Aside Posted on
Quaternions have been around for some time now. Sometimes game programmers use them for rotations in graphics – after all they (quaternions) have lots of good properties. Programmers have good properties too. As it happens people use them without understanding them… but they work all the same – another good property of theirs. Quaternions too.
Anyway, we are not here to talk about quaternions as we already know about them and if you don’t, fear not, there is tons of info on the web and if you really care there are even books… now if you are hardcore enough you can go into some geometric algebra but that’s another story…
We are going to discuss Tuturnions. So what are they?
Let’s revisit the average quaternion definition (I mean, definition as the average programmer, me, defines it):
“Quaternions – they look like 4D vectors, you know, and they rotate things in 3D, and you can interpolate them, and they take less storage than 3×3 matrix, and they are easy to invert and so on”.
So couple of days ago I was doing some 2D stuff and thinking… well doing 2D stuff definitely, thinking – not so much, but anyway, it popped in my head – if we have these quaternions that can rotate stuff in 3D how come there aren’t any cool 2D rotating entities and I’m stuck with these 2×2 matrices. So, without wasting my time with googling and internet and other spoilers like that, I put my wheel inventing hat on and there they were – Tuturnions.
Now, I’ve got to warn you, most of these things I’m going to talk about can actually be done with a 2×2 matrix (or are equivalent to it). But unless there is some secret society against ignoring 2×2 matrices when it comes to rotations we should be safe – come to think of it, I haven’t googled that either.
So, here’s the rules:
- We want to be able to convert between our Tuturnions and 2×2 rotation matrices easily
- Easy to keep the Tuturnion constrained so it can rotate stuff ( Quaternions need to be normalized to rotate stuff – right? )
- Easy to perform operations on them – inversion, rotation concatenation, interpolation, etc
In 2D if we are given an angle, lets call it theta, we can rotate a vector using the following equations:
X’ = cos(theta)*X – sin(theta)*Y
Y’ = sin(theta)*X + cos(theta)*Y
And you can easily see how that translates to a 2×2 matrix. If you don’t, you either need glasses or you don’t understand how matrices work – another excellent reason to keep reading.
Looking at these equations we can see that really, there is only one quantity involved here when it comes to rotation – the angle theta. But obviously, everyone knows that sine and cosine are quite expensive operations so we don’t really want to deal with just the angle. Also angle would be a bit tricky to interpolate continuously without some conditional statements.
But here is an idea: instead of using the angle we could use sin(theta) and cos(theta) instead – revolutionary isn’t it? So our Tuturionon will have two components called C and S. I’ll leave you guessing which one is which. Also, they do sound like they should have two components – two-turniouns.
Basically, in many respects, our Tuturion is a 2D vector. In order to represent a rotation though it needs to be normalized so if we have a Tu(c,s) following has to hold true:
C2 + S2 = 1
…and if you check your school math, like I did, you’ll remember that:
cos(theta)2 + sin(theta)2 = 1
You must admit our Tuturnions look more like 2D quaternions now. Though if we were to say they looked like unit 2D vectors we would be a lot closer to the mark but quite boring to talk about them.
Interpolation of rotation entities is important, especially if you are doing animation system or you just want to have some fun.
We can linearly interpolate a Tuturnion – it is a 2D vector after all. If you can’t interpolate a 2D vector I think you probably need to cover some more basic material first, though I admit, it might be challenging to find such material but if you got so far with this article it’s obvious that you like a challenge.
Anyway after we interpolate we need to re-normalise the Tuturnion, so it represents a rotation once again.
We could also interpolate our Tuturnions using Slerp.
Yes, turns out Slerp is not really just for quaternions (Slurprised? Me too!) and we can apply it to everything in order to interpolate. Although it kind of beats me, if it wasn’t to do anything with 3D and quaternions, why on earth would someone call it spherical linear interpolation. After all the Slerp uses sine and cosine functions of the angle between the interpolated quantities and if anything should be called circular linear interpolation – Clerp.
Bottom line: you can use Clerp to interpolate tuturnions.
This chapter seems to require a lot of 2×2 matrix visualization and my “word” skills are as advanced as my math skills. Short story long you have a sine and cosine (from your Tuturnion) and you already know how to make a 2×2 matrix from that, don’t you?
As for the 2×2 matrix to Tuturnion it should be clear by now, but if you want me to spell it for you: given a rotation matrix M you get equivalent tuturnion T(m11, m12)
Just like quaternions and rotation matrices you can perform numerous operations on a Tuturnion.
In order to rotate a 2D vector V(x,y) by a Tuturnion denoted by T(c,s) you need to (replacing from the formula above):
x’ = c*x – s*y
y’ = c*y + s*x
Looks incredibly like these equations I showed you before and the reason for that is simple: “if it works – don’t fix it”.
You need to simply multiply (transform) the two tuturnions. In essence the operation is the same as rotating a 2D vector but again our Tuturnion could be a 2D vector but that would be a wild guess.
In our example we have T1(c1,s1) and T2(c2,s2) we get the result T(cr, sr):
cr = c1*c2 – s1* s2
sr = c1*s1 + s2* c2
We could spend some time deriving that this is actually true but I’ve always found that part boring to read in articles and to be honest it looks boring to write as well. In fact I don’t see why people bother…
When you work with rotations, it happens some times, you want to rotate stuff backwards. In order to do that you need to invert your rotation entity.
With Tuturnions this is quite easy – you just need to invert your tuturnion’s S part and there you are. It may come as a surprise to you (thought if it does you probably need to stop reading and consider going into politics instead), but if you look at your 2×2 rotation matrix and transpose it, because that’s how you invert it apparently, you can see how the sign changes place but then again that could be some weird coincidence.
We can go on and provide some proof by actually inverting the angle and checking what happens to the sine and cosine functions then. I’ll leave that an exercise to the reader, though believe me, going to the gym will be a far better exercise.
If you are developing a 2D skeleton animation system you could find tuturnions useful. They will compress quite well, bearing in mind you can actually compress them pretty much in to a single entity.
Integrate a special quantity Q in the Tuturnion so they sound more complicated.
We have introduced new powerful rotation entities for 2D. They resemble quaternions in many ways (or at least one – they rotate things). They are easy to work with and they eliminate the need for 2×2 matrices when it comes to rotation – especially useful if you have “a thing” against matrices. They do have a cool name (that’s mainly what marketing is banking on) and they save a whooping 8 bytes if you use them instead of 2×2 matrix.
 That’s a long sentence, isn’t it?
 Well, you can approximate and some hardware supports them natively but where is the fun in that?
 It is important that when you are talking about Tuturnions you always use T(c,s) and not T(x,y).