In (very much) earlier articles, I described:
- using complex numbers for rotating, translating, and scaling the plane
- using Clifford algebras for rotating, translating, and scaling the plane, and
- using quaternions for rotating, translating, and scaling three-dimensional space.
Today, it is time to tackle rotating, translating, and scaling three-dimensional space using Clifford algebras.
Three dimensions now instead of two
Back when we used Clifford algebras to rotate, translate, and scale the plane, we were using the two-dimesional Clifford algebra. With the two-dimensional Clifford algebra, we represented two-dimensional coordinates as . It shouldn’t surprise you then to find we’re going to represent three-dimensional coordinates as .
As before, we will have and . Similarly, we will have . In the two-dimesional case, we showed that . By the same logic as the two-dimensional case, we also find that and . We could potentially also end up multiplying , , and all together. This isn’t going to be equal to any combination of the other things we’ve seen so we’ll just leave it written .