In this fourth lesson, we tie up several loose ends. First we use quaternions to parametrize the various loops and homotopies from Lesson Two. We then meet a virtual avatar of Bala, built in the open-source 3-D animation software Blender, who can read and perform all our previous discoveries, now in quaternion form.
We then look at how quaternions act on R³ as spatial rotations, via a surprisingly simple formula for the image of a point (x, y, z) under a quaternion rotation q. Finally, using Blender, we get a glimpse of the relations between infinitesimal rotations - also known as the Lie bracket on the Lie algebra of SO(3).
First, let's try parametrizing a simple 360° rotation loop:
Were you able to parametrize the trivial tracking loop? There are multiple approaches, but I hope you found a way that is not too compuationally challenging!
We've seen how to parametrize loops using quaternions...But how can we parametrize homotopies, or gradual deformations of loops?
Okay, let's put all this together and parametrize the gradual deformation from a 720° loop to a stationary loop.
Let's switch gears and look at a final, supremely useful application of quaternions: their action on R³ as spatial rotations:
We still haven't proved that the group of 3-D rotations is isomorphic to the group of unit quaternions modulo sign. But our formula for the action of unit quaternions on R³ is a useful starting point!
Long ago we mentioned that even infinitesimal rotations are somehow entangled. Using Blender, we can now see exactly how:
If you'd like to know more about Lie groups and Lie algebras, I recommend John Stillwell's Naive Lie Theory, which nicely complements everything we've done so far, and provides a rigorous framework for further exploration.
