Graphics Reference
In-Depth Information
Chapter 13
Conclusion
The aim of this topic was to take the reader through the important ideas and mathe-
matical techniques associated with rotation transforms. I mentioned that I would not
be too pedantic about mathematical terminology and would not swamp the reader
with high-level concepts and axioms that pervade the real world of mathematics.
My prime objective was to make the reader confident and comfortable with com-
plex numbers, vectors, matrices, quaternions and bivector rotors. I knew that this
was a challenge, but as they all share rotation as a common thread, hopefully, this
has not been too onerous for the reader.
The worked examples will provide the reader with real problems to explore. As
far as I know, they all produce correct results. But that was not always the case, as it
is so easy to switch a sign during an algebraic expansion that creates a false result.
However, repeated examination eventually leads one to the mistake, and the correct
answer emerges so naturally.
The real challenge for the reader is the next level. There are some excellent topics,
technical papers and websites that introduce more advanced topics such as the B-
spline interpolation of quaternions, the kinematics of moving frames, exponential
rotors and conformal geometry. Hopefully, the contents of this topic has prepared
the reader for such journeys.
What I have tried to show throughout the previous dozen chapters is that rotations
are about sines and cosines, which are ratios associated with a line sweeping the unit
circle. These, in turn, can be expressed in various identities, especially half-angle
identities.
Imaginary quantities also seem to play an important role in rotations, and it is just
as well that they exist otherwise life would be extremely difficult! We have seen that
complex numbers, quaternions and bivector rotors all include imaginary quantities,
and at the end of the day, they just seem to be different ways of controlling sines and
cosines. I am certain that you now appreciate that quaternions are just one of four
possible algebras that require an
n
-square identity, and that they are closely related
to Clifford algebra. Which one is best for computer graphics? I don't know. But I am
certain that if you attempt to implement these ideas, you will discover the answer.
