Have I understood Quaternions correctly?

[H_man]

Have I understood Quaternions correctly?

Hi all,

I have been trying to understand how quaternions work when rotating coordinate axes in 3-D. I think I may have finally succeeded but I would appreciate if someone who really knows how to use them could confirm if I am correct.

I think that the 3 vector coordinates represent a single point in space and so represent a vector coming from the origin to the point.

This vector actually represents an axis which coincides I suspect with the NEW x axis.

Finally, the real component of the quaternion represents the angle through which the coordinate axes is rotated about the NEW x-axis.

The thing which worries me about the way I understand it is that its too simple. Surely of all the dozens of sites I looked at someone would have put in a diagram to express it this way if it were this simple?

Thanks for any help,

H_man

 

[D H]

One typically uses unit quaternions to represent rotations of coordinate axes in 3-space. Such quaternions are closely related to eigenrotations. An eigenrotation is a rotation about some rotation axis $\hat n$ by some angle $\theta$. The unit quaternion corresponding to such an eigenrotation is
$$q = \cos\frac\theta 2 \pm \sin\frac\theta 2 \hat n$$
The reason for the uncertainty in the sign of the "imaginary" part is that the sign differs depending on whether one uses left or right quaternions to represent the rotation.

Your thinking regarding the use of unit quaternions for representing the rotation of coordinate axes in $\mathbb{R}^3$ is thus partially correct. The real part of a unit quaternion does indeed represent the angle through which the axes are rotated, but not directly. Instead, $\operatorname{Re} q = \cos\frac\theta 2$.

However, the vector or imaginary part of a unit quaternion used to represent a rotation does not represent the new X axis. The imaginary part instead represents the rotation axis but scaled such that the resulting quaternion is indeed a unit quaternion.

Unit quaternions are but a subset of the quaternions. It is helpful to understand the quaternions in their entirety. One is otherwise left with a bunch of plug-and-chug formulae without this understanding.

 

[H_man]

Thanks for the explanation!

I think based on what you have said that I now understand it. BUT, could you confirm that what I now picture in my mind is correct...

The unit vector (represented by the complex components) is a rotation axis. Thus all 3 of the coordinate axis are transformed simultaneously by being rotated about this axis. And that they are rotated by an amount "theta".

I know it may seem that I am stating the obvious but I want to make sure that I do understand it correctly.

Cheers.