What is the formal definition of a symmetry of a subset of the real plane?

Question

Consider a set $S \subseteq \mathbb{R}^2$. What does it mean for a function $f$ to be a symmetry of $S$? After all, there are usually infinitely many bijections on $\mathbb{R}^2$ that send $S$ to $S$. I would very much like some clarification of this topic.

Answer 1

The missing ingredient is that you want $f$ to preserve distances: if $d(x,y)$ is the distance between point $x$ and $y$, then you want $$d(x,y) =d(f(x),f(y))$$ for all points $x$ and $y$.

A function with this property is called an isometry. When we speak of geometric symmetries, we always require them to be isometries. It can be shown that any isometry of the plane is a reflection, a rotation, a translation, or a composition of these.