Paths and geodesics in $SO_n$ I would like to find some (easily computable) function to interpolate between special orthogonal matrices:
Let $A$, $B$ be $n\times n$ special orthogonal matrices and $0\let\le1$, then $f(A, B, t)$ must return a $n\times n$ special orthogonal matrix such that $f(A, B, 0) = A$, $f(A, B, 1) = B$ and $f$ is smooth as a function of $t$.
In particular, I'm interested in the case where $A = I$.
If possible I would like to avoid computing $O(n)$ decompositions.
 A: A (simple) rotation of $\mathbb{R}^n$ is an element of $SO_n$ that leaves point-wise fixed an $n-2$ dimensional subspace of $\mathbb{R}^n$.
Observation.
Every element of $SO_n$ is a product of simple rotations.
Indeed, let $A \in SO_n$ and denote by $v_1, \dots, v_n$ the columns of $A$. Denote by $R(v_1,e_1)$ the simple rotation of $\mathbb{R}^n$ that maps $v_1$ to $e_1$.
 Then
$$
R(v_1, e_1)A = [e_1 \; \tilde v_2 \; \cdots \: \tilde v_n],
$$
and each $\tilde v_i$ has no component in the direction $e_1$, by orthogonality. By reccurence, you find a sequence $R_1^{-1}, \dots, R_n^{-1}$ of simple rotations such that
$
R_n^{-1} \cdots R_1^{-1} A = I_n.
$
In other words,
$$
A = R_1 \dots R_n I_n.
$$
Solution. Now, denote by $R^{\theta}_i$ the rotation that moves the same 2d plane as $R_i$, but by an arbitrary angle $\theta$. You can interpolate between $A$ and $I_n$ by adjusting the angles in the product
$$
R_1^{\theta_1}\dots R_n^{\theta_n}.
$$
Notes. 


*

*This method requires computing a decomposition in $n$ matrices. In $\mathbb{R}^3$, as noted by Ethan, you can use a single rotation.

*A better solution would study the geodesics of $SO_n$.
