I'm working on a model that would require to use vectoriel functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \lVert y \lVert_2$, ie with an orthogonal Jacobian.

I can only think of trivial functions (like $f(x) = Ox + c$ for $O$ orthogonal and $c \in \mathbb{R}^n$).

Are there other functions that verify this property? What would it be if we add the constraint $\forall x \in \mathbb{R}^n$, $\lVert f(x) \lVert_2 = \lVert x \lVert_2$ ?