Essential Math Weblog

As part of the book review (see previous article), they have images from the Indexed weblog.   If you like graphical math jokes, you might find them amusing.

For those who are not aware, there is a new chapter in the book about Random Numbers.  Because it is only a 50 page chapter in a much larger work, I don’t go into as much detail as I would have liked about probability.  So I recommend a couple of longer works dedicated to the subject.  One is Grinstead and Snell’s Introduction to Probability, which I had the honor of being taught from (in a pre-release edition) by John Kemeny.  The other is Larry Gonick’s Cartoon Guide to Statistics, which is a surprisingly informative little book.  However, after seeing a review of The Drunkard’s Walk: How Randomness Rules Our Lives, by Leonard Mlodinow, I may have a third.  I recommend at least reading the review, and possibly checking out the book.

And if you’re into podcasts (and who isn’t, really?  But I jest), the BBC’s In Our Time podcast has a episode about Probability.  I haven’t had a chance to listen to it yet, but if it’s like their other programmes, I’m sure it will be quite interesting and informative.

After months of hard work, we’re finally done with the Second Edition.  It went to the presses last week, and it will be available May 16th.  Pre-order it now.

The following drove me crazy for the past few weeks, so I figured I’d better post about it just to get the word out.

For those who don’t know, VS 2005 added the notion of a “manifest” to manage DLLs. Every application must have a valid manifest, or you can’t run. In general, you can use the default settings, but if you’re running on a FAT32 partition (which I am, as I share it through BootCamp), for some reason it just doesn’t always create the manifest correctly. Beats me why since I’d think that sort of thing should be independent of what file system you have, but there you go. The solution is to check the “Use FAT32 work-around” option in the Manifest Tool section of the main project’s properties.

Now I can finally go through and update all of the examples to use our new rendering platform — it took a bit to put together, but we finally have a cross-platform renderer that supports both OpenGL and D3D9.

Sorry about the long delay between updates: things have been very busy here in Chez Jim.

Lars and I have been hard at work on the second edition, and are wrapping up the text this week to be shipped to the publisher. The major changes are:

• Brand new chapter on random numbers
• The three core rendering chapters have been completely rewritten for vertex and fragment shaders
• Direct3D 9 and OpenGL 2 support added

In addition, all the remaining chapters have been revised for clarity, corrections and updated content. Skinning and morphing unfortunately didn’t make it into the main text because of space, but may end up on the CD or may be posted on this site at some point. Availability of this edition will be sometime next year.

A few bits of other news:

• Lars and I are both at NVIDIA Corporation now. Lars is working on mobile technology, and I’m working on OpenGL drivers.
• I should be at GDC in 2008, although that may be up in the air.

That’s about it for now. I’ll update as the process continues to let you know how it’s going.

I’ve had a couple reports of issues with the OS X libraries as shipped with the book.  The problem appears to be that the libraries are compiled for Mac OS 10.2, but don’t work so well with 10.3 and particularly 10.4 and up.  And the fact that they are compiled for a PowerPC architecture doesn’t help the new Intel machines.

The solution is simple: recompile the libraries.  To do so, you’ll need to make sure that all the files are writeable.  Open a terminal window and change to the “common” directory and then type:

make clean PLATFORM=OSX
make PLATFORM=OSX

If you still have issues after doing this, let me know and we’ll try to track it down.

Note that this is the same procedure you’ll have to follow for the Linux builds, except that the platform doesn’t need to be specified.  While the libraries are compiled under an older version of Mandrake, there is no guarantee that they will work for other Linux builds or versions.

For those who are interested in the process of construction of a math library, or perhaps are interested in other people’s opinions on how such a library would be constructed, or perhaps would like to share your own opinions on such a library… er, I’ll start again.

In the sweng-gamedev mailing list, Noel Llopis started up a thread regarding the creation of a industry-wide standard math library. This has led to a very long discussion of such a library, and the creation of a wiki page. As of yet I don’t have much of an opinion on this other than the discussion is extremely verbose and so I can’t keep up with it all. However, someone out there might, and so I share its existence with you all.

Some of this may meander into the next edition of the book, particularly the management of SIMD operations which is something I’m dealing with in my company’s library. However, I can say for certain that there will not be a separate Point class in the book’s library, because a) it’s simpler to manage and b) I trust people not to do dumb things with Vectors when they’re actually Points. This may be a foolish assumption on my part.

Addendum (05/19/06): there’s also a SourceForge page to deposit potential code for said library.

Just a quick note to say that a second edition for Essential Mathematics for Games… is in the works.  If you have items that you would like to see updated, expanded, or just plain included in a new edition, please let me know.  Also, if you’re an educator and you’re using this book as part of a course, please drop me a line as well.  We may be interested in having you do a detailed review of the first edition.
You can send me email at ‘jim’ at this domain, or use the form on the front page of the site.

Thanks in advance.

I had originally intended on writing up a post about some of the science and math I’ve been seeing on TV lately. However, as I got into it, I realized it was going to end up too long, too verbose and (more importantly) without a point. A boring rant for ranting’s sake, if you will. So… y’all (or youse guys, if you prefer) get this post instead: a series of fun math- and science-related links which provide a nice diversion on slow days.
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First of all, my apologies for the lack of posts. Part of it was the demands of my paid work: For some time, I’ve been working on the PS2 version of Rainbow Six: Lockdown, and those of you in the know are aware that the ship date was pushed out from March to September. While most of that was just polish, doing that final 1% takes a fair amount of effort. So only recently have I felt like digging into new engineering challenges and blogging again. The other part was that I didn’t have much to write about, but I have some ideas which I’ll put up in the next few weeks.
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The GDC sessions have finally been announced, and I’m pretty excited about the tutorial this year. Instead of doing the usual 2-3 people talking about a variety of subjects, some of which is not their specialty, this year we have 6 people with a broad range of experience. This should allow us to have deeper coverage and present an overall better tutorial.

Here are the speakers, with their current proposed topics:

Jim Van Verth: Physics Engine Overview
Christer Ericson: Numerical Robustness
Squirrel Eiserloh: Relative Motion and Collision
Gino van den Bergen: Narrow (GJK) and Broad (Sweep and Prune) Phase Collision
Erin Catto: Constraints and Solvers
Marq Singer: Dynamic Destruction

The full session description can be found on the GDC 2006 site. Hope to see some of you there!

A couple more errata have come in since the last update. From Tim Lowery comes:

“On page 117, the figures look mislabeled, or the axis is incorrect. For example, Figure 3.5a shows x-axis rotation, but the object is rotating around the y-axis. Likewise, for Figure 3.5b.”

In this case, the captions are correct (the intent is to go in x-y-z order), but the figures are reversed.

From k. avery comes:

“At the end of p. 623, cot theta should be csc theta / sec theta, not sec theta / csc theta.”

Indeed it should be. And I had thought I caught that one once before. Evasive devils, those cotangents.

I’ve finally gotten around to creating a formal errata file, and a code update. The code update is an encrypted ZIP file; the password is the second word on page 37. The paths should be set up correctly to match the original CD. To use, simply unzip it to the directory where you copied the CD contents.

If there are issues, you can post a comment here, or write me by using my first name at this domain.

Now we come to the final part: merging Newton-Raphson and bisection together, and choosing the correct one depending on the situation.
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Due to preparing for a week of vacation, and then going on a week of vacation, and then recovering from a week of vacation, this blog has fallen from the wayside a bit. But I’m back, and I’ll try to wrap up this topic in a couple of postings.

When last we left our intrepid heroes, we were discussing Newton-Raphson root finding, how it’s used for reparameterizing a curve, and when it fails (namely when the speed on the curve approaches zero). The solution hinted at was the bisection method.
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A bit of a digression here: the notes for the Future of Curved Surfaces roundtable are up in the publications area. Overall, a pretty good discussion. I don’t think anything was really resolved, but I certainly learned quite a bit. Hopefully others did too.

We now return you to your irregularly scheduled reparameterization discussion.

I’m still optimizing memory and speed so nothing exciting to report under the lighting topic. Maybe later this week or next week.

So let’s look at a bit of code from the book, shall we?
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I had a few more topics left over from GDC, but I left my notes at work. And I’ll be on lead engineer duty this week and a bit of next so I’m not optimistic I’ll have time to post anything from there.

So instead I’ll write a bit about some lighting research I’ve been doing. Lately I’ve been interested in doing some improvements to lightmap generation — both making it efficient (in both speed and memory) and improving the quality of the lighting solution.
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One of the questions asked during the tutorial was whether matrices or quaternions were more efficient for angular rigid body dynamics. My response was that a matrix would be more efficient, because a quaternion requires a conversion to a matrix to compute the transformed inertial tensor matrix. If we analyze the operation count, this certainly appears to be true:
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Having promised to post about center-of-mass calculations, and subsequently having put it off for a week, I’m going to punt.

First, you can read the seminal Brian Mirtich paper on the subject. Then Dave Eberly responds with a more efficient method. Both rely computing solid integrals across a polytope of constant density by treating them as surface integrals across the polytope’s faces. In the end they end up with total mass (assuming a density of 1), center of mass, and the inertial tensor matrix for the polytope.

If those haven’t scared you off, Jonathan Blow has a more intuitive approach, noting that the solid integral is basically a volume calculation. By breaking the object into tetrahedrons, we can compute the total solid integral as a weighted sum of tetrahedral solid integrals. The tetrahedrons can be computed by selecting a single point — the origin or the centroid will work — and using that point together with the points on each triangular face. As he notes, it’s likely that the Eberly and Mirtich approaches are more efficient, but not as easy to understand from a geometric standpoint. Blow doesn’t provide an implementation, but you can grab a similar one from Stan Melax here.

Finally, Erin Catto pointed out to me after the tutorial that you can compute the center of mass by computing the centroids of the tetrahedrons and then performing a weighted sum of all the centroids, where the weight is the volume of a particular centroid’s tetrahedron divided by the total volume of the polytope. Blow also covers this in his paper as well. To get an intuitive sense of this, I recommend messing around with some origin-centered triangles (e.g. (1,0),(-.5,-.5),(0,-.5),(.5,-.5)) and using areas instead of volumes.