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From what I understand, quaternions are a way to represent a rotation

enter image description here

In this formula, n is the axis of rotation and theta is the angle. So if I'm trying to represent the following rotation enter image description here

The quaternion, q, would be [0, 0, sin(pi/4), cos(pi/4)]

But why would -q be the same as q? Since we are taking the negative of q, the axis of rotation and angle are flipped, so wouldn't it look something like this?

enter image description here

codingcultivator445
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  • Alternative way to look at it: quaternions represent $3D$ rotations via the conjugation map $q\mapsto t^{-1}qt$ for some unit quarternion $t$ (and the $q$ pure imaginary quaternions). Since $ (-t)^{-1}q(-t)=t^{-1}qt$ they must represent the same rotation. – Mo Pol Bol Jan 09 '23 at 20:31
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    A rotation by angle $\theta$ about the axes $\vec{n}$ is the same rotation as that by angle $-\theta$ about the axes $-\vec{n}$. And they give the same $q$, so I don't quite see that. One simple way of going from $q$ to $-q$ is to replace $\theta$ with $\theta+2\pi$, and that won't change the rotation either. Exactly what is puzzling you? – Jyrki Lahtonen Jan 09 '23 at 20:36
  • Geometrically, thats what happens: if you grip an axle with a wheel while facing the wheel and twist it counterclockwise a certain angle, you would have gotten the same effect by going to the other side of the wheel and turning the axle clockwise by the same angle. Algebraically, it's just obvious from conjugation. What's the cause of your concern? – rschwieb Jan 09 '23 at 20:41
  • Never mind, I think I've been getting confused because I've been looking at the angle of rotation of -n the same way I'm looking at the angle of rotation through n. I've been trying to understand quaternions through a sort of geometric understanding. – codingcultivator445 Jan 09 '23 at 20:49
  • The quaternions allow to define rigid movements (isometries) of the 3D space, but they are not rotations themselves. As already mentioned a non zero quaternion $q$ induces an isometry of the $3$-dim space of traceless quaternions by $z\mapsto q z q^{-1}$. It is rather obvious that $q$ and $-q$ define the same isometry even though, in fact, $q\neq-q$. – Andrea Mori Jan 09 '23 at 21:01
  • FYI - While rotations are the most common application made of quaternions, that is not what they were invented for, nor all there is to them. They were originally designed to be a three-dimensional version of complex numbers, except it turned out (after 20 years effort) that four dimensions were needed instead of three. – Paul Sinclair Jan 10 '23 at 19:12

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