Does every rotation vector has an one-to-one corresponding rotation matrix?

Question

Given a rotation vector, can we always get the same corresponding rotation matrix computed by the Rodrigue's rotation formula?

Answer 1

Mathematically a rotation vector(or axis angle) representation will always convert to the same rotation matrix.

However, multiple different rotation vectors can lead to the same rotation matrix.

A rotation vector represents a rotation by an axis and an angular rotation around that axis(axis angle).

$$ \begin{bmatrix} e_x \\ e_y \\ e_z \end{bmatrix},\theta$$

Thinking about it for a bit it is pretty easy to see where different axis angle representations will lead to the same rotation.

One is when the angle($\theta$) is the same angle but in the opposite direction. E.g $\frac{\pi}{2}$ and $\frac{-3\pi}{2}$.

The other is when the axis points in the opposite direction, and the angle is also in the opposite direction.

$$ \{\begin{bmatrix} e_x \\ e_y \\ e_z \end{bmatrix},\theta\}=\{\begin{bmatrix} -e_x \\ -e_y \\ -e_z \end{bmatrix},-\theta\}$$

Finally for numerical reason oftentimes the Rodrigues formula will also set some values constant/Identity if the rotation is small enough. So while mathematically they should lead to distinct values, on your computer they might end up being the same thing.