科学与数学eater.net编辑精选
可视化四元数：可探索视频系列
Visualizing quaternions: An explorable video series
2018 — 2025，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
这是 eater.net 上的《Visualizing Quaternions》可探索视频系列，用视频和交互式可视化讲解四元数以及三维旋转。它把抽象的四元数运算和旋转直觉拆成可视、可操作的步骤。
为什么经久不衰：四元数是游戏和图形学中常用却难有直觉的数学对象，这套可视化讲解提供了少见的可探索入口，所以多年反复被推荐。
HN 评论精选
期待有可交互配套，并讨论用 manim 做可探索版本。
唉，我看 3b1b 视频时总想要一个配套的交互套件……我知道这要求很不识好歹 ;P 我本来很希望这个就是，但它只是该系列的简明课程形式……不过也很棒！我是 Eater 和 Sanderson 全部作品的超级粉丝。3b1b 会发布生成视频的代码，我以前多次克隆过 manim；想用一个简单的 localhost Python 服务来动态生成帧，也许这周末再试试。学新数学概念时我喜欢自己写源码，但通常要等动手调模型或常数之后，才真正对底层概念有直觉。
justifier查看英文原文
oof, the one thing i am wont to want with 3b1b videos is an interactive suite accompaniment.. unreasonably ungrateful i know ;P i had high hopes this would be it, but this is just a concise lesson form for the series.. which is great! i am a huge fan of both eater's and sanderson's entire ouvre what's great is that 3b1b releases the code that generates these videos and i have cloned the manim(o) library a number of times in the past when a video had an idea i wanted to play around with but the effort usually gets a low priority and i get distracted with more pressing projects i figure a simple localhost python server serving up dynamic frames generated by manim could do the trick, maybe i'll work at it again this weekend when i want to learn a new mathematical concept i like to write the source myself, which is great for getting at the nuts and bolts but it is usually after i have done this and start to tweak the models or constants that i begin to gain a real intuitive understanding of an underlying concept (o) https://github.com/3b1b/manim
双四元数更适合描述随时间变化的 3D 空间，且与线性逻辑有神秘联系。
我知道我这么说有点烦，但双四元数更古怪。它们是推理随时间发展的三维空间的最佳形式。这里有个很酷的演示……它们似乎对应线性逻辑，这太疯狂了。
adamnemecek查看英文原文
I know I’m getting annoying with this but dual quaternions are even whackier. They are the best formalism for reasoning about 3D space developing over time. Here’s a cool demo http://www.chinedufn.com/dual-quaternion-shader-explained/ They seem to correspond to linear logic which is insanity.
通过复数基向量乘法与线性分解来建立四元数直觉。
我一直很难可视化这些高维对象。我想这是因为我其实有相当好的可视化和空间意识，但一到 4D 就崩了，所有具体能力都不那么有用。后来看了数学、更理解复数之后，我对基向量乘法有了些直觉，觉得可以通过线性分解（分配律）来扩展直觉。乘以一个复基向量就像一对旋转：一个在实数与该向量张成的平面内，另一个是“正交”于该向量、由另外两个基向量张成的旋转。这帮助我把复数中的缩放和旋转直觉迁移到四元数。
nh23423fefe查看英文原文
I always had a hard time trying to visualize these higher dimension objects. I think its because I actually have quite good visualization and spacial awareness. But going to 4d it just breaks and all the concrete ability isn't so useful. eventually after looking at the math and understanding more about complex numbers, i got some intuition for the multiplication of the basis vectors and i feel like I can sort of extend the intuition by linear decomposition (distributivity) multplication by a complex basis vector is like a pair of rotations one of the rotations is in plane spanned by the reals and vector itself and the other is a rotation "orthogonal" to the vector spanned by the 2 other basis vectors. This helped me recover the intuition around scaling and rotating from the complex numbers to quats.
游戏开发中把四元数当作轴角旋转已经足够实用。
尽管作为游戏开发者我一直使用四元数，但我从未真正花时间从数学上完全理解它们。我的捷径是把它们想成绕一个轴的旋转。一旦这么想，即使没有更深数学理解，我也能用它们应对各种用途和情境。
MattRix查看英文原文
Despite using quaternions all the time as a game developer, I’ve never really taken the time to fully understand them mathematically. My shortcut has been to think of them as a rotation around an axis. Once I thought of them that way, I was able to use them for all kinds of purposes and situations despite not having a deeper mathematical understanding.
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