- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Vol: 72 Issue: 1
- Invariants of a mapping of a set to the two-dimensional Euclidean space
Invariants of a mapping of a set to the two-dimensional Euclidean space
Authors : Djavvat Khadjiev, Gayrat Beshimov, Idris Ören
Pages : 137-158
Doi:10.31801/cfsuasmas.1003511
View : 9 | Download : 4
Publication Date : 2023-03-30
Article Type : Research
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