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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 $\theta$.

enter image description here

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.

enter image description here

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

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    $\begingroup$ It is the same, since you have $n_1^2+n_2^2+n_3^2=1$. $\endgroup$ – user10354138 May 21 at 1:57
  • $\begingroup$ 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. $\endgroup$ – amd May 21 at 2:26
  • $\begingroup$ oh sick, I forgot about the unit vector constraint. thanks all $\endgroup$ – Carpetfizz May 21 at 2:48

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