3D graphics eBook - Course Materials Repository

Quaternions and spatial rotation 133
Quaternion arithmetic in practice
Let's show how we reached the previous result. Let's develop the expression of f (in two stages), and apply the rules
It gives us:
which is the expected result. As we can see, such computations are relatively long and tedious if done manually;
however, in a computer program, this amounts to calling the quaternion multiplication routine twice.
Explaining quaternions' properties with rotations
Non-commutativity
The multiplication of quaternions is non-commutative. Since the multiplication of unit quaternions corresponds to
the composition of three dimensional rotations, this property can be made intuitive by showing that three
dimensional rotations are not commutative in general.
A simple exercise of applying two rotations to an asymmetrical object (e.g., a book) can explain it. First, rotate a
book 90 degrees clockwise around the z axis. Next flip it 180 degrees around the x axis and memorize the result.
Then restore the original orientation, so that the book title is again readable, and apply those rotations in opposite
order. Compare the outcome to the earlier result. This shows that, in general, the composition of two different
rotations around two distinct spatial axes will not commute.