查看“︁几何代数”︁的源代码

{{for|一般代数结构|克利福德代数}}{{Distinguish|代数几何}}
[[数学]]中，'''几何代数'''（也称作实[[克利福德代数]]）是[[初等代数]]的推广，用于处理[[向量]]等几何对象。几何代数由加法与几何积两种基本运算组成，向量的乘积是更高维对象，称作[[多重向量]]。与其他处理几何对象的形式相比，几何代数在支持不同维度的对象的向量除法与加法方面具有优势。

几何积最早由[[赫尔曼·格拉斯曼]]简单提及，{{sfn|Hestenes|1986}}{{rp|6}}他的兴趣主要在于发展与之紧密相关的[[外代数]]。1878年，[[威廉·金顿·克利福德]]大大扩展了格拉斯曼的工作，形成现在所谓[[克利福德代数]]以纪念他（虽然克利福德自己称之为“几何代数”）。克利福德将克利福德代数及其积定义为[[格拉斯曼代数]]和哈密顿的[[四元数代数]]的统一。加上格拉斯曼外积的[[对偶 (数学)|对偶]]（“相遇”）就可以使用[[格拉斯曼–凯莱代数]]，后者的[[共形几何|共形版本]]与共形克利福德代数一起产生了[[共形几何代数]]（CGA），为[[克莱因几何|经典几何]]提供了框架。{{sfn|Li|2008}}{{rp|411}}实践中，这些运算和一些可派生运算可将代数的元素、子空间、运算同几种几何解释对应起来。几十年来，几何代数有些被忽视了，因为当时为描述电磁学产生的[[向量分析]]挤占了几何代数的地盘。1960年代，“几何代数”由[[大卫·黑斯廷斯]]重新发掘出来，主张其对相对论物理学的重要性。{{sfn|Hestenes|1966}}

标量和向量有其通常的解释，并构成几何代数的不同子空间。[[二重向量]]可更自然地表示[[向量分析]]中的伪向量，如有向面积、旋转的有向角度、挠、角动量与电磁场。三重向量可表示有向体积，等等。称作[[刃 (几何)|刃]]的元素可用于表示''V''的子空间，及其上的[[投影_(线性代数)#正交投影|正交投影]]。旋转与反射也可用元素表示。不同于向量分析，几何代数可自然地容纳任何维度和任何二次型，如[[相对论]]中的二次型。

几何代数在物理学中的应用有[[时空代数]]（及不太常见的[[物理空间代数]]）与[[共形几何代数]]。[[几何微积分]]是几何代数的推广，包含了[[微分]]和[[积分]]，可用于形成其他理论，如[[复分析]]和[[微分几何]]，例如用克利福德代数代替[[微分形式]]。大卫·黑斯廷斯{{sfn|Hestenes|2003}}和Chris Doran{{sfn|Doran|1994}}等人一直主张将几何代数作为[[物理学]]的主要数学框架。支持者声称，几何代数为包括[[经典力学]]、[[量子力学]]、[[电磁学]]、[[相对论]]等许多领域提供了紧凑而直观地描述。{{sfn|Lasenby|Lasenby|Doran|2000}}几何代数还被用作[[计算机图形学]]{{sfn|Hildenbrand|Fontijne|Perwass|Dorst|2004}}和[[机器人学]]的计算工具。

== 定义与符号 ==
几何代数有多种定义。黑斯廷斯最初的定义是公理化的、{{sfn|Hestenes|Sobczyk|1984}}{{rp|3–5}}“充盈着几何意义”，等价于泛克利福德代数。{{sfn|Aragón|Aragón|Rodríguez|1997}}{{rp|101}}给定[[域 (数学)|域]]''F''上的有限维向量空间''V''，并配备对称双线性形式（即内积，如欧氏或[[伪黎曼流形|洛伦兹度量]]）<math>g : V \times V \to F</math>，则[[二次型#介绍|二次空间]]<math>(V, g)</math>的'''几何代数'''是[[克利福德代数]]<math>\operatorname{Cl}(V, g)</math>，成员称作多重子或多重向量（多重向量一词更常用于指外代数的具体元素）。按领域内的通常做法，本文将只考虑实数情形，即<math>F = \R</math>。符号<math>\mathcal G(p,q)</math>（分别为<math>\mathcal G(p,q,r)</math>）将用于表示双线性形式''g''具有[[度量符号|符号]]<math>(p,q)</math>（分别是<math>(p,q,r)</math>）的几何代数。

代数中的本质积称作几何积，包含的外代数的积称作外积（更多叫[[楔积]]{{efn|几何代数的“外积”（outer product）与其他数学领域中的同名异义}}）。标准写法分别是并列（省去任何符号）和楔形<math>\wedge</math>。几何代数的上述定义是抽象的，因此我们用下面一组公理概括几何积的性质。对于多子<math>A, B, C\in \mathcal{G}(p,q)</math>，几何积具有如下性质：
* <math>AB \in \mathcal{G}(p,q)</math>（[[闭包 (数学)|封闭]]）
* <math>1A = A1 = A</math>，其中<math>1</math>是单位元（[[单位元]]的存在）
* <math>A(BC)=(AB)C</math>（[[结合律]]）
* <math>A(B+C)=AB+AC</math> and <math>(B+C)A=BA+CA</math>（[[分配律]]）
* <math>a^2 = g(a,a)1</math>，其中''a''是代数子空间''V''的任意元素。

外积具有相同的性质，只是最后一条改为<math>a \wedge a = 0,\ \forall a \in V</math>。

注意，在上述最后一个性质中，若''g''不是正定的，则实数<math>g(a,a)</math>不必是非负的。 几何积的一个重要性质是元素有乘法逆元：<math>\forall\vec{a}</math>，若<math>a^2 \ne 0 </math>，则<math>a^{-1}</math>存在，且等于<math>g(a,a)^{-1}a</math>。代数的非零元不一定有乘法逆元，例如若<math>\vec{u}\in V</math>，且使<math>u^2 = 1</math>，则元素<math>\textstyle\frac{1}{2}(1 + u)</math>既是非平凡幂等元素，也是非零[[零除子]]，于是没有逆。{{efn|给定<math display="inline">u^2 = 1</math>，可知<math display="inline">(\tfrac{1}{2}(1 + u))^2</math> <math>= \tfrac{1}{4}(1 + 2u + uu)</math> <math> = \tfrac{1}{4}(1 + 2u + 1)</math> <math> = \tfrac{1}{2}(1 + u)</math>，说明<math display="inline"> \tfrac{1}{2}(1 + u)</math>是幂等的，且<math>\tfrac{1}{2}(1 + u)(1 - u)</math> <math>= \tfrac{1}{2}(1 - uu)</math> <math>= \tfrac{1}{2}(1 - 1) = 0</math>，表明它是非零零除子。}}

通常将<math>\R</math>、''V''与其在自然[[嵌入 (数学)|嵌入]]<math>\R \to \mathcal{G}(p,q)</math>、<math>V \to \mathcal{G}(p,q)</math>下的像视作等同的。本文中，标量和向量分别指<math>\R</math>、''V''的元素（及它们在此嵌入下的像）。

=== 几何积 ===
{{main|对称双线性形式|外代数}}
[[File:GA parallel and perpendicular vectors.svg|200px|right|thumb|给定两向量''a''、''b''，若其几何积<math>ab</math>{{sfn|Hestenes|2005}}反交换，则是垂直的，因为<math>a \cdot b = 0</math>；若是交换的，则是平行的，因为<math>a \wedge b = 0</math>。]]
{{multiple image
 | left
 | footer = 实外代数中''n''次元素的几何解释：<math>n = 0</math>（有符号点）、<math>1</math>（有向线段或向量）、<math>2</math>（有向面元）、<math>3</math>（有向体积） 。''n''个向量的外积可直观视作任何''n''维形状（如''n''-[[平行六面体#超平行体|超平行体]]、''n''-[[椭球]]）；其大小（[[勒贝格测度|超体积]]）和[[定向 (向量空间)|方向]]由(n-1)维边界上的方向和内部哪一边的方向定义。{{sfn|Penrose|2007}}{{sfn|Wheeler|Misner|Thorne|1973}}{{rp|83}}

 | width1 = 220
 | image1 = N vector positive.svg
 | caption1 = 由有序向量集定义的方向
 | width2 = 220
 | image2 = N vector negative.svg
 | caption2 = 反转方向相当于对外积取负
}}
可将任意两向量''a''、''b''的几何积写成对称积与反对称积之和：
: <math>ab = \frac{1}{2} (ab + ba) + \frac{1}{2} (ab - ba)</math>

于是可以定义内积{{efn|这是[[伪欧几里得空间|伪欧几里得向量空间]]标量积的同义词，指1-向量子空间上的对称双线性形式，而不是[[赋范向量空间]]上的[[内积]]。有人会将内积推广到整个代数，但实际上对此几乎没有共识。即使是有关几何代数的文章中，这个术语也不常用。}}
: <math>a \cdot b := g(a,b),</math>
于是，对称积可写作
: <math>\frac{1}{2}(ab + ba) = \frac{1}{2} \left((a + b)^2 - a^2 - b^2\right) = a \cdot b</math>

反之，''g''完全由代数决定。反对称部分是两个向量的外积，即含[[外代数]]部分之积：
: <math>a \wedge b := \frac{1}{2}(ab - ba) = -(b \wedge a)</math>

那么从简单加法就能有：
: <math>ab=a \cdot b + a \wedge b </math>几何积的非广义或向量形式。

内外积与标准向量代数中的相应概念有关。几何上，若''a''和''b''的几何积等于其内积，则就是[[平行]]的；若等于其外积，则就是[[垂直]]的。在几何代数中，非零向量的平方都是正的，因此两向量的内积可视作标准向量代数的[[点积]]。两向量外积可用向量形成的[[平行四边形]]所包围的[[有向面积]]来表示。3维中具有正定二次型的两向量之[[叉积]]与其外积密切相关。

大多数相关几何代数的实例都具有非退化[[二次型]]。若二次型是完全退化的，则任意两向量的内积总是零，几何代数就是简单的外代数。除非另有说明，本文只讨论非退化几何代数。

外积可自然推广为代数中任意两元素之间的结合双线性算子，且满足
: <math>\begin{align}
                          1 \wedge a_i &= a_i \wedge 1 = a_i \\
  a_1 \wedge a_2\wedge\cdots\wedge a_r &= \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} \operatorname{sgn}(\sigma) a_{\sigma(1)}a_{\sigma(2)} \cdots a_{\sigma(r)},
\end{align}</math>

其中，和是对指数的所有排列，<math>\operatorname{sgn}(\sigma)</math>是排列的符号，<math>a_i</math>是向量（不是代数的一般元素）。由于代数中的每个元素都可表示为这种形式的积之和，这也就定义了代数中每对元素的外积。从定义中可以看出，外积构成[[交替代数]]。

克利福德代数的等价结构方程为{{sfn|Wilmot|1988a}}{{rp|2338}}{{sfn|Wilmot|1988b}}{{rp|2346}}
: <math> a_1 a_2 a_3 \dots a_n = \sum^{[\frac{n}2]}_{i=0} \sum_{\mu\in{}\mathcal{C}}
(-1)^k Pf(a_{\mu_1}\cdot a_{\mu_2},\dots,a_{\mu_{2i-1}} \cdot a_{\mu_{2i}})
a_{\mu_{2i+1}}\land\dots\land a_{\mu_n}</math>
其中<math>Pf(A)</math>是''A''的[[普法夫值]]，
<math>\mathcal{C} = \binom{n}{2i}</math>提供了将''n''个索引分为{{mvar|2i}}、{{mvar|n-2i}}两部分的[[组合]]<math>\mu</math>，''k''是[[组合]]的[[奇偶性 (数学)|奇偶性]]。

普法夫值为外代数提供了度量。另外，正如Claude Chevalley指出的，克利福德代数可还原为二次型为零的外代数。{{sfn|Chevalley|1991}}从几何角度看，可从[[单纯形]]出发，发展克利福德代数，来理解普法夫值所起的作用。{{sfn|Wilmot|2023}}这种推导为[[杨辉三角]]和单纯形之间提供了更好的联系，因为提供了对杨辉三角第一层一个1的解释。

=== 刃、次、规范基 ===
多重向量是''r''个线性独立向量的外积，称作一个刃（blade），次数为''r''（grade）。{{efn|次（grade）是齐性元素之次的同义词，是在[[分次代数|作为代数的次]]与外积（<math>\mathrm{Z}</math>-分次）下的次，而非在几何积下的次。{{efn|提到几何积下的分次时，文献一般只关注<math>\mathrm{Z}_2</math>-分次，即分为奇数与偶数的<math>\mathrm{Z}</math>-次。<math>\mathrm{Z}_2</math>是几何积完整的<math>\mathrm{Z}_2{}^n</math>-分次的一个子群。}}}}''r''次刃之和形成的多重向量称作（齐性）''r''次多重向量。根据公理与闭包，几何代数中的多重向量都是刃之和。

考虑''r''次线性独立向量集合<math>\{a_1,\ldots,a_r\}</math>，跨越向量空间的''r''维子空间，之后就可定义实[[对称矩阵]]（与构造[[格拉姆矩阵]]的方法相同）：
: <math>[\mathbf{A}]_{ij} = a_i \cdot a_j</math>

根据[[谱定理]]，<math>\mathbf{A}</math>可由[[正交矩阵]]<math>\mathbf{O}</math>对角化为[[对角矩阵]]<math>\mathbf{D}</math>：
: <math>\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{A}]_{kl}[\mathbf{O}^{\mathrm{T}}]_{lj}=\sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{O}]_{jl}[\mathbf{A}]_{kl}=[\mathbf{D}]_{ij}</math>

定义一组新的向量<math>\{e_1, \ldots,e_r\}</math>，称作正交基向量，是由正交矩阵变换的向量：
: <math>e_i=\sum_j[\mathbf{O}]_{ij}a_j</math>

由于正交变换保内积，所以<math>e_i\cdot e_j=[\mathbf{D}]_{ij}</math>，<math>\{e_1, \ldots, e_r\}</math>垂直。也就是说，两不同向量<math>e_i \ne e_j</math>的几何积完全由外积决定，更一般地说
: <math>\begin{array}{rl}
e_1e_2\cdots e_r &= e_1 \wedge e_2 \wedge \cdots \wedge e_r \\
&= \left(\sum_j [\mathbf{O}]_{1j}a_j\right) \wedge \left(\sum_j [\mathbf{O}]_{2j}a_j \right) \wedge \cdots \wedge \left(\sum_j [\mathbf{O}]_{rj}a_j\right) \\
&= (\det \mathbf{O}) a_1 \wedge a_2 \wedge \cdots \wedge a_r 
\end{array}</math>

于是，''r''次刃都可写作''r''个向量的外积。更一般地，若允许退化几何代数，则正交矩阵将被替换为非退化块中正交的[[分块矩阵]]，对角阵的零值项沿退化维度分布。若非退化子空间的新向量是归一化的单位向量：
: <math>\hat{e}_i=\frac{1}{\sqrt{|e_i \cdot e_i|}}e_i,</math>
则这些归一化向量必须平方为<math>\pm 1</math>。[[西尔维斯特惯性定理]]指出，沿对角阵的<math>+1</math>、<math>-1</math>的总数是不变的。推而广之，平方得<math>+1</math>的向量总数''p''、得<math>-1</math>的向量总数''q''也是不变的。（平方为零的基向量总数也不变，若允许退化情形，则可能不为零。）记此代数为<math>\mathcal{G}(p,q)</math>。例如，<math>\mathcal G(3,0)</math>是3维[[欧氏空间]]的模型，<math>\mathcal G(1,3)</math>是相对论[[时空]]。<math>\mathcal G(4,1)</math>是3维空间的[[共形几何代数]]。

索引依次递增的''n''个正交基向量的所有可能积集合，包括作为空积的<math>1</math>，构成了整个几何代数的基（类似于[[PBW定理]]）。例如，下面是几何代数<math>\mathcal{G}(3,0)</math>的基：
: <math>\{1, e_1, e_2, e_3, e_1e_2, e_2e_3, e_3e_1, e_1e_2e_3\}</math>
这样形成的基称作'''规范基'''，''V''的任何其他正交基都会产生另外的规范基。每个规范基都有<math>2^n</math>个元素，几何代数的每个多重想来那个都可表为规范基元素的线性组合。若规范基元素是<math>\{ B_i \mid i \in S \}</math>，其中''S''是索引集，则任意两多重向量的几何积是
: <math> \left( \sum_i \alpha_i B_i \right) \left( \sum_j \beta_j B_j \right) = \sum_{i,j} \alpha_i\beta_j B_i B_j .</math>

在描述只含1次元素的多重向量时，常用“<math>k</math>-向量”。高位空间中，有些这样的多重向量不能视作刃（不能分解为''k''个向量的外积）。举例来说，<math>\mathcal{G}(4,0)</math>中的<math> e_1 \wedge e_2 + e_3 \wedge e_4 </math>不能分解，不过通常情况下，代数中这类元素不能被几何解释为对象，尽管它们可能代表诸如旋转之类的几何量。只有<math>0,1,(n-1),n</math>-向量在''n''-空间中还是刃。

=== 次投影 ===
== 另见 ==
* [[克利福德代数]]
* [[格拉斯曼–凯莱代数]]
* [[时空代数]]
* [[旋量]]
* [[四元数]]
* [[物理空间代数]]
* [[泛几何代数]]

== 注释 ==
{{notelist}}

== 脚注 ==
{{reflist}}

== 参考文献 ==
{{refbegin|2}}
: ''时间顺序排列''
* {{citation |year=1844 |last=Grassmann |first=Hermann |author-link=Hermann Grassmann |title=Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert |url=https://books.google.com/books?id=bKgAAAAAMAAJ |location=Leipzig |publisher=O. Wigand |oclc=20521674 |accessdate=2023-12-31 |archive-date=2023-10-09 |archive-url=https://web.archive.org/web/20231009095552/https://books.google.com/books?id=bKgAAAAAMAAJ |dead-url=no }}
* {{cite journal |last1=Clifford |first1=Professor |title=Applications of Grassmann's Extensive Algebra |journal=American Journal of Mathematics |year=1878 |volume=1 |issue=4 |pages=350–358 |doi=10.2307/2369379|jstor=2369379 }}
* {{citation |orig-year=1957 |last=Artin |first=Emil |author-link=Emil Artin |title=Geometric algebra |series=Wiley Classics Library |publisher=Wiley |year=1988  |isbn=978-0-471-60839-4 |mr=1009557 |doi=10.1002/9781118164518 }}
* {{citation |year=1966 |last=Hestenes |first=David |author-link=David Hestenes |title=Space–time Algebra |publisher=Gordon and Breach |isbn=978-0-677-01390-9 |oclc=996371 }}
* {{citation |year=1973 |last1=Wheeler |first1=J. A. |last2=Misner |first2=C. |last3=Thorne |first3=K. S. |title=Gravitation |publisher=W.H. Freeman |isbn=978-0-7167-0344-0|title-link=Gravitation (book) }}
* {{citation |year=1980 |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Eléments de Mathématique. Algèbre |publisher=Hermann |chapter=Ch. 9 "Algèbres de Clifford" |isbn=9782225655166 }}
* {{cite book |year=1984 |last1=Hestenes |first1=David |author-link1=David Hestenes |last2=Sobczyk |first2=Garret |title=Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics |url=https://archive.org/details/cliffordalgebrat0000hest |publisher=Springer Netherlands |isbn=9789027716736 |mode=cs2 }}
* {{citation |year=1986 |last1=Hestenes |first1=David |title=Clifford Algebras and Their Applications in Mathematical Physics |chapter=A Unified Language for Mathematics and Physics |author-link1=David Hestenes |editor1=J.S.R. Chisholm |editor2=A.K. Commons |doi=10.1007/978-94-009-4728-3_1 |isbn=978-94-009-4728-3 |publisher=Springer |series=NATO ASI Series (Series C) |volume=183|pages=1–23 }}
* {{citation |year=1988a|last=Wilmot|first=G.P.|title=The Structure of Clifford algebra. Journal of Mathematical Physics|volume=29|pages=2338–2345 }}
* {{citation |year=1988b|last=Wilmot|first=G.P.|title=Clifford algebra and the Pfaffian expansion|journal=Journal of Mathematical Physics|volume=29|pages=2346–2350|doi=10.1063/1.528118 }}
* {{citation |year=1991 |last=Chevalley |first=Claude |title=The Algebraic Theory of Spinors and Clifford Algebras, Collected Works |publisher=Springer |volume=2 |isbn=3-540-57063-2 }}
* {{citation |year=1994 |last=Doran |first=Chris J. L. |author-link=Chris J. L. Doran |title=Geometric Algebra and its Application to Mathematical Physics |type=PhD thesis |publisher=[[University of Cambridge]] |oclc=53604228 |doi=10.17863/CAM.16148 |hdl=1810/251691}}
* {{citation |orig-year=1996 |editor=Baylis, W. E. |title=Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering |publisher=[[Birkhäuser]] |isbn=9781461241058 |year=2011 }}
* {{citation |year=1997 |last1=Aragón |first1=G. |last2=Aragón |first2=J.L. |last3=Rodríguez |first3=M.A. |title=Clifford Algebras and Geometric Algebra |journal=Advances in Applied Clifford Algebras |volume=7 |issue=2 |pages=91–102 |doi=10.1007/BF03041220|s2cid=120860757 }}
* {{citation |year=1999 |last=Hestenes |first=David |author-link=David Hestenes |title=New Foundations for Classical Mechanics |edition=2nd |publisher=Springer Verlag | isbn=978-0-7923-5302-7 }}
* {{citation |year=2000 |last1=Lasenby |first1=Joan |last2=Lasenby |first2=Anthony N. |last3=Doran |first3=Chris J. L. |author-link3=Chris J. L. Doran |title=A Unified Mathematical Language for Physics and Engineering in the 21st Century |url=http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/00RSocMillen.pdf |archive-url=https://web.archive.org/web/20150319085814/http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/00RSocMillen.pdf |archive-date=2015-03-19 |url-status=live |journal=[[Philosophical Transactions of the Royal Society A]] |volume=358 |issue=1765 |pages=21–39 |bibcode=2000RSPTA.358...21L |doi=10.1098/rsta.2000.0517|s2cid=91884543 }}
* {{citation |year=2002 |last=Baylis |first=W. E. |title=Electrodynamics: A Modern Geometric Approach |edition=2nd |publisher=[[Birkhäuser]] |isbn=978-0-8176-4025-5 }}
* {{citation |year=2002 |last=Dorst |first=Leo |chapter=The Inner Products of Geometric Algebra | publisher=[[Birkhäuser]] |pages=35–46 |editor-last=Dorst |editor-first=L. |editor2-last=Doran |editor2-first=C. |editor3-last=Lasenby |editor3-first=J. |title=Applications of Geometric Algebra in Computer Science and Engineering |isbn=978-1-4612-0089-5 |doi=10.1007/978-1-4612-0089-5_2 }}
* {{citation |year=2003 |last1=Doran |first1=Chris J. L. |author-link=Chris J. L. Doran |last2=Lasenby |first2=Anthony N. |title=Geometric Algebra for Physicists |url=http://assets.cambridge.org/052148/0221/sample/0521480221WS.pdf |archive-url=https://web.archive.org/web/20090106022223/http://assets.cambridge.org/052148/0221/sample/0521480221ws.pdf |archive-date=2009-01-06 |url-status=live |publisher=Cambridge University Press |isbn=978-0-521-71595-9 }}
* {{citation |year=2003 |last=Hestenes |first=David |author-link=David Hestenes |title=Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics |journal=Am. J. Phys. |volume=71 |issue=2 |pages=104–121 |url=http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf |archive-url=https://web.archive.org/web/20100617123323/http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf |archive-date=2010-06-17 |url-status=live
 |bibcode=2003AmJPh..71..104H |doi=10.1119/1.1522700 |citeseerx=10.1.1.649.7506 }}
* {{citation |year=2004 |last1=Hildenbrand |first1=Dietmar |last2=Fontijne |first2=Daniel |last3=Perwass |first3=Christian |last4=Dorst |first4=Leo |journal=Proceedings of Eurographics 2004 |title=Geometric Algebra and its Application to Computer Graphics |url=http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf |archive-url=https://web.archive.org/web/20150906043232/http://www.gaalop.de/dhilden_data/CLUScripts/eg04_tut03.pdf |archive-date=2015-09-06 |url-status=live |doi=10.2312/egt.20041032 }}
* {{citation |year=2005 |last1=Hestenes |first1=David |title=Introduction to Primer for Geometric Algebra |url=http://geocalc.clas.asu.edu/html/IntroPrimerGeometricAlgebra.html |accessdate=2023-12-31 |archive-date=2023-06-09 |archive-url=https://web.archive.org/web/20230609085238/http://geocalc.clas.asu.edu/html/IntroPrimerGeometricAlgebra.html |dead-url=no }}
* {{cite book |last1=Selig |first1=J.M. |year=2005 |title=Geometric Fundamentals of Robotics |url=http://link.springer.com/10.1007/b138859 |publisher=Springer New York |isbn=978-0-387-20874-9 |series=Monographs in Computer Science |location=New York, NY |language=en |doi=10.1007/b138859 }}
* {{citation |year=2006 |last=Bain |first=J. |title=The ontology of spacetime |editor=[[Dennis Dieks]] |section=Spacetime structuralism: §5 Manifolds ''vs.'' geometric algebra |page=54 ''ff'' |isbn=978-0-444-52768-4 |publisher=Elsevier |url=https://books.google.com/books?id=OI5BySlm-IcC&pg=PT72 |accessdate=2023-12-31 |archive-date=2023-10-09 |archive-url=https://web.archive.org/web/20231009095112/https://books.google.com/books?id=OI5BySlm-IcC&pg=PT72 |dead-url=no }}
* {{citation |year=2007 |last1=Dorst |first1=Leo |last2=Fontijne |first2=Daniel |last3=Mann |first3=Stephen |title=Geometric algebra for computer science: an object-oriented approach to geometry |url=http://www.geometricalgebra.net/ |publisher=Elsevier |isbn=978-0-12-369465-2 |oclc=132691969 |accessdate=2023-12-31 |archive-date=2020-05-04 |archive-url=https://web.archive.org/web/20200504112116/http://www.geometricalgebra.net/ |dead-url=no }}
* {{citation |year=2007 |last=Penrose |first=Roger |author-link=Roger Penrose |title=The Road to Reality |publisher=Vintage books |isbn=978-0-679-77631-4 |title-link=The Road to Reality }}
* {{citation |year=2008 |last1=Francis |first1=Matthew R. |last2=Kosowsky |first2=Arthur |title=The Construction of Spinors in Geometric Algebra |journal= [[Annals of Physics]] |volume=317 |issue=2 |pages=383–409 |arxiv=math-ph/0403040v2  |bibcode=2005AnPhy.317..383F |doi=10.1016/j.aop.2004.11.008|s2cid=119632876 }}
* {{citation |year=2008 |last=Li |first=Hongbo |title=Invariant Algebras and Geometric Reasoning |publisher=World Scientific |isbn=9789812770110 |url=https://books.google.com/books?id=bUEcbyfW55YC }}. [https://web.archive.org/web/20110629144727/http://www.worldscibooks.com/etextbook/6514/6514_chap01.pdf Chapter 1] as PDF
* {{citation |year=2008 |last=Vince |first=John A. |title=Geometric Algebra for Computer Graphics |publisher=Springer |isbn=978-1-84628-996-5 }}
* {{cite arXiv |year=2009 |last1=Lundholm |first1=Douglas |last2=Svensson |first2=Lars |title=Clifford Algebra, Geometric Algebra and Applications |eprint=0907.5356v1 |mode=cs2 |class=math-ph }}
* {{citation |year=2009 |last=Perwass |first=Christian |title=Geometric Algebra with Applications in Engineering |volume=4 |publisher=Springer Science & Business Media |isbn=978-3-540-89068-3 |bibcode=2009gaae.book.....P |doi=10.1007/978-3-540-89068-3 |series=Geometry and Computing }}
* {{cite journal |year=2000 |last=Selig |first=J.M. |title=Clifford algebra of points, lines and planes |url=https://www.cambridge.org/core/product/identifier/S0263574799002568/type/journal_article |journal=Robotica |language=en |volume=18 |issue=5 |pages=545–556 |doi=10.1017/S0263574799002568 |s2cid=28929170 |issn=0263-5747 }}
* {{citation |year=2010 |last=Bayro-Corrochano |first=Eduardo |title=Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action |publisher=Springer Verlag |isbn=9781848829299 }}
* {{citation |year=2010 |editor-last=Bayro-Corrochano |editor-first=E. |editor-last2=Scheuermann |editor-first2=Gerik |title=Geometric Algebra Computing in Engineering and Computer Science |publisher=Springer |url=https://books.google.com/books?id=MaVDAAAAQBAJ |isbn=9781849961080 |accessdate=2023-12-31 |archive-date=2023-10-09 |archive-url=https://web.archive.org/web/20231009134639/https://books.google.com/books?id=MaVDAAAAQBAJ |dead-url=no }} Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html {{Wayback|url=http://geocalc.clas.asu.edu/html/UAFCG.html |date=20230609080529 }} #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
* {{citation |year=2010 |last=Goldman |first=Ron |title=Rethinking Quaternions: Theory and Computation |publisher=Morgan & Claypool |isbn=978-1-60845-420-4 |at=Part III. Rethinking Quaternions and Clifford Algebras}}
* {{citation |year=2011 |last1=Dorst |first1=Leo. |last2=Lasenby |first2=Joan |title=Guide to Geometric Algebra in Practice |publisher=Springer |isbn=9780857298119 }}
* {{citation |year=2011 |last=Macdonald |first=Alan |title=Linear and Geometric Algebra |publisher=CreateSpace |url=http://faculty.luther.edu/~macdonal/laga/ |isbn=9781453854938 |oclc=704377582 |accessdate=2023-12-31 |archive-date=2024-03-10 |archive-url=https://web.archive.org/web/20240310010558/http://www.faculty.luther.edu/~macdonal/laga/ |dead-url=no }}
* {{citation |year=2011 |last=Snygg |first=John |title=A New Approach to Differential Geometry using Clifford's Geometric Algebra |publisher=Springer  |isbn=978-0-8176-8282-8 }}
* {{citation |year=2012 |last=Hildenbrand |first=Dietmar |title=Foundations of Geometric Algebra computing |journal=Numerical Analysis and Applied Mathematics Icnaam 2012: International Conference of Numerical Analysis and Applied Mathematics |volume=1479 |issue=1 |pages=27–30 |isbn=978-3-642-31793-4 |bibcode=2012AIPC.1479...27H |doi=10.1063/1.4756054 |series=AIP Conference Proceedings |citeseerx=10.1.1.364.9400 }}
* {{citation |year=2014 |last=Bromborsky |first=Alan |title=An introduction to Geometric Algebra and Calculus |url=http://mathschoolinternational.com/Math-Books/Books-Geometric-Algebra/Books/An-Introduction-to-Geometric-Algebra-and-Calculus-by-Alan-Bromborsky.pdf |archive-url=https://web.archive.org/web/20191015203752/http://mathschoolinternational.com/Math-Books/Books-Geometric-Algebra/Books/An-Introduction-to-Geometric-Algebra-and-Calculus-by-Alan-Bromborsky.pdf |archive-date=2019-10-15 |url-status=live }}
* {{citation |year=2014 |last=Klawitter |first=Daniel |title=Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics |publisher=Springer |isbn=9783658076184 }}
* {{citation |year=2015 |last=Kanatani |first=Kenichi |title=Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics  |publisher=CRC Press |isbn=9781482259513 }}
* {{cite arXiv |year=2015 |last1=Li |first1=Hongbo |last2=Huang |first2=Lei |last3=Shao |first3=Changpeng |last4=Dong |first4=Lei |title=Three-Dimensional Projective Geometry with Geometric Algebra |eprint=1507.06634v1 |mode=cs2 |class=math.MG }}
* {{cite journal |year=2016 |last=Hestenes |first=David |title=The Genesis of Geometric Algebra:A Personal Retrospective |journal=Advances in Applied Clifford Algebras |volume=27 |issue=1 |pages=351–379 |doi=10.1007/s00006-016-0664-z |s2cid=124014198 }}
* {{citation |year=2016 |last=Dorst |first=Leo |title=3D Oriented Projective Geometry Through Versors of <math>\mathbf{R}^{3,3}</math> |publisher=Springer |isbn=9783658076184 }}
* {{cite book |year=2016 |last=Lengyel |first=Eric |title=Foundations of game engine development |series=Mathematics |volume=1 |location=Lincoln, California |publisher=Terathon Software LLC |isbn=978-0-9858117-4-7 }}
* {{citation |year=2016 |last1=Vaz |first1=Jayme |last2=da Rocha |first2=Roldão |title=An Introduction to Clifford Algebras and Spinors |publisher=Oxford University Press |bibcode=2016icas.book.....V |isbn=978-0-19-878292-6 }}
* {{cite journal |year=2017 |last1=Du |first1=Juan |last2=Goldman |first2=Ron |last3=Mann |first3=Stephen |title=Modeling 3D Geometry in the Clifford Algebra R(4,4) |journal=Advances in Applied Clifford Algebras |volume=27 |issue=4 |pages=3039–3062 |doi=10.1007/s00006-017-0798-7 |s2cid=126166668 }}
* {{cite book |year=2018 |last=Bayro-Corrochano |first=Eduardo |series=Geometric Algebra Applications |volume=I |title=Computer Vision, Graphics and Neurocomputing|url=https://books.google.com/books?id=SSVhDwAAQBAJ|publisher=Springer|isbn=978-3-319-74830-6 }}
* {{cite thesis |year=2018 |last= Breuils |first=Stéphane |title=Structure algorithmique pour les opérateurs d'Algèbres Géométriques et application aux surfaces quadriques |type=PHD |publisher=université-paris-est |url= http://monge.univ-mlv.fr/~breuils/manuscritBreuils.pdf |archive-url=https://web.archive.org/web/20190714145624/http://monge.univ-mlv.fr/~breuils/manuscritBreuils.pdf |archive-date=2019-07-14 |url-status=live }}
* {{cite book |year=2018 |first1=Carlile |last1=Lavor |first2=Sebastià |last2=Xambó-Descamps |first3=Isiah |last3=Zaplana |title=A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry |url=https://books.google.com/books?id=avBjDwAAQBAJ&pg=PA117 |publisher=Springer |isbn=978-3-319-90665-2 |pages=1– }}
* {{cite journal |year=2018 |last1=Hrdina |first1=Jaroslav |last2=Návrat |first2=Aleš |last3=Vašík |first3=Petr |title=Geometric Algebra for Conics |url=https://doi.org/10.1007/s00006-018-0879-2 |journal=Advances in Applied Clifford Algebras |language=en |volume=28 |issue=3 |pages=66 |doi=10.1007/s00006-018-0879-2 |s2cid=253589145 |issn=1661-4909 }}
* {{cite book |year=2019 |first1=Miroslav |last1=Josipović |title=Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics |url=https://books.google.com/books?id=4uG_DwAAQBAJ |publisher=Springer International Publishing;Birkhäuser |isbn=978-3-030-01756-9 |pages=256 }}
* {{citation |year=2020 |last1=Hadfield |first1=Hugo |last2=Lasenby |first2=Joan |title=Constrained Dynamics in Conformal and Projective Geometric Algebra |url=http://dx.doi.org/10.1007/978-3-030-61864-3_39 |work=Advances in Computer Graphics |pages=459–471 |access-date=2023-10-03 |place=Cham |publisher=Springer International Publishing |isbn=978-3-030-61863-6 |series=Lecture Notes in Computer Science |volume=12221 |doi=10.1007/978-3-030-61864-3_39 |s2cid=224820480 }}
* {{cite web |year=2023 |last=Wilmot |first=G.P. |title=The Algebra Of Geometry |website=[[GitHub]] |url=https://github.com/GPWilmot/geoalg}}
{{refend}}

== 外部链接 ==
{{Wikibooks|Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space}}
{{Wikiversity|Investigating 3D geometric algebra}}
{{refbegin}}
* [http://faculty.luther.edu/~macdonal/GA&GC.pdf A Survey of Geometric Algebra and Geometric Calculus] {{Wayback|url=http://faculty.luther.edu/~macdonal/GA%26GC.pdf |date=20230728222046 }} [http://faculty.luther.edu/~macdonal/ Alan Macdonald] {{Wayback|url=http://faculty.luther.edu/~macdonal/ |date=20240312181817 }}, Luther College, Iowa.
* [http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf Imaginary Numbers are not Real – the Geometric Algebra of Spacetime] {{Wayback|url=http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf |date=20231009095333 }}. Introduction (Cambridge GA group).
* [http://geometry.mrao.cam.ac.uk/category/lecture/ Geometric Algebra 2015, Masters Course in Scientific Computing] {{Wayback|url=http://geometry.mrao.cam.ac.uk/category/lecture/ |date=20231009094745 }}, from Dr. Chris Doran (Cambridge).
* [http://www.iancgbell.clara.net/maths/ Maths for (Games) Programmers: 5 – Multivector methods] {{Wayback|url=http://www.iancgbell.clara.net/maths/ |date=20240212095101 }}. Comprehensive introduction and reference for programmers, from [[Ian Bell (programmer)|Ian Bell]].
* [http://www.visgraf.impa.br/Courses/ga/ IMPA Summer School 2010] {{Wayback|url=http://www.visgraf.impa.br/Courses/ga/ |date=20231014130139 }} Fernandes Oliveira Intro and Slides.
* [https://web.archive.org/web/20110722080341/http://sinai.apphy.u-fukui.ac.jp/gcj/pubs.html University of Fukui] E.S.M. Hitzer and Japan GA publications.
* [https://groups.google.com/group/geometric_algebra Google Group for GA] {{Wayback|url=https://groups.google.com/group/geometric_algebra |date=20121104140601 }}
* [http://www.jaapsuter.com/geometric-algebra/ Geometric Algebra Primer] {{Wayback|url=http://www.jaapsuter.com/geometric-algebra/ |date=20240418101815 }} Introduction to GA, Jaap Suter.
* [http://bleyer.org/dw/doku.php?id=geometric_algebra Geometric Algebra Resources] {{Wayback|url=http://bleyer.org/dw/doku.php?id=geometric_algebra |date=20240228110049 }} curated wiki, Pablo Bleyer.
* [https://mat-web.upc.edu/people/sebastia.xambo/A18/EP.pdf Applied Geometric Algebras in Computer Science and Engineering 2018] {{Wayback|url=https://mat-web.upc.edu/people/sebastia.xambo/A18/EP.pdf |date=20190506143837 }} Early Proceedings
* [https://bivector.net/  GAME2020] {{Wayback|url=https://bivector.net/ |date=20240611025134 }} Geometric Algebra Mini Event
* [https://www.youtube.com/playlist?list=PLsSPBzvBkYjyWv5wLVV7QfeS_d8pwCPv_ AGACSE 2021 Videos] {{Wayback|url=https://www.youtube.com/playlist?list=PLsSPBzvBkYjyWv5wLVV7QfeS_d8pwCPv_ |date=20231009095114 }}
'''English translations of early books and papers'''
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/combebiac_-_tri-quaternions.pdf G. Combebiac, "calculus of tri-quaternions"] {{Wayback|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/combebiac_-_tri-quaternions.pdf |date=20240317010547 }} (Doctoral dissertation)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/markic_-_tri_and_quadri-quaternions.pdf M. Markic, "Transformants: A new mathematical vehicle. A synthesis of Combebiac's tri-quaternions and Grassmann's geometric system. The calculus of quadri-quaternions"] {{Wayback|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/markic_-_tri_and_quadri-quaternions.pdf |date=20240629153239 }}
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf C. Burali-Forti, "The Grassmann method in projective geometry"] {{Wayback|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf |date=20181005151509 }} A compilation of three notes on the application of exterior algebra to projective geometry
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"] {{Wayback|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf |date=20181005151456 }} Early book on the application of Grassmann algebra
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/grassmann_-_mechanics_and_extensions.pdf H. Grassmann, "Mechanics, according to the principles of the theory of extension"] {{Wayback|url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/grassmann_-_mechanics_and_extensions.pdf |date=20230326030849 }} One of his papers on the applications of exterior algebra.

'''研究组'''
* [https://web.archive.org/web/20110722080105/http://sinai.apphy.u-fukui.ac.jp/gcj/gc_int.html Geometric Calculus International]. Links to Research groups, Software, and Conferences, worldwide.
* [https://web.archive.org/web/20011129095049/http://www.mrao.cam.ac.uk/~clifford/ Cambridge Geometric Algebra group]. Full-text online publications, and other material.
* [https://web.archive.org/web/20050603081056/http://www.science.uva.nl/ga/ University of Amsterdam group]
* [http://geocalc.clas.asu.edu/ Geometric Calculus research & development] {{Wayback|url=http://geocalc.clas.asu.edu/ |date=20230627094458 }} (Arizona State University).
* [http://gaupdate.wordpress.com/ GA-Net blog] {{Wayback|url=http://gaupdate.wordpress.com/ |date=20240503191422 }} and [https://web.archive.org/web/20110722080156/http://sinai.apphy.u-fukui.ac.jp/GA-Net/archive/index.html newsletter archive]. Geometric Algebra/Clifford Algebra development news.
* [http://www.gdl.cinvestav.mx/edb/ Geometric Algebra for Perception Action Systems. Geometric Cybernetics Group] {{Wayback|url=http://www.gdl.cinvestav.mx/edb/ |date=20180712042836 }} (CINVESTAV, Campus Guadalajara, Mexico).
{{refend}}

{{线性代数}}

{{DEFAULTSORT:几何代数}}
[[Category:几何代数a| ]]