# Special values for 3D rotations matrices

I wanted to know if there are other "special" angles when using the rotation matrices in 3D. Looking at them I see that if the value of $\theta = 0$ or $\theta = 2\pi$ then the rotation matrices are the identity matrix, therefore vector doesn't change at all.

Are there other "special" values for $\theta$ to consider?

The rotation matrices are as follow:

$$R_x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \\ \end{pmatrix}$$

$$R_y = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{pmatrix}$$

$$R_z = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\[3pt] \sin \theta & \cos \theta & 0\\[3pt] 0 & 0 & 1\\ \end{pmatrix}$$

• What's $\theta$? – joriki Mar 3 '13 at 18:03
• The angle of the rotation... Check the update :D – BRabbit27 Mar 3 '13 at 19:17
• Well I'd certainly consider $\theta=\pi$ special. Beyond that, much depends on what you mean by "special". – Andreas Blass Mar 3 '13 at 22:43
• Ok, i tried with $\theta=\pi$ and I see it changes the sign in the rotating axes. For instance, making a rotation about $Rx$ of the vector $a(x,y,z)$ transform the vector into $a'(x,-y,-z)$, is this a kind of reflection? I think we have the same meaning of "special" :D – BRabbit27 Mar 4 '13 at 6:53

For me case $\theta=i\pi/2$ where $i=0,1,2,3$ is a special one because it generates matrices with values exclusively $\{-1,0,1\}$. Compositions of these matrices also give matrices with $\{-1,0,1\}$ values (there are such $24$ cases) so they are good for simple exercises with rotation matrices and for checking different rotation formulas for example Rodrigues' rotation formula.