The purpose of this book is to examine both the properties and applications of
quaternions, and, in particular, to explore visual representations that help to develop
our intuition about quaternions and their exploitation. Of all the natural
advantages of quaternions, several stand out clearly above all others:
• Normalized quaternions are simply Euclidean four-vectors of length one,
and thus are points on a unit hypersphere (known to mathematicians as the
three-sphere) embedded in four dimensions. Just as the ordinary unit sphere
has two degrees of freedom, e.g., latitude and longitude, the unit hypersphere
has three degrees of freedom.
• There is a relationship between quaternions and three-dimensional rotations
that permits the three rotational degrees of freedom to be represented exactly
by the three degrees of freedom of a normalized quaternion.
• Because quaternions relate three-dimensional coordinate frames to points
on a unit hypersphere, it turns out that quaternions provide a meaningful
and reliable global framework that we can use to measure the distance or
similarity between two different three-dimensional coordinate frames.
• Finally, again because they are points on a hypersphere, quaternions can be
used to define optimal methods for smooth interpolation among sampled
sets of three-dimensional coordinate frames.