数学2020arxiv.org
几何代数（Geometric Algebra）
Geometric Algebra (2012)
进阶 · 约 75 分钟读完前置知识：线性代数、向量空间、基础几何
导读
这篇论文是一篇几何代数（Geometric Algebra）导论，介绍如何用统一的代数结构描述向量、旋转等几何对象，使传统向量数学中显得不完整的概念获得更自然的表达。它偏重数学框架与概念说明，而非面向工程实现的代码示例。
HN 关注点：几何代数是否真的实用：它在简单旋转中很优雅，但在游戏引擎等工程场景中处理平移、缩放和统一变换时是否足够简单、是否有实际代码示例，以及为何迟迟没有广泛采用。
HN 评论精选
这几条评论里：质疑 3 补充 1
质疑工程实践中统一变换很复杂，GA 尚不成熟，难以替代矩阵。
GA 的主要问题是：如果想要统一的变换层级，几何代数会很快变得非常复杂，因为需要 projective GA 处理平移、dual-quaternion 处理非均匀缩放。GA 在简单旋转情形很漂亮，也能很好解释简单向量数学中不完整的概念，但在通用游戏引擎的真实应用里，你没法两小时就写出带统一变换的场景图。更别说即便 GA 圈内人对更高级的 GA 做法（奇怪的 inversion 等）也不全一致。奇怪的是 YouTube 上那么多 GA 教程视频却很少谈这个，bivector 网站论坛信息也少。相比之下，矩阵表示虽然对几何概念映射较弱，但以更统一的接口处理一切，不会太复杂。就此而言 GA 数学还远不够理想。可能存在一个尚未发现的更好版本 GA，更好处理平移和缩放且大家能立刻认同。在那之前 GA 大概不会广泛使用。
hohohmm查看英文原文
The main problem with GA is if you want a unified transformation hiearchy, geometric algebra could quickly get very complicated, as it requires projective GA to handle translation and dual-quaternion to handle non-uniform scaling. Geometry algebra appears beautiful to simple rotation cases and explains very well concepts that feel incomplete in simple vector math, but in real application for genric game engine, you can't really code up a scene graph with unified transforms in like 2 hours. Not to mention that even the GA people do not all agree with each other how to do the more advanced GA stuff(strange inversions and ext). Surprisingly very few people talk about this on Youtube with all those GA tutorial videos, and you can find scant information on the bivector site forum. In comparison, despite matrix represetntion's weaker mappng to geometric concepts, it handles everything in a more unified interface without too much complications. For that matter the GA math feels much to be desired. There probably exists a undiscovered better version of GA that handles translation and scaling better and everybody could instantly agree on. Until that day GA probably won't see much general usage.
质疑批评 GA 导论普遍缺少实际代码和应用题示例。
粗略浏览，这篇导论和我见过的所有其他 GA 导论有同样问题：完全没有实际例子。哪里有导论展示如何用 GA 写出更好的计算代码？或者做应用题，像教孩子代数/微积分那样？即便在 Foundations of Game Engine Development 第一卷这种号称实用的书里，除了 GA 章节之外全书都有代码示例。
skybrian查看英文原文
From a brief skim, it looks like this introduction has the same problem as every other introduction to geometric algebra that I've seen: a complete lack of practical examples. Where is there an introduction that shows you how to use geometric algebra to write better code to perform calculations? Or maybe to do word problems, like when we teach kids algebra or calculus? Even in a supposedly practical book like Foundations of Game Engine Development, Volume 1, there are code examples throughout the book, except for the chapter about geometric algebra.
补充补充资料：将几何代数扩展到流形微积分。
对同时喜欢 GA 和广义相对论的人：我以前的学生 Joey Schindler 写了一篇很漂亮的论文，把 GA 扩展到流形上的微积分。链接：https://arxiv.org/pdf/1911.07145.pdf
AnthonyAguirre查看英文原文
For those who happen to like both geometric algebra and general relativity, my former student Joey Schindler wrote this quite beautiful paper extending geometric algebra to calculus on manifolds. https://arxiv.org/pdf/1911.07145.pdf
质疑采用阻力在于现有工具够用、迁移收益有限，GA 难以被吸收。
好主意会很快流行，GA 不是。可能因为已有还行方案（点积/叉积），切换收益增量很小，对一年只用几次点叉积的人净成本甚至为负。范畴论是统一理论成功例子，花时间但我能较快吸收。GA 我什么都记不住——我有其他够用工具（做游戏，经常接触 3D，但传统微积分教材工具大多够用，需要高级概念时多来自微分几何）。我记得另一场大战是推动 gauge integrals 进微积分教材（作为 Riemann 积分替代/补充，极大扩展可积范围），我猜没成，但是勇敢斗争。我对叉积的突破是把它理解成奇怪但实用的工具，而不是根本美丽几何概念。两向量叉积对我没有内在意义（不认为它有柏拉图身份），但对方向/取直角/共面有用。可以想象某天教育者相信 GA 有用，事情会翻转。
jan_Inkepa查看英文原文
There are good ideas that gain traction super quickly. GA is not that. Possibly because there's already an ok solution (dot/cross-products), the benefits of switching are pretty incremental (or negative in terms of net effort for people who use dot/cross products a few times a year). Category Theory is an example of a unifying theory that caught on. It took time, but my brain was able to assimilate to it quite quickly. With GA none of it sticks for me - I have other tools that suffice (I work on games and have reasonably regular 3D exposure, but the traditional tools of calculus textbooks mostly suffice, and if I need fancy concepts they tend to come from differential geometry). I remember another big war being to push gauge integrals into Calculus textbooks (as an alternative/augmentation to Riemann integrals that vastly increase the scope of what can be integrated). I assume that went nowhere, but that was a valiant struggle. My breakthrough with the cross product was coming to understand it as a weird but practical tool rather than representing some fundamentally beautiful geometrical concept. It might be one of the most well known weird mathematical tools. The cross product of two vectors has no real meaning to me intrinsically (I don't think it has or deserves a platonic identity) but it's useful for talking about orientations/picking right-angles/planarity. I can imagine at some point educators might become convinced about the utility of GA and things flipping, but it feels far from that tipping point still.
相关资源
从这篇论文所有 HN 讨论帖的全部评论里整理出来的链接。
历次讨论
- 2020Hacker News 讨论帖147 分 · 57 评
同领域还值得读
一种新的相关系数
A New Coefficient of Correlation
提出了一种新的相关性系数（由 Sourav Chatterjee 提出），能够检测变量之间任意形式（不限于线性）的相关关系，且计算简单高效，主要基于排序和距离计算。
数学2021 · 682 分 · 158 评 · 引用 1进阶 · 约 60 分钟编辑精选数学的两种文化
The Two Cultures of Mathematics (2000)
这是数学家 Timothy Gowers 于 2000 年发表的一篇随笔，他把数学研究大致分成两种文化：一类是“解题者”，热衷于攻克具体难题、发展技巧（如组合数学）；另一类是“理…
数学2009 · 307 分 · 102 评入门 · 约 35 分钟编辑精选重新定义统计显著性
Redefine statistical significance
提议将统计显著性阈值从常用的 p<0.05 收紧到 p<0.005，以降低假阳性率、缓解可重复性危机。
数学2017 · 303 分 · 114 评入门 · 约 45 分钟编辑精选统计建模：两种文化（2001）
Statistical Modeling: The Two Cultures (2001)
讨论统计学中两种建模文化：一种假设数据由某种随机模型生成，侧重推断与系数解释；另一种把数据生成机制视为未知，侧重预测精度，主张使用算法建模。
数学2015 · 254 分 · 36 评进阶 · 约 60 分钟编辑精选通信的数学理论 (1948)
A Mathematical Theory of Communication (1948)
这是香农 1948 年的经典论文《通信的数学理论》，奠定了信息论的基础。
数学2016 · 169 分 · 53 评入门 · 约 90 分钟编辑精选