In this third lesson, we introduce the quaternion number system, an extension of the real numbers whose algebraic structure perfectly captures the behavior of spatial rotations.
The quaternions give us a symbolic language for writing down all of our discoveries from the previous lessons, and will help us discover new relationships that are virtually impossible to discover purely through movement.
To begin with, we need to be a bit more precise in how we talk about 3-D rotations, as well as how they relate to spatial orientations.
Let's take a quick look at 2-D and 3-D rotation matrices, with all their advantanges and disadvantages.
For inspiration, we'd better review the complex numbers, which perfectly capture 2-D rotation.
Can we somehow generalize the complex numbers, in order to capture 3-D rotation? Why yes, we can!
Perhaps the coolest thing about quaternions is that their product beautifully captures composition of spatial rotations. Let's see some exmaples of quaternion multiplication in action:
The fact that composition of spatial rotations is equivalent to multiplication of quaternions can be stated more precisely as follows:
The proof of this remarkable result is (just barely) beyond the scope of this workshop. However, in the next lesson, we develop key ideas needed for understanding why this isomorphism holds.
