Invariants of a mapping of a set to the two-dimensional Euclidean space

Year 2023, Volume: 72 Issue: 1, 137 - 158, 30.03.2023

Djavvat Khadjiev , Gayrat Beshimov İdris Ören

https://doi.org/10.31801/cfsuasmas.1003511

Abstract

Let $E_{2}$ be the $2$-dimensional Euclidean space and $T$ be a set such that it has at least two elements. A mapping $\alpha : T\rightarrow E_{2}$ will be called a $T$-figure in $E_{2}$. Let $O(2, R)$ be the group of all orthogonal transformations of $E_{2}$. Put $SO(2, R)=\left\{ g\in O(2, R)|detg=1\right\}$, $MO(2, R)=\left\{F:E_{2}\rightarrow E_{2}\mid Fx=gx+b, g\in O(2,R), b\in E_{2}\right\}$,
$MSO(2, R)= \left\{F\in MO(2, R)|detg=1\right\}$.
The present paper is devoted to solutions of problems of $G$-equivalence of $T$-figures in $E_{2}$ for groups $G=O(2, R), SO(2, R)$, $MO(2, R)$, $MSO(2, R)$. Complete systems of $G$-invariants of $T$-figures in $E_{2}$ for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of $G$-invariants are given for these groups.

Keywords

Euclidean geometry, invariant, figure

Supporting Institution

The Ministry of Innovative Development of the Republic of Uzbekistan and The Scientific and Technological Research Council of Turkey

Project Number

UT-OT-2020-2 and 119N643

Thanks

This work is supported by The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan) under Grant Number UT-OT-2020-2 and The Scientific and Technological Research Council of Turkey (T\"{U}B{\.I}TAK) under Grant Number 119N643.

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