Examples of functions mapping rotation matrices to rotations

Question

I am wondering if anyone has any good examples of non-linear functions that act pointwise on the entries of a rotation matrix and map it back to a rotation matrix. Explicitly, I am looking for, if they exist, functions $f: \mathbb{R} \to \mathbb{R}$ such that for $R \in SO(n)$, $[f(r_{ij})]_{i,j} \in SO(n)$.

Answer 1

There are such functions for $n=2$. Indeed, take any function $g:[0,\pi/2]\to [0,\pi/2]$ satisfying $g(0)=0, g(\pi/4)=\pi/4$ and $g(\pi/2-x)=\pi/2-g(x)$. To construct such a function take any function $[0,\pi/4]\to [0,\pi/4]$ which maps $0\to 0$ and $\pi/4$ to $\pi/4$ and then extend it on $[\pi/4,\pi/2]$: $g(x)=\pi/2-g(\pi/2-x)$.

Now extend this function $g$ symmetrically to $[-\pi/2,0]$: $g(-x)=-g(x)$.

Now for every $x\in [-1,1]$ represent $x$ as $\sin(y)$, $y\in[-\pi/2,\pi/2]$, and define $f(x)$ as $\sin(g(x))$.

Now if you take any matrix of rotation $\begin{pmatrix}\cos x &\sin x\\ -\sin x &\cos x\end{pmatrix}$, its image is $$\begin{pmatrix}f(\cos x) &f(\sin x)\\ f(-\sin x) &f(\cos x)\end{pmatrix}$$ which is $$\begin{pmatrix}\sin(\pi/2-g(x)) &\sin g(x)\\ \sin g(-x) &\sin(\pi/2-g(x))\end{pmatrix}=$$

$$\begin{pmatrix}\cos(g(x)) &\sin g(x)\\ -\sin g(x) &\cos g(x))\end{pmatrix}$$ which is again a matrix of rotation.