Now that we’re starting to get somewhere with Vectors and moving around in 3D, let’s turn our attention to the next topic, rotations. (Get it? “Turn our attention.”) Rotations in 3D are pretty awesome, it’s surprising to me how many clever ways there are to represent them. Unfortunately they all pretty much suck for writing a Physics Engine, but let’s take a look at them and discuss the pros and cons. If you want to jump to the answer, it’s Quaternions, but let’s see why.
Caveat Emptor: I am not a Math major, I am self taught on this stuff. As I’ve learned new things, I’ve often found that my previous understandings were incorrect. The web is awash in articles about 3D rotation that are simply wrong, this may be just another example.
Euler Angles
Apparently this is pronounced “Oiler,” who knew? These are appealing because they’re the simplest to understand. If you’ve played any flight simulators or spaceship fighting games then you already understand them, they’re pitch, yaw, and roll. If you haven’t played those kinds of games then I feel sorry for you. Here’s a refresher, imagine you’re flying an airplane, these angles are:
- Pitch – is the nose pointing up or down
- Yaw – this isn’t used much in airplanes, but it’s turning left and right while keeping the wings level. It’s how a car turns
- Roll – when planes turn one wing goes up and the other goes down, this is roll. Hopefully it’s never used in a car.
This is probably a good a time as any to try to start thinking in terms of X, Y, and Z axis. For our discussions the X axis moves left to right. The Y axis moves up and down. The Z axis moves forward to back. Pretty simple stuff, right? For Euler Angles to make sense we also have to think about the planes that the axes define. In this case I mean plane like a flat surface like a table top or a wall, not the kind you fly in. Take a moment to convince yourself that any 2 axes define a plane. The X and Z axes together define planes like the floor or table top. The XY plane would be a TV you’re watching. The YZ plane defines the walls to your right or left side.
So anyway, if you say your pitch is 45 degrees up from the XZ plane (a table top), 45 degrees to the right of the YZ plane (a wall to your left or right) and then 180 degrees of roll, that would mean you’re facing up and right, and you’re flipped upside down. That’s Euler Angles. Like I said, it’s not very hard to understand. However it suffers from some pretty serious limitations. The first of which is that the order of application is important. If you do the roll first or last you will end up in a completely different place. In our previous example if you started by rolling 180 degrees then you’d be upside down and your concept of “up” would be different. So from there if you went “up” and “right” you’d end up in a different place then if you did the roll last. So if you’re tempted to write a rotation function like
void rotate( float pitch, float yaw, float roll )
Just stop! Nothing about that interface says in which order the operations will be defined, so you’re almost guaranteed to forget and change your assumption and end up with a horrible bug.
The second major problem with Euler Angles is “Gimbal Lock.” Go to Wikipedia for more information on this one, it’s fascinating but a bit beyond the scope of this. The short version is that while Euler Angles can define every possible rotation, they can’t always be combined. So while you’re perfectly safe representing your current rotation with Euler Angles, if you then want to turn the ship a bit, and maybe spin when a missile hits and you try to combine all of these rotations it might work… or you might end up in a state where the next rotation will be jarring and random.
The third major problem with Euler Angles is Interpolation. This means if you have two different rotations and you want to move some percentage of the way between them, it’s difficult to figure out where that should be. How could this be useful? Well if you know you’re rotating 50 degrees per second and a half a second has passed, how far should you rotate? 25 degrees sounds like an obvious answer. But trying doing this when you’re rotating around all 3 axes at the same time? That’s Interpolation and it’s very hard to do with Euler Angles.
Axis and Angle
This one here is my own personal favorite because it’s just so darned handy. It takes a little bit longer to get used to, but it’s worth the study time. For this system you start with a Vector, and then you rotate around it by the Angle specified. Makes sense, right? Think about a door. They rotate around the axis defined by the hinges, a Vector that points straight up and down. How far they rotate is specified by the Angle. What about if you’re standing straight up and you want to bow? The Axis would be a straight line from your left to right, forming a hinge through your hips. How far you bow around that Axis is the Angle. These start to get a little confusing when you want to rotate around an Axis that isn’t so easy to picture. Like what if the Axis is (1,1,1) that’s sort of a diagonal axis that’s up, right, and back so rotating around that is a little harder to imagine. I often end up holding my hands in front of my face with my fingers pointing in different directions and trying to rotate my arms around them. My wife calls this “Invisible Knitting.”
Another great advantage of Axis and Angle is that it works really well with Cross Products. Imagine you’re pointing directly forward and you want to turn to your right. If you take your starting Vector (forward) and get the Cross Product with your destination Vector (right) you will end up with a new Vector that points straight up. It turns out that this new Vector is exactly the Axis you need to rotate around. And it’s length is proportional to how far you have to rotate, ie the Angle. So yeah, Cross Product and Axis and Angle are made for each other.
So to compare Axis and Angle with Euler Angles I would say that Euler Angles are a little simpler to understand. But that’s about their only win. Axis and Angle may be a bit harder to grasp, but once you become comfortable with them they’re easy. Furthermore they don’t suffer from bugs due to order of execution, and they work really well with Cross Products. Big wins. As far as gimbal lock goes, I think they don’t have this problem. I have to confess I don’t understand all the math on this one, so we’ll chalk that one up as a half point in their favor. Alas Interpolation is still very difficult with Axis and Angle.
Rotation Matrices
I’m afraid if you’ve come here hoping for a deeper understanding of Rotation Matrices then you’ve come to the wrong place. In all my Physics Engines over all my years I have made it a point to never use these. A 3×3 rotation matrix simply has too many numbers in it to make any sense to me. I have no intuitive feel for how Matrices work, so debugging them is basically impossible. The Dubious Engine uses exactly one matrix, and it’s required to pass rotations to OpenGL. I don’t understand it, I just know how to dump my rotation into it.
Here’s the important thing I do know about Rotation Matrices, they also suffer from Gimbal Lock and are hard to Interpolate. So as far as comparisons go, they’re impossible to understand, and suffer from the same exact failures as Euler Angles and Axis and Angles. So why bother learning them?
Quaternions
Enter the Quaternion. For most of my Game Engine life I considered Quaternions to be unknowable. I couldn’t figure out how they worked or how to imagine them, so I just copied some equations I learned online, tested them enough to convince myself they worked, and moved on. However I am pleased to report that after some more study they’ve started to make some sense to me. I’ve recently gone back through my engine and smoothed out some misunderstandings with Quaternions. None of the math changed, so I don’t think I fixed any bugs, but it does make a bit more sense now, which is worthwhile. It’s also a little slower, which is a shame, but worth it for the clarity.
I’m not going to try to write another article explaining Quaternions. I’ve found two that did the trick for me, so I’m just going to send you there and let you learn the same way I did. However I will fill in the starting point that I was missing. I was familiar with “Imaginary Numbers,” which I hope you are too. If not, they’re usually represented as i and are defined as:
i = sqrt(-1) or i * i = -1
What I did not know was that there is another kind of number called “Complex Numbers” that are defined as a real number plus an imaginary:
a + bi
I had no idea these existed, and most Quaternion articles start assuming you know them. Luckily not the ones I recently found.
So, at this point, push the pause button on this article, and spend some time learning from these two sources. I am humbled by how well they explain the subject, they do it better then I ever will. When you’re done, feel free to return and we can look at how we can use Quaternions in a game engine:
- https://www.youtube.com/watch?v=mHVwd8gYLnI – as taught by an actual University Professor. Unfortunately while attempting to speak and write on the blackboard he occasionally messes up an equation, so don’t copy the Math verbatim. Still an Amazing lecture that will get you up to speed very quickly.
- https://www.3dgep.com/understanding-quaternions/ – the big one. Nothing about it is easy, you will have to re-read it a bunch of times, but it is complete and awesome. In fact, his whole site is awesome, I will be aspiring to be half as good.
Okay, you know know what I do about Quaternions. Here’s how you can represent one in C++
struct Quaternion {
float w;
Vector v;
};
From the 3D Game Engine Programming article, we know that in a lot of Quaternion Math, the imaginary component acts as a Vector (with dot products, cross products, scalar multiplication, etc). Luckily we already have a Vector class defined that does these things, so it’s easy to reuse. In fact, all of the math for a standard Quaternion is pretty straight forward, so my Quaternion class is relatively simple, here’s a link, there shouldn’t be any surprises.
How do we use them? Well really for 3D rotations it’s a Unit Quaternion we want to use. These are the kind that represent rotations. They can easily be created from an Axis and Angle like this:
Unit_quaternion::Unit_quaternion( const Unit_vector& axis, float angle )
{
w = cosf( angle / 2.0f );
v = Vector(axis) * sinf( angle / 2.0f );
}
So now we can convert from something we’re comfortable with, Axis and Angle, into a Unit Quaternion. From there the math for combining multiple Quaternions to produce an output Unit Quaternion is fairly easy to code up. Lastly any Unit Quaternion can be pretty easily converted back to three axis: X, Y, and Z.
I’ll tell you one major drawback of working with these things though. There is just no way to get an intuitive sense of them. Here’s a pop quiz for you to see how your 3D intuition is developing. Point (with your finger) to Vector (1, 1, 0) and rotate 45 degrees forward around it. If you pointed diagonally up and right and then kind of rolled forward and left a bit, congrats! It might not be perfect, but at least you have a sense of it. Okay, now imagine Quaternion (1, (1, 1, 0)) and rotate around that. That real bit has something to do with half the cosine of the angle, and the imaginary bit is related to half the sine… I have absolutely no idea where to go. So imagine trying to debug this code. Like you notice in your simulation that some cubes are rotating “oddly.” So you set your breakpoint, look at your rotation and see it’s (0.43255, (0.97666432, 0.43234, 0.124535)). What do you do with that?
So in summary, let’s compare Quaternions to the other Rotation representations we’ve discussed. On the plus side, they do not suffer from Gimbal Lock, or bugs due to order of execution. We didn’t specifically discuss Interpolation, but they do it very well, it’s called “Spherical Linear Interpolation” or SLERP and it’s easy to code up. On the negative side they’re very difficult to understand and debug. However this is somewhat smoothed over by the simplicity in converting between Unit Quaternions and Axis and Angle. All in all, they’re a big win, which is probably why most Physics Engines use them.
Here’s the links:
- Quaternion – The one and only
- Unit Quaternion – And the other one
- SLERP – Not as messy as it sounds
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