What Was Up With That Rotation Trick? June 10th, 2009
Patrick Stein

In my prior post about using Clifford algebras to do plane rotations, I finished with a non-intuitive step at the end. Rather than multiplying on the right by an element representing a rotation of angle \theta, I multiplied on the left by an element representing a rotation of angle \frac{\theta}{2} and multiplied on the right by an element representing a rotation of angle -\frac{\theta}{2}.

Why did I do this? Well, I mentioned it would be awkward for the two-dimensional case, but that it will be important when we get to three or more dimensions. Well, work for a moment with \frac{\theta}{2} being a quarter rotation (ninety degrees, \frac{\pi}{2} radians). This means our total rotation is going to be a half turn (180 degrees, \pi radians).

For that \frac{\theta}{2}, r = e_1e_2 and so \overline{r} = -e_1e_2. Let’s just look at what it does to our unit vectors e_1 and e_2 to multiply on the left by r and on the right by \overline{r}.

For e_1, we get -e_1e_2e_1e_1e_2 = -e_1e_2e_2 = -e_1. Similarly, for e_2, we get -e_1e_2e_2e_1e_2 = -e_2.

So far, we were only working in two dimensions. As such, there wasn’t any e_3 to worry about. But, what if there were? What happens to the z-coordinate of something if you rotate things parallel to the xy-plane? It remains unchanged.

Well, what would happen if we multiplied e_3 on the right by \cos\theta + \sin\theta e_1e_2? We would end up with \cos\theta e_3 + \sin\theta e_3e_1e_2 = \cos\theta e_3 + \sin\theta e_1e_2e_3. We’ve ended up scaling e_3 and adding in a trivector e_1e_2e_3. We’ve made a mess.

Let’s try it instead with our trick. We’re going to start with -e_1e_2e_3e_1e_2. Every time we transpose elements with different subscripts, we flip the sign. Every time we get two elements next to each other with the same subscript, they cancel out. So, switching the e_3 with the second e_1, we get e_1e_2e_1e_3e_2. From there, we can switch the first two elements to get -e_2e_1e_1e_3e_2 which is just -e_2e_3e_2. We can switch the e_3 with the second e_2 to get: e_2e_2e_3 which is just e_3. So, our trick leaves e_3 unchanged.

In the above, there is nothing special about the subscript three. It would work for any subscript except one or two. So, the trick allows us to break the rotation up into two parts that still do what we want with e_1 and e_2 but leave our other directions unchanged (or, maybe it’s easier to think of them as changing them and then changing them right back).