance that moves
the family F under the action of quasi-isometric mapping f:
ρ(F,f∘F)≤logLl,
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 isk-point family inRn,
f:F(I)→Rn is quasi-isometric mapping;Z—a set of k-point families.IfF∉Zand logLl<ρ(F,Z), then f∘F∉Z.
(Z is set of families considered the property of not having).
Also we provide a general scheme of finding the value ρ(F,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 log2.
Keywords:Delaunay's condition of empty ball, quasiisometrique mapping, triangle nondegeneracy, meshes.
Document Type:
Article
UDC:
517.5+514.174
BBC:
22.15+22.16
Language: Russian
Citation:
A. Yu. Igumnov, “Metrization in space families of points in Rn and adjoining questions”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 6(37), 40–54