3D Math Primer for Graphics and Game Development (2011) | Hacker News Reader

3D Math Primer for Graphics and Game Development (2011)

gamemath.com

131 pointswsc98144 comments

It is funny that the author says to recycle the 1st edition and never show it to anyone. I actually failed to optimally solve an interview question because a topic covered in the 1st edition was removed in the 2nd and other graphics books only have the O(N log N) solution instead of the optimal O(N) solution.

The problem was about vertex welding. Some books, like Real-Time Rendering, only mention the N log N solution.

I was annoyed after the interview and thought, which book would I have had to read to get this question right? The only 2 books I could find that had the optimal answer were the 1st edition of this math book and Real-Time Collision Detection.

I suppose if I was smarter I could have figured out the "bucket" trick. I find interviews too stressful to think clearly.

What a petty interview :( Does the candidate have the desire and skill to find the correct answer given access to the right resources? That's what they need to ask. Not what is in the candidate's brain on the day of the interview.

I've forgotten more than I know, but I know where to find it.

Depending on the job, this might be (somewhat) reasonable. I'd assume that one could still get hired even if one failed that particular question, but showed some sense of understanding about the problem space.

For many problems, someone tackling it might not even understand that there is a problem, and might have no clue where to start searching for answers.

I've seen many developers who burn a crazy amount of hours by going down the wrong path, and then try to StackOverflow or Google their way out of the mess they initiated. It takes experience (either gained or learned) to avoid such traps.

Then again, it really depends on the job. The interview question suggests this was not for a CRUD job, though :)

It was for a 3D graphics programming job.

Another question was about some low level assembly performance instruction trick. Afterwards I did the same thing and tried to figure out which text book I would have had to read to know the answer. I only found one Computer Architecture book that mentioned the instruction and that wasn't the book I read in university.

I do not Envy a graphics programmer interview. There's at least 3 types of graphics programmers in the modern day and the questions couldn't be more night and day. Imagine studying these specific mathematical approaches but instead you're thrown shader trivia. Or super specific DirectX API calls.

I've interviewed for 3D graphics jobs and interviewed others. I was definitely only interested in whether the person had the skill to find a solution.

For my first 3D game dev job I just turned up with a floppy which had my 3D engine on it and got them to stick it in a PC. That was all that it needed to get hired. That and be incredibly naive about salary negotiations :D

I also turned down a possible position at Argonaut at Star Fox time. I got that one by arguing repeatedly with Jez San on Usenet lol

> Not what is in the candidate's brain on the day of the interview.

A close sibling of this mentality lives in academia as well, manifested as that professor who bases 50%+ of one's grade on the final. Also, pass/fail filtering exams (and the cultures that endorse them) generally, such as the MCAT.

About 80% of my grades in uni were 100% based on the finals. AFAIAA this is the norm for STEM in Germany.

This was less true for my Master's and is generally less true for non STEM.

I kind of liked it tbh

Companies don't tell interviewee's why the didn't hire them so are you sure it was these questions that lead to them passing on you? Often you are one among many interviewing at the same time and you may have been given a thumbs up yet someone with more experience is hired.

Clearly someone who read the first addition and provided the O(N) solution would be "more experienced" in this scenario.

Meanwhile I wouldn't even know where to start if asked such a question. Is this a tech artist or animation programmer position?

It was a C++ graphics programming job.

I agree that it isn't the sole reason why I didn't get the job. If I did great on the other 5 interviews, then my O(n log n) solution probably would have been fine. However, not getting the optimal solution didn't help.

Can you share what the bucket trick is, I don't seem to be able to find anything by googling it.

Thanks!

I also struggled to find the answer online. I found it in only 2 books.

Chapter 12 of Real-Time Collision Detection has a section on vertex welding.

The 1st edition of 3D Math Primer has it in Chapter 14.4.2: Vertex Welding.

If you want to merge vertices that are within WELD_EPSILON then you can create a 3D grid of cells with cell sizes being at least as large as 2*WELD_EPSILON and put each vertex in its cell. For each cell, you only need to check the 8 neighboring cells to see if the vertex should be merged with a nearby vertex.

The worst case could be worse than O(N) if most of the vertices end up in the same grid cell but don't require merging.

Awesome thanks, grids work really well for a lot of optimization tricks.

Nice. Reminds me of the PC Game Programmer's Encyclopedia from back in the 90s/early 2000s: http://qzx.com/pc-gpe/index.php

I had that book! I was like 16. I was so blown away by the blend between math, programming, and graphics. I didn't write a game until a decade later though.

I read some similar books when I was around that age too. And now I’m over 30 years old and still didn’t write a proper game yet :p

One day though

You should try game jams! I found the external pressure really helpful. Shipped half a dozen (tiny) games this year.

If you work on something small and simple on a constrained timescale with a set deadline first—like a Pac-Man clone, or an Arkanoid clone, or a Vampire Survivors clone—it will take you longer than you would've expected, but it still won't take too long, and you'll come away from the experience with some code you can reuse, some ideas of how to do things better next time, and a ton of confidence due to having explored the problem domain somewhat.

"Punching above your weight" and taking on an experimental or otherwise complex game design for a game development project is a good way to learn, but it can be frustrating and you can get burned out when you run into unforeseen design and implementation issues. I think this is the way most people go about independent game development, and it's how I started out when I was much younger, too. "Punching below your weight" and choosing a smaller-scoped project—ideally one with a predefined and tested design—is extremely underrated. Unless you've done something like this many times before, it might seem "beneath you" (especially if you have some ideas of the ideal long-term game project you want to work on that you're excited about), but it's actually a great way to solidify your skills (especially your design skills, as you play around with modifying an existing game design once you've implemented the basics of it), and prove to yourself that you can finish something you set out to do. Plus, in the process, you might find something unexpected in terms of either tech or design that you can take with you to your next "real" project.

My game was a simple tetris-like puzzle in 3D on Windows Phone (using C# and XNA) ;) https://www.windowscentral.com/gaming/this-square-enix-jrpg-...

I personally prefer Essential Mathematics for Games and Interactive Applications.

https://www.essentialmath.com/book.htm

This is still my go-to math book for game dev. Excellent writing style.

I'd also advise it for anyone getting into ML or LLMs, the intuition it can help you build up around the linear algebra stuff like vectors and matrices is very helpful for understanding what's going on in ML.

The cool thing about graphics programming is that often times you aren't even doing "linear algebra" strictly.

A lot of the math for 3D graphics programming uses the concepts of affine spaces/transformations since standard 3x3 matrices don't have enough information to support translation/projection. I had no clue about this branch of math at all until I started learning graphics programming--in fact I think graphics programming requires you to learn the most math of just about any discipline of computer science outside theoretical computer science. The amount of math you need to truly "understand" path tracing is immense.

Failed all my maths exams all the way through school. Us kids: "Teacher when we will we ever use this crap in real life?"

Me: *becomes 3D game developer, has to write 3D entire 3D engines from scratch*

They always say school is wasted on kids :D

Homogeneous coordinates is still linear algebra in a 4D space instead of the obvious 3 you would naively pick. Kinda like quaternions where adding an extra degree of freedom suddenly works around some expressivity problem you didn't really know you had but clearly it solves (e.g. gimble lock on euler angles). But the term linear algebra covers high dimensional uses like training neural networks, its the thing that gives us the matrix formulation of stuff.

This looks awesome! Does anyone know of something similar but for 2D games?

If you understand 3D you understand 2D, and in fact explanations for 3D usually start by explaining the equivalent in 2D first.

best tool for 3D math is complex numbers. most languages removed support for them.

Julia supports Complex numbers as a built-in type.

are numbers types?

It's certainly possible to interpret numbers as types, but it's not particularly common AFAIK, since there are generally more convenient interpretations of arithmetic for a given problem domain (von Neumann sets, Dedekind cuts, IEEE 754 floating point values, &c &c).

imaginary numbers to me are more real than real numbers, they are the numbers of geometry - i like them being first class numbers

Personally, I've always found that the non-rigidity of the complex numbers (i.e. that when considered as a field, they admit a non-trivial automorphism), which requires the introduction of the Re and Im operators (i.e. projection back to the reals) to banish, to be strongly suggestive that the reals have a somehow more "real" ontological status.

Can't say I see a lot of bolted on complex number support in graphics or game dev.

Even the posted primer suggests you can skip the section that mentions complex numbers.

What do you use them for?

reality

Ironic.

Quaternions are an extension of complex numbers and are used ubiquitously in graphics programming for both 2D and 3D applications. Most math libraries support them.

If your gripe is over programming languages not having support for 3D math, then I think that should be expected. Most languages try to keep their standard libraries small, and their built-in types smaller. A language that has built-in support for such things would be incredible niche. Maybe JAI is what you're looking for? I'm not sure if that even contains built in support however.

J and APL are what I am looking for. JAI is not my thing.

I think the first edition he wants pulped is the book that got me through GAM302 Games Programming Math lmao.

um, out of print, and the pdf is available 'soon'? What's even the point of this link then if we can't actually get the book in any way?

however it can be found at anna's archive

the book is free to read online

Easy to overlook the link above the image (at least on mobile)...

This is the direct link to the online content

https://gamemath.com/book/

The image itself links to https://gamemath.com/book/intro.html by the way.

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