# Rotation matrix from axis-angle representation

I came across the following representation of a rotation matrix given an axis of rotation $$\begin{bmatrix}n_1 & n_2 & n_3\end{bmatrix}$$ and an angle $$\theta$$.

I'm trying to figure out where it comes from, and first thought to check the Rodrigues rotation formula.

$$R = I + \sin \theta K + (1 - \cos \theta)K^2$$

where $$K$$ is the skew symmetric matrix of the axis of rotation.

I used MATLAB's symbolic math library to compute the Rodrigues formula given $$K$$ and $$\theta$$ to see if it matches the expression in the paper - but it's indeed different.

Anyone have an idea of where equation (8) comes from?

• It is the same, since you have $n_1^2+n_2^2+n_3^2=1$. – user10354138 May 21 at 1:57
• Well, they should be the same for the above reason, but for a sign error in (8). The lower-left entry should be $n_1n_3(1-\cos\theta)-n_2\sin\theta$. With that correction, you can see that they are indeed the equal by subtracting one from the other and simplifying. – amd May 21 at 2:26
• oh sick, I forgot about the unit vector constraint. thanks all – Carpetfizz May 21 at 2:48