Game Physics Engine Development, 2nd Edition

170 Chapter 9 The M athematics of Rotations

where

⎡

⎢

⎣

x

y

z

⎤

⎥

⎦

is the axis, and θ is the angle, as before.

Quaternions are not merely a four-element vector, however; their mathematics are

more exotic. If you are allergic to mathematics, then feel free to skip this explanation

and head for the implementation in the next section.

You ma y remember in high school mathematics learning about the square root of

−1, the so-called imaginary number (in contrast to the real numbers), often w ritten

as i or j.So,i

2

=−1. A complex number is then made up of both a real number

and some multiple of i,intheforma + bi. If your mathematical memory is very

good, you might recall drawing complex numbers as coordinates in 2D (the Argand

diagram), and deriving lots of their