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

Authors : Djavvat KHADJİEV, Gayrat BESHİMOV, İdris ÖREN

Pages : 137-158

Doi:10.31801/cfsuasmas.1003511

View : 18 | Download : 7

Publication Date : 2023-03-30

Article Type : Research Paper

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 $Oinsert ignore into journalissuearticles values(2, R);$ be the group of all orthogonal transformations of $E_{2}$. Put $SOinsert ignore into journalissuearticles values(2, R);=\\left\\{ g\\in Oinsert ignore into journalissuearticles values(2, R);|detg=1\\right\\}$, $MOinsert ignore into journalissuearticles values(2, R);=\\left\\{F:E_{2}\\rightarrow E_{2}\\mid Fx=gx+b, g\\in Oinsert ignore into journalissuearticles values(2,R);, b\\in E_{2}\\right\\}$, $MSOinsert ignore into journalissuearticles values(2, R);= \\left\\{F\\in MOinsert ignore into journalissuearticles values(2, R);|detg=1\\right\\}$. The present paper is devoted to solutions of problems of $G$-equivalence of $T$-figures in $E_{2}$ for groups $G=Oinsert ignore into journalissuearticles values(2, R);, SOinsert ignore into journalissuearticles values(2, R);$, $MOinsert ignore into journalissuearticles values(2, R);$, $MSOinsert ignore into journalissuearticles values(2, R);$. 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