Basic skeletal animation with geometric algebra

https://www.jeremyong.com/klein/case_studies/ga_skeletal_animation/

by atomlib • 6 years ago

170 36 6 years ago

modeless5• 6 years ago

Geometric algebra looks cool, and this library seems like exactly what is needed for graphics programmers to use it.

What I'd really like to see is a comprehensive cheat sheet with formulas for common tasks in computer graphics with an absolute minimum of theory and jargon; even less than this article. Of course it's great to learn the theory too, but most people just call slerp in a quaternion library without learning the theory of quaternions. It should be possible to use a geometric algebra library in a similar way.

iod• 6 years ago

Maybe you want to check out a javascript implementation of GA math with Ganja.js. While not yet comprehensive to all the tasks you probably would want to see, it does have some good examples that you can play around with

https://enkimute.github.io/ganja.js/examples/coffeeshop.html...

jiggawatts3• 6 years ago

That library pisses me off on a visceral level.

The author "Enki Mute" is no doubt a brilliant mathematician, his library looks awesome, the online demos are unreal, but he's squandered his talent.

The code he writes is unreadable gibberish.

Zero useful comments. One-character variable names. Stringly-typed programming. Regex soup. It's psychotic.

I tried to reverse-engineer it in order to write a SIMD-optimised Rust version. But fuck me. It's like disassembling the machine code of an obfuscated anti-cheat library. Even with good tooling and industrial-strength refactoring, I couldn't make enough sense of it to make any useful headway.

Just try and read this. Seriously: https://github.com/enkimute/ganja.js/blob/master/ganja.js

What saddens me is that every other library I looked at was ancient and no longer maintained. Half wouldn't even compile. Most started with an explicit assumption that only 2D, 3D, or non-degenerate metrics are needed. Very few have efficient code-generation, and those that do are invariably C++ only, but I'm never going back there, even at gunpoint. Just shoot me now, I'd rather eat a bullet than face another STL compilation error that I have to decode from a linker error to do with __malloc() or some garbage I don't care about.

Enki Mute's Ganja.js is the only recent GA library I've seen that ticks all the checkboxes, but when I opened it up, I faced only the unspeakable horror of yet another dead end.

GA is going nowhere because everyone cooks up their own special version, the terminology is unique and special to each researcher, and most libraries stop just short of being useful.

What we need is standardisation around a well-documented and extensible library with polyglot code-gen. Something with expression simplification, SIMD, and both maths-centric and 3D graphics optimised capabilities.

Ganja.js is not it, unfortunately.

modeless5• 6 years ago

Ganja.js is exactly the opposite direction from what I want. The generality is not useful, the performance will be abysmal for graphics code, and the non-standard syntax is totally bonkers. It's a really cool experiment but not suitable for production use.

ninepoints10• 6 years ago

Original author here, it is absolutely my goal that the library should be usable without needing to understand the ins and outs of GA. To this end, I've started adding a bunch of "helper" functions that do tasks like projecting a point onto a plane, or identifying the line through a pointer parallel to another line, etc.

I've been (slowly) working on additional documentation in the meantime and didn't expect to see the animation article hit the frontpage :). That said, in the meantime, there is a GREAT cheatsheet for all the relevant formulae here: https://bivector.net/tools.html (scroll down to the bottom). You can get the PDF version here: https://bivector.net/3DPGA.pdf

auggierose2• 6 years ago

I was wondering, once you've abstracted from the GA bits enough, is there any point of using GA at all? I like GA as well, just wondering what it gives you once you've abstracted it away. Or in other words, are there aspects of GA that cannot be abstracted away like that?

modeless5• 6 years ago

Yes, something like that cheat sheet but with code instead of formulas. And more helper functions would be great.

ninepoints10• 6 years ago

I'm the author of Klein and was surprised to see a sudden burst of traffic so I checked the usual suspects and ended up here :)

My goals for authoring Klein was to provide a library for performing all manner of geometric operations using the language of Geometric Algebra. The benefits are that the formulism is exception-free (meaning, for example, parallel planes have a well-defined line of intersection, that projections to higher/lower grade entities make sense, etc), and works equally well on points, lines, and planes. Practitioners coming from robotics/animation will also find the entirety of the Lie algebra/group inside GA as well for smooth interpolation, etc.

As a graphics engineer, I found most libraries out there unsuitable because of performance reasons. We're used to having nicely packed SIMD optimized quaternions for example. Klein fills this gap and (hopefully) in a manner where you can use it even if you don't understand the Geometric Algebra (yet!). Over time, I'd like to round out the documentation to explain the underlying theory, which I find unreasonably elegant and has shifted my thinking in the last few years.

Let me know if you have any questions about Klein or geometric algebra!

throwaway17_172• 6 years ago

I saw a post in the last month or so on the front page of HN that linked a video lecture and a paper/presentation on GA, it’s still sitting in my watch/read folder unfortunately. So, since you volunteered for questions, I’ll ask a few:

- is there a generally accepted, or just personally approved, set of introductory materials on GA, to get one up to speed? I have a decent mathematics background, but I have been focusing on other areas recently so I am a bit rusty.

- I think your goals for Klein look interesting, and since you are a graphics engineer, I assume you are aware of the prior art (your comment indicates this is correct), so, is the main draw of using GA as an implementation theory the elegance of the underlying math alone, or does the math create a new space for potential optimization in the graphics space (I am assuming that your mention of performance penalty in existing implementations is due to a lack of focus on performance not unsuitability of GA to a performance domain).

throwaway17_172• 6 years ago

As a quick note: I think your last comment up thread answered my second question reasonably well, it posted while I was writing my comment. But any additional details you want to share will be welcome.

ninepoints10• 6 years ago

Thanks for the questions.

re: introductory materials

The 3 links I compiled here (https://www.jeremyong.com/klein/references/) I consider an absolute must for starting to consume GA. The book (ref #3) has one chapter undergoing revision regarding the usefulness of the degenerate metric for modeling projective geometry. My understanding is the chapter will be released for free once finished. From this alone, I was able to piece together enough to write one working implementation of GA that I was able to use to verify the computations and my understanding. Klein is my third GA library which I wrote once I was pretty sure I was comfortable with the formalism. Another great set of resources is anything posted on bivector.net.

re: question 2

The elegance of the math does help me identify better optimizations in a number of cases, but in other cases, it makes a whole series of operations well-defined (for which there was no good analog before). I posted some examples in another comment, but essentially, quats and dual-quats generally only work on points, and in a somewhat awkward manner. I can't easily compose 2 rotations, a translation, a dual quat, and a quat, and then expect it to work on a line for example. In GA, this is ... completely trivial. And because it all fits nicely in the framework, all the above benefits from the same SIMD optimizations.

I believe the reason GA was not considered well-suited for production/perf reasons is twofold. First, prior art focused on GA with arbitrary dimensionality/metric (results in either increased compilation time, or reduced performance). Second, there wasn't much overlap between practicitioners of GA, and graphics/animation engineers on the other side. I view Klein as "low-hanging fruit" in that sense, and consider myself fortunate to have come across the theory in the manner I did. I had written it off as just "an easier to understand Quaternion," but once I realized how much I (and perhaps some of my colleagues) were missing, I opted to actually try and productionize GA to "fill the gap" so to speak.

(FYI tried to send this over an hour ago but HN keeps rate limiting me :( )

bufferoverflow• 6 years ago

How do you write an article on skeletal animation, and have zero examples of what that skeletal animation looks like?

ninepoints10• 6 years ago

Working on them! That said though, the audience of this article was originally meant to be animation engineers that have already "earned some stripes" and wanted to see the GA formulation.

uglycoyote3• 6 years ago

I'm in the intended audience group then, as a game developer who works on animation regularly but only has a passing familiarity with geometric algebra. You could say I am "GA curious". But I'm afraid I didn't get much out of that article. I had hoped I would get more information about the benefits of using GA over standard matrix and quaternion methods. Instead, at the first mention of something GA related (which was motors) it was glossed over, as a hand wavy "don't worry about this". Later, pages of difficult math (which I skimmed over trying to find the punch line) were devoted to, if I understood, trying to do the equivalent of slerp that everyone is familiar with and is easy with quaternions (though perhaps just as complicated to derive). What I would have liked to have seen is an article that starts off with some motivation about what benefit one gets from looking at skeletal animation through this lense, for example, are there problems that vex people using standard methods which just don't occur when using GA?

modeless5• 6 years ago

Might I suggest some WebGL demos? You should be able to compile Klein with emscripten fairly easily.

enkimute2• 6 years ago

Just a heads up - @ninepoints let me know he got rate limited and will answer as soon as allowed again ;)

duchess_toffee• 6 years ago

Ninepoints is getting repeatedly rate-limited. He invites anyone with questions to ask questions in the Klein discord (https://discord.gg/gkbfnNy)

ninepoints10• 6 years ago

The mods emailed me a couple hours ago and said my account was mistakenly flagged as a "spam" account but its been since corrected. Thanks!

Basic skeletal animation with geometric algebra

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