Rotation Vectors

Rotation vectors are a more compact way to describe rotation than the Rotation Matrix.

• $\text{SO}(3)$ rotation matrices have 9 quantities, but a 3D rotation only has 3 DOFs. Similarly, the Transform Matrix expresses a 6DOF transformation with 16 quantities.
• The rotation matrix and transformation matrix are very constrained by the fact that they must be an orthogonal matrix with determinant 1, making optimization/solution more difficult.

A rotation vector expresses rotation with only 3 quantities; its direction parallel with the axis of rotation, and the length is equal to the angle of rotation.

Consider a rotation represented by rotation matrix $\mathbf{R}$. If described by a rotation vector, assuming that the rotation axis is a unit-length vector $\mathbf{n}$ and the angle is $\theta$, then the vector $\theta \mathbf{n}$can also describe this rotation. The conversion is given by the formula:

$$\mathbf{R}=\cos \theta \mathbf{I}+(1-\cos \theta )\mathbf{n}{\mathbf{n}}^T+\sin \theta {\mathbf{n}}^{\wedge }$$

(Recall that the ${}^{\wedge }$ symbol denotes a skew-symmetric matrix).

The conversion from rotation matrix to a rotation vector can also be found. The angle $\theta$ can be found by taking the trace of both sides:

$$\begin{array}{rl}\text{tr}(\mathbf{R})& =\cos \theta \text{tr}(\mathbf{I})+(1-\cos \theta )\text{tr}({\mathbf{nn}}^T)+\sin \theta \text{tr}({\mathbf{n}}^{\wedge })\\ & =3\cos \theta +(1-\cos \theta )\\ & =1+2\cos \theta \\ \therefore \theta & =\arctan \left(\frac{\text{tr}(\text{R})-1}{2}\right)\end{array}$$

For the axis $\mathbf{n}$, since the rotation axis does not change after the rotation, we have:

$$\mathbf{Rn}=\mathbf{n}$$

Therefore, the axis $\mathbf{n}$ is the eigenvector corresponding to the matrix $\mathbf{R}$‘s eigenvalue $1$. Solving this equation and normalizing it gives the axis of rotation.