Quaternion Finder
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and build the cube.
Thanks to Tomasz Wierzicki for the typesetting.
The cube can also be used for finding (the rotations of)
transformations between different coordinate systems.
Suppose you need the rotation from coordindate system C1 to C2:

Align the cube with coordinate system C1.

Find the position of (1; 0,0,0) on the cube.

Align the cube with coordinate system C2.

The quaternion can be read off from the place where
(1; 0,0,0) was found in Step 2.
Example
What quaternion represents the eye coordinates
of a pilot, relative to the coordinate system of the plane?
Assume that you are the pilot. Airplane coordinates have
X pointing forward, Y to the right (starboard), Z down.
In this orientation,
(1; 0,0,0) is at the bottom of the cube to the right.
In your eye coordinates, X will be to the right, Y will be upwards,
Z will be pointing backwards.
If you align the cube, bottom right now contains the
quaternion (1; 1,1,1).