Isometry between two identical shape in $\mathbb{R}^2$ I know that it could be a silly question, but if I have two identical convex compact shapes  in $\mathbb{R}^2$ (e.g. two identical equilateral triangles with side length equal to $1$), what is the isometry between them? I know they are isometric up to a rigid motion, but I'd like an explicit formula of such isometry. 

Isometry is for small quadrilateral differential lengths as part of larger surfaces. . In the case of equilateral triangles it is a congruence or identity.
