Euler Quaternions and the Trouble with Y-Up Space, Part 1

Yesterday I did something that I didn’t think I was capable of. It involved quaternions, matrices and a decent amount of math. Let’s start with the problem definition:

• We have a system that tracks rotations using quaternions
• We would like people to be able to use Euler coordinates (Tait-Bryan angles if you’re picky)
• Our world is Y-UP

Doesn’t sound too hard. In fact, half of it was fairly easy. Turning a set of Euler coordinates into a quaternion is as simple as creating three quaternions using angle axis pairs derived from the incoming Euler rotation. For us, the three Quaternions were as follows:

• [Angle: Yaw, Axis: [0, 1, 0]]
• [Angle: Pitch, Axis: [0, 0, 1]]
• [Angle: Roll, Axis: [1, 0, 0]]

We create a combined rotation quaternion by conjugating the three quaternions together in the following order:

Roll * (Pitch * (Yaw) * Pitch^-1) * Roll^-1

This is all that is required to produce a quaternion from a set of Euler coordinates. Now, we also had to retrieve Euler coordinates from a given quaternion. This is more difficult than taking Euler coordinates and converting them into a quaternion as the task requires a fairly solid understanding of the math. This too would be simple, if it weren’t for the fact that the entire Internet agrees that Z-Up is the way to go for Euler/Quaternion conversions. From what I could tell, the equations for converting a rotation quaternion into a Y-Up Euler coordinate do not exist! (On the internet).

To my shock, horror and dismay, this meant that I had to derive the equations myself. Needless to say, this was a daunting task. But, after an arm and a leg… and the other arm… and the other leg… and parts of my face/body and an assortment of pieces from whichever organs remain… I was victorious. My glorious victory over the maths was quite euphoric.

First, let’s talk about why Y-Up Euler coordinates are so different from Z-Up Euler coordinates. The idea behind Euler coordinates is that given three angular values, you apply three rotational transformations in a predetermined order around three dynamic or constant axes (depending on convention) and arrive at a destination rotation.

Rotational transformations are not commutative, that means that applying them in a different order will produce a different rotation all-together. I won’t get into why that is but it has to do with the fact that the space of all 3D rotations is something akin to the surface of a 4 dimensional hypersphere and applying a rotation is like traveling from one point on the surface of that sphere to another surface point while always remaining on the surface. (Feel like going for a dip in some advanced math? Read here).

The effect is that a rotation quaternion composed from rotations around axes in the order ZYX is fundamentally different from one composed from rotations in the order YZX, which is what we want. So, the equations must be derived. I won’t delve very much into why the following works as that could take several blogs. First, we must build a rotation matrix that will compose our Euler coordinates in the order that we want. This is as simple as getting the angle-axis matrix formula and plugging some numbers into it. That formula is the following:

Where c is cos(θ), s is sin(θ) and C is 1-c, θ is the angle of rotation and (x,y,z) are the vector components of the axis. This is taken from here . By plugging in axis and angle, we construct three rotation matrices:

Yaw, about [0, 1, 0], angle sign Ψ (Psi):

Pitch, about [0, 0, 1], angle sign Θ (Theta):

Roll, about [1,0,0], angle sign Φ (Phi):

Now, we multiply the matrices together in the order (Roll)(Pitch)(Yaw), producing the following:

Next step is to refer to the Quaternion -> Rotation matrix formula:

This formula was taken from here. The quaternion components are [w, x, y, z] -> [0, 1, 2, 3]

Finally, to retrieve the Euler coordinates we must match the matrix form of the rotation quaternion to our RPY rotation matrix exploiting cells where the values are easy to retrieve. In this case, the values are as follows (Notation, Matrix _{Row Column} where row and column are in the range [1,3]):

• Φ, roll: atan2(RPY₃₂, RPY₂₂) = atan2(2(q₀q₁ + q₂q₃), q₀q₀ – q₁q₁ + q₂q₂ – q₃q₃)
• Θ, pitch: asin(-RPY₁₂) = asin(2(q₀q₃ – q₁q₂))
• Ψ, yaw: atan2(RPY₁₃, RPY₁₁) = atan2(2(q₀q₂ + q₁q₃), q₀q₀ + q₁q₁ – q₂q₂ – q₃q₃)

This is almost enough. What is left is to test for pitch singularities (+90, -90) and treat them specially. Since this blog has gone on for far too long, I will stop now as what has been given is certainly enough to get a similar system working. In my next blog, I will detail how we could cheaply test for the singularities and what we could do in that case. You can do some advance reading on that here.

A Scalable Solution Part One: Local Reference Frame

Blog Reading Requirement: you need to know what a reference frame is, from wiki:

“A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer. It may also refer to both an observational reference frame and an attached coordinate system as a unit…”

Read the source article here.

In this series of blog posts, I will be talking about scaling 3D objects in their local reference frame and in the world reference frame. In part one, I will be introducing the topic and detailing scaling in the local reference frame. In part two, I will be introducing scaling in the world reference frame as well as detailing some of the differences between the two types of scaling. I will also be detailing some of the complexities as well as a possible implementation.

As an artist, the ability to scale groups of objects is fairly important when working with 3D models. When scaling an object, it is done against a known axial reference frame; typically, the two most useful reference frames are the world reference frame and the local reference frame and they are vastly different. The local reference frame changes as the object is rotated while the world reference frame never changes. I will take this blog to elaborate on some of the challenges and requirements involved in implementing these manipulations as well as giving a possible implementation.

Local reference frame scaling is the simpler of the two as it is performed against the local reference frame of the object that is being scaled. With groups of objects, it is expected that each object in the group is scaled against its own local reference frame as opposed to the reference frame of the group. It is also expected that the positions of the objects in the group will not be affected by the scaling operation.

This scaling method is simple because it only involves applying a scaling transform against the 3 known axes of any given object. This means that the scale parameters can be stored as 3 floats. During frame rendering, the transform could be applied after position and rotation have been applied; this fits in with the rendering pipeline of most 3D engines and means that one could simply alter the scale parameter present on a renderable object in most 3D engines.

In part two, I will be introducing world reference frame scaling and what complexities may arise from allowing the combination of the two methods. Thank you for reading.

Release 0.4 – Not so much a bang but a whisper

Here we are at the bottom end of OSD600. This course has been fun and has taught me a lot. This is why it is all the more unfortunate that my 0.4 release is not nearly as atomic as my 0.3. For my 0.4, I had 2 projects to work on alongside another course. One project was migrating XSS Me from the current tab-based threadpool implementation to Web Workers. This didn’t go so well. The conclusion of my research into Web Workers deemed them unusable for XSS Me’s requirements. So instead, a very minor fixup was in order.

As for Paladin, as usual, quite a lot of work went into the configurator. The configurator is now closer than ever to landing but still has not landed. This iteration, quite a number of changes went into the configurator. I’ve itemized the changes below:

• Store/load now accept named parameters
• Configurator objects were receiving undefined guids, now test suite fails if that happens
• ConfNode is now it’s own require module
• Configurator constructor is now not stored on engine
• Configurator attempts to test for db caps on engine startup
• Logger is now accepted as a construction option
• Configurator now picks up default.js
• /gladius/gameID is now just /id
• Error handlers now introduced to indexeddb function hierarchy
• Try/catch around sensitive areas

I will continue to work on XSS Me and paladin.

Release Details:

XSS Me:

commit: 731fe1563f97ab3951a5c799257a9acaea945cd3

issue thread: https://github.com/SecurityCompass/XSSMe/issues/3

pull request: https://github.com/SecurityCompass/XSSMe/pull/4

Paladin:

commit: 4539a66ee3a5ac6a38c420c089781f66e4120eb7

issue thread: https://github.com/alankligman/gladius/issues/21

pull request: https://github.com/alankligman/gladius/pull/52

Release 0.3!

Whew! It’s been a while but my 0.3 has now been released! In this release, I’ve been working on Paladin/configurator subsystem. By now, it has become quite mature and achieves all required functionality. Also being used is IndexedDB as the load/store backend. Also important is that the codebase has now been brought up-to-date with the rest of the engine.

Changelog:

• Added comments to all of the Configurator tests that didn’t have any. Improved some of the comments that were there before as well as improving some test structures slightly.
• Removed Cookie.js from the engine. Cookies are no longer used as load/store backend, now replaced with IndexedDB.
• Many functions had vague local variable names; their names have been improved.
• load/store are now both asynchronous.
• Added config/default.js for local configurations that need to be baked into the engine.
• Configurator now uses gameID to load/save configurations.
• Brought Configurator codebase up-to-date.

commit: a94ea288daabd84ea222a5a08da78ec4095479b3

issue thread: https://github.com/alankligman/gladius/issues/21

pull request: https://github.com/alankligman/gladius/pull/52

I also have some work to do for XSS Me but that has taken a back-seat to the work on Paladin. Currently, the plan is to dedicate most of 0.4 towards XSS Me. We’ll see how this all pans out.

X-Men. On netflix.

In other news, X-Men.

JACKPOT! AGAIN! And this time for something more/less nerdy:

X-MEN!!!! From your childhood. On netflix. So much win. I clicked play today and I was so impressed by how this show beats the shit out of every other kid’s show on the air right now. Voice acting is godly and the jokes are hilarious. WATCH EPISODE 1 NOW!

Non-Netflix Member Link: https://signup.netflix.com/movie/X-Men/70172490
Netflix Member Link: http://ca.movies.netflix.com/WiMovie/X-Men/70172490?trkid=4022489

For some strange reason, if you’re signed into netflix, you can’t view the non-netflix link and vice versa with the other link.

Rome

Rome is awesome. Rome is a build server that is speedy and hooked up to a fat pype ( I was getting ~5.5MB/s downloading from github ( ! )  ). I got acquainted with Rome just tonight when I asked @humph for an account on it ( which he kindly obliged <3 ). Long story short, complete make do nothing cycle in ~52 seconds; this is down from ~10 minutes on my supercomputer of a laptop. To get started, follow the following easy steps:

1. Ask @humph to get you an account on Rome.
2. Configure your environment to forward X11 to a local X11 server. On Linux, this should be as simple as passing -CX to ssh when connecting. On Windows, this is a bit more involved but boils down to the following steps:
   1. Get Xming
   2. Start Xming.
   3. Start PuTTy.
   4. In the options on the left, select Connection -> SSH -> X11
   5. On this window, click “Enable X11 forwarding” and in the box labeled “X display location” enter 127.0.0.1:0.0
   6. You probably want to save your configuration options. Switch back to the session options by selecting Session in the options on the left.
   7. Enter a reasonable name for this configuration, something like 01 – Rome should be fine.
   8. Enter rome.proximity.on.ca as the server address.
   9. Hit save and you should now be set to login and forward X11!
3. Execute any application that you desire. In the case of our firefox build, you will probably want to call the debug firefox binary. This should be as simple as navigating to the top-level of your mozilla-central repo and calling:
   [whatever your build directory is]/dist/bin/firefox -ProfileManager -no-remote
   • If you already have a profile that you like to use on this machine for nightlies, you can substitute -ProfileManager on the commandline with -P [whatever the profile name is]
   • Obviously, you will need to build firefox first before you can call the binary
4. Sit back, relax and enjoy the lightning speed builds.

0.3 Coming Up Fast

So the due date for 0.3 is coming up soon and I’m honestly just a tad bit worried… The project that I’ve picked up for 0.3 ( migrating XSS Me to web workers ) is proving to be too big for the time that I have left for 0.3 so I’m thinking that I’ll push most of the work that has to go into it to 0.4 with some preliminary work done for 0.3 but I’m wondering if that will be enough…

As for Paladin, unfortunately the guys are fairly busy so for now I’m just waiting till they have some time to take another look at the issue thread. Still, I’m confident that my questions will get some attention at some point and until then… I only have to wait.