There are three types of rigid transformations that can be combined to create an isometry: reflection, translation, rotation. The former two can be applied to any function to create a new (or trivially, identical) function in isometric correspondence, whereas the latter cannot in general (only to bijections) although can be applied to non-invertible relations. Can these be expressed as a mapping $f$? Let $\{f\}$ denote the set of all mappings $f$ from one set-relation to all its isometries, where $c$ and $d$ represent fixed constants (causing a translation) and a negative sign ($-$ from $±$) before one or more of the variables represents a reflection.
$\stackrel{A}{\{(x,\:y)\}_1}\quad\stackrel{\{f\}}{\mapsto}\quad\big\{\{(±x+c,\:±y+d)\}_{\{≅A_1\}}\big\}$
Does this look right (for two variables, without rotation)? If so, does it extend without issues to more dimensions?, and How can rotations be applied (without, and if possible also with, precluding multiple-to-one mappings with or without restrictions)?