Quaternions and spatial rotation

Quaternions are non intuitive. A point along the surface of a 4-dimensional sphere, isn’t really something which is imaginable. But if you’re more interested in the application rather than research, well this article might just be for you.

When I started out researching quaternions and their application I was working with animation in games. There's a lot to it, and quite a bit of nonsense too. I found that a bit tiresome so I'll try to get right to it.

The thing about quaternions (I'll refer to them as quats or simply quat, from now on) is that they are defined in a 4-dimensional space referred to as H. This article is about application in 3-dimensions, so one thing you need to understand is that a quat compared to an scalar or vector is never the same thing. One way to look at quats is to visualize them as a rotation around an axis. However, this can lead to incorrect assumptions and wont make much sense in end. The way I chose to picture it, is more like a function with arguments. Input and output in this case is simply something useful in euclidean space (E for short).

A naive camera rotation generally suffers from a gimbal lock, a simple way of avoiding this is by representing the rotation by each axis as two separate quats. Creating these two quats is straight forward, now combine these rotations using the quaternion (Grassmann) product and then extract the rotation matrix from this product.

This was the step that initially confused me. But the trick is to create quats in H space from E values, then do the math (simple product) and finally converting back into E space (extracting the rotation matrix from the quaternion). The result is a rotation matrix which does not suffer from gimbal lock and can transform E space values for you.
In order for the calculations to work you need to use unit quats only.

Vector3 v = {0,0,1}
Q yaw = fromE({0,1,0}, yawAngle) // commonly known as the axis/angle representation
Q pitch = fromE({1,0,0}, pitchAngle)
Q result = mul(yaw, pitch) // magic!
Matrix3x3 m = fromQ(result) // convert back to E space from H space
Vector3 v1 = mul(m, v)

The result is a vector v1, rotated about the two angles yawAngle and pitchAngle around the origin ({0,0,0}). Most people when the first try to rotate things, end up rotating objects around a stick because they fail to realize that rotation is the result of rotation around the origin and translation. If you want the appearance of rotating around some other point you need to translate between that point and the origin before applying the rotation.

There are many great code examples out there and it shouldn't be a problem for you to figure out the precise math involved. Once you get going, it's easy. And hopefully this will help you in getting there faster.

Comments Alongside Code (CAC)

While working on a game I’m currently developing I started to think about how poor some text editors really are. I usually stick with Visual Studio as my preferred environment and it’s a very powerful IDE. But one thing I’m currently missing is the ability to treat any certain range of characters across different lines as a group of text.

Allow me to explain. Back in the old days, any terminal had a fixed number of characters one could display on a single line (typically 72) and basically because of this; you limited any single line to that number of characters. While we still try to avoid writing long lines of code (for readability purposes) we tend to use more characters per line now a days and it’s more a question of what’s suitable rather than what’s readable. I tend to average 80 or 100 characters per line and the readability is still great. But with my screen resolution I can fit up to 148 characters per line and thus wasting about 50 characters, for something which I believe, can be put to better use.

Any modern text editors worth mentioning have a way of doing syntax highlighting (colors to certain keywords or expressions) most text editors also do code folding or outlining which is a way of grouping lines together (this requires some sort of semantic awareness). But I have yet not seen a text editor which can take any rectangular selection as a group of text. I think this is best illustrated by an example.

// A Direct 3D device object other than the one that returned this code caused the 
// hardware adapter to be reset by the OS. Delete all video memory objects (surfaces, 
// textures, state blocks) and call Reset() to return the device to a default state. 
printf( "IDirect3DDevice9::Present D3DERR_DEVICELOST\n" );

One can for instance write comments preceding a certain action which is very common and not wrong in any way at all. But what about writing comments like this:

g_Globals.BeginD3DDReset(); // NOTE to self: this is blocking. 
                            // Blocking the game loop becuase the graphics device
                            // is currently unavailable will totally mess up the
                            // fixed game updates, just ignore the rendering step
                            // until the device is successfully reset.

This is not ideal all the time but this way of writing a comment will allow me to write comments alongside code and I think that rather useful. For this to be really practical, I would expect a text editor to recognize the above comment as comments alongside code (CAC) and conveniently indent the whole thing if any code following the above statement exceed current indentation (see below). It should also adjust the input mode so that these whitespaces used to align the comments alongside code never interfere with the actual writing of additional code. One way to think of this, is to picture the text as a layer on top of your code. This layer, is directly synonymous with the actual text itself and is just a feature by the text editor for positioning comments alongside code.

g_Globals.BeginD3DDReset();             // NOTE to self: this is blocking. 
                                        // Blocking the game loop becuase the graphics device
DoMoreCode( NULL, NULL, NULL, NULL );   // is currently unavailable will totally mess up the
                                        // fixed game updates, just ignore the rendering step
                                        // until the device is successfully reset.

One additional thing which came to mind while writing this is that manual word wrapping is a tedious task. That being said, any implementation of comments alongside code should have the ability to word wrap a comment group through a simple user action. The idea is to augment the experience while still working with text files and never adding any clutter to the actual code. Whitespaces can thus not be considered clutter but storing information about a comment block inside the actual comment would be considered clutter. However, it would also be useful as it could provide sensible metadata about the comment itself. E.g. one could chose to treat a comment beginning with a certain word or pattern as a certain marker.

I think it’s obvious at this point why this would be a very useful and powerful feature. Not only as an extension but it could theoretically provide a lot more functionality than just comments alongside code when working with plain text files.

Hopefully I will be able to revisit this topic with a working demo sometime in the future but I leave no such promises.

To all you Emacs and VIM users, if you think you got it, then you're wrong. I use VIM myself and VIM does not do what I want to do. You can do rectangular text selection, but the editor needs to recognize the comments as more than just comments. It a block which needs to adjust to it's surroundings.

Game loops – Not to be taken lightly

Not too far back, I started developing games. Back then I decided to hit the books for a while before getting upfront with the actual programming and one of my questions was about game loops. While googling the web I came across this article (a must read for anyone not familiar with different game loops) and must say that I was a bit surprised how a rather simple thing as a loop turned into something a bit more complex, but for the better!

I must begin to thank Koen Witters for his post about game loops as it helped me a lot. However, it still took me some time to wrap my head around the constant game speed independent of variable FPS loop, and decided to write this in order to work out some of the questions I had.

This is a modified version of the suggested game loop by Koen Witters:

typedef T_TIMEINFO struct {
    int updates_per_second;
    int frequency;
    int skip_ticks;
    int max_frameskip;
    int next_game_tick;
    int loops;
    float interpolation;
} TIMEINFO;

TIMEINFO t;
ZeroMemory( &t, sizeof( TIMEINFO ) );

t.updates_per_second = 25;
t.frequency          = get_frequency();
t.skip_ticks         = t.frequency / t.updates_per_second;
t.max_frameskip      = 5;
t.next_game_tick     = get_time();

while( true ) 
{
    t.loops = 0;

    while( get_time() > t.next_game_tick && 
        t.loops < t.max_frameskip ) 
    {
        update_game();

        t.next_game_tick += t.skip_ticks;
        t.loops++;
    }

    t.interpolation = float( get_time() + t.skip_ticks – 
        t.next_game_tick ) / float( t.skip_ticks );

    update_render( t.interpolation );
}

The variables here have been moved into a struct to make them more easily managed and accessible. Also, time is treated as an incremental value and only through the frequency can you find a relation to actual seconds. This is important since high-resolution timers come with a variable frequency (Hz) while the GetTickCount() function returns a discreet number of milliseconds (1000 Hz).

So without further due I present to you this diagram of the above running game loop:

Let’s start with the scheduling as it’s essential to any game loop.

Each game_update (red dot) occur after deadline (black dot) and thus there is always a difference between game_update and deadline denoted here by overdue. This relatively small value can be used in various ways e.g. in a multiplayer game to check that game state is synchronized among participants, as the overdue value is simply a metric of how far game_update is behind. Note that this value is only accumulated if game_update requires more time than skip_ticks and if so, another game_update is scheduled immediately after.

I bet you went – “Eh? What the… the timer is all backwards!?” when you first looked at the above code. It sure isn’t trivial but it’s essential. You have to think about it as two separate time lines; one continuous and one sampled. While we can plan ahead exact moments in time it’s unlikely that we’ll be that punctuational. If we’d assume we’re always late and plan our next deadline always skip_ticks after last deadline we not only prevent drifting, we also make up for any subtle interrupt game_update might cause. In practice we will either schedule game_update a little before or little after actual skip_ticks but as long as this overdue error remains small, it’s perfectly fine.

It’s possible that game_update might stall (be interrupted) and take very long to finish due to some unexpected behavior. When this occurs the next deadline might be scheduled within the overdue error and immediately run game_update until either caught up with real time or max_frameskip force game_render. A max_frameskip is good, it will give the impression that your game is actually still running, slow, but running. However, getting behind and accumulating an overdue is very bad and you should be aware of problems that might arise from this.

And that’s it! Now all the remaining time between each game_update is spent on game_render. The game_render have to implement prediction and interpolation if you want to smooth results. This is fairly simple and Witters covers that which you need to know in his article. This interpolation value is conveniently always between 0.0 and 1.0 depending on the intermediate step of the current iteration. So the only thing you need to be concern about is how to represent a progression between each fixed game_update.

Ad free Windows Live Messenger

It's actually very simple. Windows Live Messenger get's all it's ad information from a host called rad.msn.com (remote advertisement?) and since it's possible to tell Windows to look up invalid IP addresses through the host file, one can quickly get rid of the ads in Messenger.

You need administrative privileges for this, and if you are running Vista with UAC enabled you need to make sure you launch both Notepad.exe and Cmd.exe with elevation (Shift right-click them in the start-menu and choose run as administrator).

• First, find you host file, Microsoft Windows (NT/2000/XP/2003/Vista) is located in %SystemRoot%\system32\drivers\etc by default.
• Second, open Notepad.exe as administrator (and with elevation if you use Vista and UAC) choose File > Open... and navigate to the host file (make sure you show all files) and open it. It's called "hosts" without a file extension.
• Now add the following new line at the end of the file: "0.0.0.0 rad.msn.com". Save the file and close Notepad.

It will tell Messenger to look up an invalid IP address for ads, and thus display no ads. You need to flush the DNS cache for this to take immediate effect, and it can be done by opening the command prompt Cmd.exe (as administrator) and type "ipconfig /flushdns". If you wait long enough Messenger should stop displaying ads but the quickest path to salvation is restarting Messenger as well.

Enjoy your Ad free Windows Live Messenger.