Relationship between equal-magnitude quaternions with different components

Question

What is the relationship called between two quaternions that are equal in magnitude, but different in component values? They are not necessarily unit quaternions.

Answer 1

Typically all quaternions we use for rotation are equal in magnitude, since we're most often using only unit quaternions.

There are two different unit quaternions corresponding to any 3D orientation you might choose, and all of them have a magnitude of 1, so just knowing that two of these quaternions have the same magnitude does not tell us anything about the orientations they represent.

More usefully, you can think of a unit quaternion as representing a rotation of \$\theta\$ radians about an axis indicated by a unit vector \$\vec a = (a_x, a_y, a_z)\$ as having components...

$$w = \cos \frac \theta 2\\ x = a_x \sin \frac \theta 2\\ y = a_y \sin \frac \theta 2\\ z = a_z \sin \frac \theta 2$$

So you if you want to examine the relationship between two quaternions, you can compare the directions of their imaginary x, y, z components to determine their axes of rotation, and compare the arc cosines of their real w components to determine their rotation angles.

The same applies to non-unit quaternions too, just normalize them first. The absolute magnitude doesn't change the orientation they represent. Some systems use this value for scale, other systems discard it entirely.