## Clifford Algebras for Rotating, Scaling, and Translating SpaceJuly 6th, 2009 Patrick Stein

In (very much) earlier articles, I described:

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 $(x,y)$ as $xe_1 + ye_2$. It shouldn’t surprise you then to find we’re going to represent three-dimensional coordinates $(x,y,z)$ as $xe_1 + ye_2 + ze_3$.

As before, we will have $e_1e_1 = 1$ and $e_2e_2 = 1$. Similarly, we will have $e_3e_3 = 1$. In the two-dimesional case, we showed that $e_1e_2 = -e_2e_1$. By the same logic as the two-dimensional case, we also find that $e_1e_3 = -e_3e_1$ and $e_2e_3 = - e_3e_2$. We could potentially also end up multiplying $e_1$, $e_2$, and $e_3$ 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 $e_1e_2e_3$.

## Quaternions for Rotating, Scaling, and Translating SpaceJune 11th, 2009 Patrick Stein

In earlier posts, I described how complex numbers can be used to rotate, scale, and translate the plane, how Clifford algebras can be used to rotate, scale, and translate the plane, and why I resorted to an awkward trick for the Clifford algebra rotations of the plane. In this post, I am going to explain what the quaternions are and describe how they can be used to represent a rotation in three-dimensional space.

### What are the quaternions

Okay, remember how we got the complex numbers? We needed something that was the square root of negative one.

Now, imagine that you are Sir William Rowan Hamilton. The year is 1843. It is springtime. You know how to use the complex numbers to represent points in the plane. And, you know that when you do that, you can use complex numbers to rotate, scale, and translate the points. That’s all well and good, but you don’t live in a two-dimensional world. How are you going to do the same sort of thing with three-dimensional space? How are you going to multiply triples?

You spend months on this. If only you could say, How about I let there be another number that is different from $i$ (and from $-i$) that has the same property that its square is negative one? You fight with this for months. You try to represent a point with coordinates $(x,y,z)$ as $x + yi + zj$. But, nothing you come up with makes any sense.

Your kids are harassing you, Daddy, did you figure out how to multiply triples yet? You have to answer them every morning with a polite, No, not yet.

Then, you’re walking along the Royal Canal in Dublin. It’s mid-October already. My, how the year has flown by. Bam, it hits you. If you add a third number like $i$ and $j$ which is equal to $i\cdot j$, everything works out. You get so excited, that you carve your equations into a stone bridge over the canal:

$i^2 = j^2 = k^2 = ijk = -1$