编程与软件工程marctenbosch.com
让我们从每个 3D 引擎中移除四元数（2018）
Let's remove quaternions from every 3D engine (2018)
2018 — 2021，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇文章主张用几何代数（Geometric Algebra）中的 Rotor 取代四元数来表示 3D 旋转，认为四元数是难以直观理解的历史遗留物，而 Rotor 具有更清晰的几何意义且数学本质相同。文章面向图形编程者，系统讲解了二重向量与 Rotor 的原理。
为什么经久不衰：它把'四元数到底为什么这样工作'这个图形学的经典困惑放在更大的几何代数框架下解释，多年后仍是理解 3D 旋转表示法的最佳入门之一。
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实际项目中全面改用几何代数代价高昂且往往不必要，团队知识壁垒是真实问题
我曾在一家游戏引擎工作，两位数学控程序员迷上了几何代数，用同样的'该取代四元数'论据把引擎的数学代码全部用 GA 重写，角色绑定系统也去掉了矩阵。结果带来几个大问题：GPU 接口全是矩阵，到处都要做转换，性能略降；全工作室只有他们俩懂 GA，还不肯教别人，其他程序员都绕着变换相关代码走；而且整个重写根本没必要——四元数解决万向节锁只是引擎里很小的一角，矩阵对 98% 的代码都工作得很好。
dahart查看英文原文
> As a side note, Geometric Algebra contains more than just Rotors, and is a very useful tool to have in one's toolbox. I worked in a game engine where a pair of mathy programmers fell in love with Geometric Algebra, and use this same argument that quarternions ought to be replaced to overhaul all of the math code in the engine using Geometric Algebra. The character rigging system removed matrices and used GA instead. This caused several large problems in the code base: For one, it slowed the code down a little because the GPU interface is all matrices, so there were conversions to matrices all over the place, rigging in particular. And only two guys in the studio knew Geometric Algebra, and they didn’t invest time in teaching it or helping people understand it, they just hoisted it on everyone. All the rest of the programmers knew matrix math but not Geometric Algebra, so they would end up avoiding touching any of the GA code, i.e., any code that dealt in transformations. The two guys ended up with a lot of support of their own creation, but they were short with their answers, in part because they got so many questions, so the problem never went away. The third problem is this whole rewrite was unnecessary. Fixing gimbal lock with quaternions is a tiny corner of the game engine, whereas using GA throughout is a massive rewrite. Matrices work really well for 98% of the code, and it’s not really a huge problem to have one or two routines that convert to quaternions and back while they do a rotation. It is a problem when any transform at all involves bivectors and rotors and you have no idea what the hell those are nor do you have time in your schedule to take a math class at work. Personally, I’m intrigued by GA and have wanted to learn it for a while, but having used it in production, I’m mildly against replacing quaternions with GA, and very wildly against replacing matrices with GA.
质疑文章实质是'用一种难懂的东西替换另一种难懂的东西'
我只粗略浏览了文章，但它似乎在说：与其使用你不理解的东西并花时间去理解它，不如使用另一种你不理解的东西，把时间花在学习那个上面。作者后文也说 3D Rotor 看起来很像四元数，代码和数学基本一样，只是 i、j、k 换成了 y∧z、x∧z、x∧y。我理解得对吗？
ColinWright查看英文原文
I've only skimmed the article, but it seems to be saying this: > Instead of using this thing you don't understand and investing the time to understand it, why don't you use this other thing you don't understand, and invest the time to learn that instead. Indeed, later in the article the author says: > We can notice that 3D Rotors look a lot like Quaternions ... In fact the code/math is basically the same! The main difference is that i, j, and k get replaced by y∧z, x∧z and x∧y, but they work mostly the same way. Have I got that right?
历史上'取代四元数'的尝试最终往往殊途同归，产出相同代码
题外话：2000 年曾有过一场争论，讨论是否应该放弃四元数、改用普通矩阵上的等价运算。争论最终平息，因为支持者在优化完自己的实现后发现，产出的代码和现有的四元数实现一模一样，唯一的区别只是推导路径。
corysama查看英文原文
Tangential: Back in 2000 there was some debate about dropping quaternions in favor of a set of equivalent operations on plain matrices. https://www.gamedev.net/articles/programming/math-and-physics/do-we-really-need-quaternions-r1199/ The debate petered out when the proponents finished optimizing their implementation and found they had produced exactly the same code as the existing quaternion implementations. The only difference was the approach used to derive it.
GA 的价值主要在楔积，几何积本身概念上站不住脚
几何代数的教材总是一开头说'我们要用某种直观、自然、优雅的东西取代向量分析'，然后立刻引入'几何积'——它在我看来既不直观也不优雅也不自然，其性质靠公式推演得出，结果像是'神奇地'成立。二重向量在概念上很棒，问题在于几何积。据我所知 GA 95% 的用处来自楔积 ∧（它被严重低估，到处以伪装形式出现），最后 5% 才是用几何积做向量旋转。
ajkjk查看英文原文
Texts on GA seem to start out saying "we're going to replace vector analysis with something intuitive and natural and eloquent and..." and then immediately introduce the "geometric product", which is neither intuitive, eloquent, or, as far as I can tell, natural. Its properties are found via formula-wrangling, and the results you get from it seem to just magically work, rather than being intuitive. Bivectors in general are conceptually great; it's just the geometric product which I think is conceptually flimsy. And it's surprisingly hard to compute or make sense of the geometric product in general which is why the writer of the first book cited on the OP wrote "I do not think it possible to give a quick definition of the general geometric product."[1] As far as I can tell 95% of the usefulness of geometric algebra is the usefulness of the wedge product ∧, which is absolutely under-appreciated and appears in loads of places in disguise (for instance, the determinant of a matrix is the wedge product of all of its rows or columns together). The last 5% of the usefulness of GA comes from using the geometric product in vector-rotation via -ava^-1, which is admittedly very useful (it's why physics and computer graphics represent rotation this way, albeit in disguised forms like Pauli matrices and quaternions.) You can express that form without the geometric product, but I haven't found a way that strikes me as very elegant. I don't mean to condemn GA - I think what it's doing is massively important. The mathematical language for vector analysis deficient compared to what we could be using, and a lot of things are more intuitive and natural in better language. I just suspect that the, uh, ideal form of this stuff will look slightly different than GA, and might not include the geometric product at all, but will definitely include the wedge product absolutely everywhere. (I've spent a lot of free time trying to figure this out but I don't really have a compelling result yet. I've been meaning to try blogging about it, though, since it's basically my favorite thing to study.) [1] https://math.stackexchange.com/questions/444988/looking-for-a-clear-definition-of-the-geometric-product
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