• 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 .

Using the identities like for , we can shuffle any combination of , , and so that all of the indexes are in order. We may have to change the sign, but we won’t have to change anything else. For example:

Recall that was a bivector. Here, we have added one more vector: , two more bivectors: and , and a trivector . The general element of the three-dimensional Clifford algebra looks like:

For the two-dimensional Clifford algebra, we had terms in the general element—the Clifford algebra is an algebra on a four-dimensional vector space. For the three-dimensional Clifford algebra, we have terms in the general element. If we backtrack here to the one-dimensional Clifford algebra, we see that the general element looks like meaning there are terms in the general element of the one-dimensional Clifford algebra. If we backtrack even further, we see that for a zero-dimensional Clifford algebra, the general element looks like meaning there is term in the general element of the zero-dimensional Clifford algebra.

This relation is going to hold for Clifford algebras in general. If is a non-negative integer, then there are terms in the general element of the -dimensional Clifford algebra. The -dimensional Clifford algebra is an algebra over a -dimensional vector space. There are degrees of freedom, if you like. We won’t be using all of those degrees of freedom because we are restricting ourselves to scaling, rotating, and translating vectors.

Translating and Scaling

We are going to represent our points in three-dimensional space using the vector portion of our three-dimensional Clifford algebra . So, a point with coordinates will be .

Just as we did in the two-dimensional case, we can scale three-space away from the origin by a factor of just by multiplying each point by the real number . So, .

Similarly, we can translate the point by units in the x-direction, units in the y-direction and units in the z-direction by creating a translation vector and adding it to each point so that .

Rotating three-space

When we used Clifford algebras to rotate the xy-plane, we let and let . Then, to rotate a point we multiplied out .

Three-dimensional space contains the xy-plane. We want to be able to rotate things in the xy-plane in just the same way we did in the two-dimensional case. We even spent some time in another article verifying that these rotations in the xy-plane would leave coordinates on other axises alone.

Rotating space with the quaternions, we found that could rotate around an arbitrary axis. If we pick an axis that points from the origin to where , then we used the quaternion (where and just like we used the two-dimensional Clifford algebra rotation. We let and . Then, we could rotate a point by evaluating .

With Clifford algebras, we don’t mush together translations and rotations both into the vector portion. We keep translations in the vector portion, but we move rotations to the bivector portion. We’re going to rotate around the axis from the origin to (where by an angle by letting
and where and . Then, we can rotate a point just by multiplying on the left by and on the right by so that .

I wrote a little differently here than I did above for the general element. If I were to keep totally consistent with what I wrote above, I should write . Certainly, we can add in any order, so it is totally cosmetic to write it one way or the other. I chose to keep , , and in order rather than , , and . I also chose to keep them all positive at the expense of using instead of .

Before we get into why we used instead of for the , let’s look at the case of rotating around the z-axis. In that case, would be . This should degenerate to exactly what we had for rotating the xy-plane with Clifford algebras. Sure enough, if we plop those values in for , , and we find that exactly as we had in the two-dimensional case. Bonus!

Why should the end up with instead of ? The way that our axis system worked with the quaternions is that we had the x-axis going east, the y-axis going north, and the z-axis going up. If we are going to rotate around the y-axis, then we need to rotate the z-axis toward the x-axis as opposed to rotating the x-axis toward the z-axis. The rotation around the z-axis takes the x-axis toward the y-axis. This is represented with the . is the x-axis and it is moving toward the y-axis . If we reverse the order, , then we are moving the y-axis toward the x-axis. Using our transposing tricks, . Thus, rotating the y-axis toward the x-axis is just negatively rotating the x-axis toward the y-axis. In better English, that is: rotating the y-axis toward the x-axis is just rotating the x-axis away from the y-axis.

If we had chosen instead of , then we would have messed up how our axises fit together. To keep the axis of rotation obvious, we used . We can still rewrite the result in any form we like. If we were given , that would be a rotation around the axis from the origin to the point . I think it’s more intuitive to write as because then you can read the axis of rotation right off of the coefficients.

Next time

Next time, we’ll tackle rotating, scaling, and translating hyperspace (four-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 (and from ) 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 as . 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 and which is equal to , everything works out. You get so excited, that you carve your equations into a stone bridge over the canal:

Now, instead of trying to represent as , it is represented as . Immediately, this looks like an improvement. Before, things were not symmetric. Squaring something that was only in the x-direction was fundamentally different than squaring something that was only in the y- or z-direction because , but . Now, at least, there is no singled-out axis.

Let’s just explore the part for a minute. If we multiply both sides (on the right) by , we see that: . If, instead, we multiply by on the left, we see that . If we take that and multiply on the right by , we get . We’re going to show in a minute that . So, if take , , and in that order and take two in a row (wrapping around if needed), the product is the next one: , , and .

So, how do we show that ? Well, it’s pretty simple to show that . We already know that . So, . We know that , so . And, we know that , so . And, since we already know that , we have . We can use this, then to manipulate . . We can do a similar thing to show that .

This should look familiar from the the Clifford algebras. In the Clifford algebras, . Here, . It turns out that this is extra important in three-dimensions compared to two.

In two dimensions, if you rotate by an angle and then by and angle , it is the same thing as if you rotate by the angle and then by the angle . In three-dimensions, you have more options. In addition to picking the angle, you can also pick the axis of rotation. (Actually, for later on, it may be helpful to think of picking the plane parallel to which the rotation will take place rather than thinking of it as an axis.) In the picture at the right, the die in the back is the original position of a die. The die on the left is the result of rotating the original die a quarter turn on its left face (orange arrow) and then a quarter turn on its right face (blue arrow). The die on the right is the result of rotating the original die around its right face (blue arrow) and then its left face (orange arrow). As you can see, the resulting positions are different.

[Ugh... I just realized the free model of a die that I downloaded is not an actual, legal die. If it were, the opposite faces would sum to seven. Clearly, however, the two and the five are on adjacent faces of this die, as are the three and four, and thus also the one and six. Erf. So, you can't repeat this experiment on your desktop unless you fabricate your own die. Sorry.]

Like the Clifford algebras and the complex numbers, addition and subtraction are simply adding and subtracting (respectively, of course) the corresponding parts.

Armed with the above identities, we have enough information to multiply together quaternions. Rather than do the full general case of , I’m going to do a simpler case to give you the flavor without wasting a ton of space in this article beating a horse that needn’t be beaten that much. Let’s just try: . We start with the distributive law, twice, and then combine like terms.

There is one other notation we’re going to need below. In the complex numbers, we used the notation to mean the number with the sign of its imaginary part reversed. So, if , then . With the quaternions, we are going to use to mean the number with the sing of all of its imaginary parts reversed. So, if , then .

Transforming Space

Say you have some list of points in three-dimensional space: , , , . We are going to represent those as pure quaternions: , , , .

If we want to translate them all by two units along the x-axis, three units along the y-axis and five units along the z-axis, we can set and then add to each of our points: .

If we want to scale three-space out from the origin by a factor of seven, we can set and multiply each point by : .

Let’s say now, that we want to rotate our points parallel to some plane through the origin by an angle . The first thing we need to find is some way to describe that plane. In three-dimensions, we can simply use the surface normal of the plane. It may make more sense to you to think of this as the axis of rotation. In two- and three-dimensions, that will be fine. In higher numbers of dimensions, this will just confuse things. Let’s say our normal starts at the origin and goes through the point . To make things nice, let’s also assume it is of unit length: . We can represent our rotation as . Then, we can rotate our points by multiplying on the left by and on the right by : .

By way of an example, let us suppose we are going to rotate the point by an angle , parallel to the xy-plane. We will expect that our z-coordinate remains the same, but that the other two coordinates change. First, let us let and . The normal to the xy-plane is just the z-axis. So, our is going to be . This leaves us to multiply out (and it’s going to get messy):

Notice how the and the cancel out. Also, notice how . So, we can simplify to an extent like this:

If we go back to the double angle identites, we find that while . So, our final formula is:

Denouement

I know that in the complex number case and in the Clifford algebra case, I used . In this case, I used (swapping the roles of and ). I could have used either in the complex number case since the complex numbers are commutative: for all complex numbers and . Thus, . I probably should have used for the complex case and still used for the Clifford algebra case.

There is a technical reason for the difference. This may sail right over your head if you haven’t done much hardcore linear algebra. But, when we are using the complex numbers or the quaternions to represent rotations, we are really jamming a linear form into a vector and pretending like they’re the same thing. We are glossing over the difference between a vector space and its dual space by just pretending they are the same. We can get away with that in the complex numbers and the quaternions, but we could not get away with it in the Clifford algebra case. This is because the three-dimensional bivectors make up the dual space of the vectors. So, when we’re using three-dimensional bivectors, we are explicitly using the dual space instead of taking advantage of the isomorphism between the dual space and the original vectors. When we get to four dimensions or more, the rotations aren’t even in the dual space.

For a more English, less Math, version of the above, look at it this way. Really, a rotation is something fundamentally different than a vector. We lucked out with the complex numbers and the quaternions in that we were still able to represent rotations with something like a vector (though in the quaternion case, it was a vector plus a scalar). The luck in the quaternion case isn’t perfect, it is off by a few signs here and there. In more dimensions, we won’t be so lucky. In three dimensions, there are three degrees of freedom for a rotation. In four dimensions, there are six degrees of freedom. We are never going to jam those six degrees of freedom into our puny four-dimensional vector.

Next time

Next time, we will do these transformations of three-dimensional space using Clifford algebras instead of quaternions.