I'm reading a paper and came across the following definition of a rotation vector.
$P_r = 2sin\frac{\theta}{2}\begin{bmatrix} n_1 & n_2 & n_3 \end{bmatrix}^T, 0 \leq \theta \leq \pi$
and "$R$ is a simple function of $P_r$ without any trigonometric functions"
$R = (1 -\frac{|P_r|^2}{2})I + \frac{1}{2}(P_rP_r^T+\alpha \cdot Skew(P_r))$
$\alpha = \sqrt{4 – |P_r|^2}$
I'm not sure where these two equations come from. I am, however familiar with the matrix formulation of the Rodrigues formula for rotation.
$R = I + Skew(n)sin(\theta) + Skew(n)^2(1-cos(\theta))$
where $n$ is the axis of rotation and $\theta$ is the angle of rotation.
I think the two equations are somehow related, but I don't know a whole lot about how the first one was derived.
Best Answer
We can get the Rodrigues formula:
$v' = (\cos \theta) v + (\sin \theta) n \times v + (1 - \cos \theta) n ( n \cdot v)$
Or in matrix form:
$v' = ((\cos \theta) I + (\sin \theta) Skew(n) + (1 - \cos \theta) n n^T) v$
From the equation that you have. First notice that:
$\| P_r \| = 2 \sin \theta /2$
$\| P_r \|^2 = 2(1 - \cos \theta)$
So the first term is just:
$(1 - \| P_r \|^2/2) I = (\cos \theta) I$
Using the same identity, the term $P_r P_r^T$ is just the same as $2(1 - \cos \theta) n n^T$
The third term $\alpha Skew (P_r)$ is just $(2 \sin \theta) Skew(n)$ since:
$\alpha Skew (P_r)= \alpha (2 \sin \theta/2) Skew (n)$
$\alpha = \sqrt{4 - 4\sin^2(\theta/2)} = 2 \sqrt{1 - \sin^2(\theta/2)} = 2 \cos \theta/2$
Using the identity $\sin \theta = 2 \cos(\theta/2) \sin(\theta/2)$:
$\alpha Skew (P_r) = 2 \sin \theta Skew (n)$
Joining the three pieces we finally get:
$R = (\cos \theta) I + (\sin \theta) Skew(n) + (1 - \cos \theta) n n^T$
Which is Rodrigues formula