Graphics Reference
In-Depth Information
other topics. Complex numbers are extremely useful from two perspectives: the first
is that they pave the way to the idea of a rotational operator, and second, they play
an intrinsic part in quaternions and multivectors. Vectors provide a mechanism for
representing oriented lines, and together with complex numbers form the basis for
quaternions, which provide a mechanism for rotating points about an arbitrary axis.
Lastly, multivectors introduce the concept of oriented areas and volumes, and pro-
vide an algebra for undertaking a wide range of geometric operations, especially
rotations.
1.3 The Reader
This is an introductory topic and is aimed at readers studying or working in com-
puter graphics who require an overview of the mathematics behind rotation trans-
forms. They are probably the same people I have encountered asking questions on
Internet forums about Euler angles, quaternions, gimbal lock and how to extract a
direction vector from a rotation matrix.
Many years ago, when writing a computer animation software, I encountered
gimbal lock and had to find a way around the problem. Today, students and pro-
grammers are still discovering gimbal lock for the first time, and that certain mathe-
matical techniques are not completely stable, and that special cases require detection
if their software is to remain operational.
1.4 Aims and Objectives of This Topic
The aim of this topic is to take the reader through the important ideas and mathemat-
ical techniques associated with rotation transforms, without becoming too pedantic
about mathematical terminology. My objective is to make the reader confident and
comfortable with the algebra associated with complex numbers, vectors, matrices,
quaternions and rotors, which seems like a daunting task. However, I believe that
this is achievable, and is why I have included a large number of worked examples,
and shown what happens when we ignore important rules.
1.5 Assumptions Made in This Topic
I only expect the reader to be competent in handling algebraic expansions, and to
have a reasonable understanding of trigonometry and geometry. They will probably
be familiar with vectors but not necessarily with matrices, which is why I have
included chapters on these topics.
