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kanition committed Apr 16, 2021
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16 changes: 16 additions & 0 deletions bibliography.bib
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Expand Up @@ -170,4 +170,20 @@ @inbook{10.5555/90767.90913
booktitle = {Graphics Gems},
pages = {498–515},
numpages = {18}
}

@misc{enwiki:1014110231,
author = {{Wikipedia contributors}},
title = {Quaternion --- {Wikipedia}{,} The Free Encyclopedia},
year = {2021},
howpublished = {\url{https://en.wikipedia.org/w/index.php?title=Quaternion&oldid=1014110231}},
note = {[Online; accessed 16-April-2021]}
}

@misc{enwiki:1013104981,
author = {{Wikipedia contributors}},
title = {Division ring --- {Wikipedia}{,} The Free Encyclopedia},
year = {2021},
howpublished = {\url{https://en.wikipedia.org/w/index.php?title=Division_ring&oldid=1013104981}},
note = {[Online; accessed 16-April-2021]}
}
118 changes: 70 additions & 48 deletions content/chap02extra.tex
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\section{译者补充：四元组}\label{sec:译者补充：四元组}
\emph{本节内容不是原书内容，而是译者自学补充的，请酌情参考和斧正。}
\section{译者补充：四元数}\label{sec:译者补充：四元数}
\begin{remark}
本节内容不是原书内容，而是译者自学补充的，请酌情参考和斧正。
\end{remark}

本节内容主要依据文献\citep{10.5555/90767.90913}整理而成，
给出四元组相关数学推导，具体介绍
四元组的定义、性质及其在几何变换中的运用。
本节内容主要依据文献\citep{10.5555/90767.90913,enwiki:1014110231,enwiki:1013104981}整理而成，
给出四元数相关数学推导，具体介绍
四元数的定义、性质及其在几何变换中的运用。

\subsection{四元组的定义}\label{sub:四元组的定义}
\subsection{四元数的定义}\label{sub:四元数的定义}
\begin{definition}
\keyindex{四元组}{quaternion}{}记作
\keyindex{四元数}{quaternion}{}记作
\begin{align}
{\bm q}=c+x{\rm i}+y{\rm j}+z{\rm k}\, ,
{\bm q}=c+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\, ,
\end{align}
其中$c$，$x$，$y$，$z$是实数，${\rm i}$，${\rm j}$，${\rm k}$是虚数。
也可简记为
\begin{align}
{\bm q}=c+{\bm u}\, ,
\end{align}
其中${\bm u}=x{\rm i}+y{\rm j}+z{\rm k}$称为四元组的\keyindex{纯部}{pure part}{}，
$c$称为\keyindex{实部}{real part}{}。
\end{definition}
其中$c,x,y,z$是实数，
$\mathbf{i},\mathbf{j},\mathbf{k}$是\keyindex{基四元数}{basic quaternion}{quaternion四元数}，
也称\keyindex{基元}{basis element}{quaternion四元数}。

设$Q$为四元组集合，且在基$\{1,{\rm i},{\rm j},{\rm k}\}$上定义了加法和乘法两种运算。
这里基满足
\begin{align}
\left\{
\begin{aligned}
{\rm i}^2={\rm j}^2={\rm k}^2=-1\, , \\
{\rm ij}={\rm k},\quad {\rm ji}={\rm -k}\, , \\
{\rm jk}={\rm i},\quad {\rm kj}={\rm -i}\, , \\
{\rm ki}={\rm j},\quad {\rm ik}={\rm -j}\, .
\end{aligned}
\right.
\end{align}
四元组加法为
\begin{align}
{\bm q}+{\bm q}'=(c+c')+(x+x'){\rm i}+(y+y'){\rm j}+(z+z'){\rm k}\, .
\end{align}
乘法展开可得
\begin{align}
{\bm q}{\bm q}' & =(c+x{\rm i}+y{\rm j}+z{\rm k})(c'+x'{\rm i}+y'{\rm j}+z'{\rm k})\nonumber \\
& =(cc'-xx'-yy'-zz')+(yz'-y'z+cx'+c'x){\rm i}\nonumber \\
& \quad+(zx'-z'x+cy'+c'y){\rm j}+(xy'-x'y+cz'+c'z){\rm k}\, .
\end{align}
也可以简记为
基四元数可视为分别指向三个空间轴向的单位向量。
此时${\bm q}$可以看作由一个标量和一个向量构成：
其中$c$称为\keyindex{实部}{real part}{}或\keyindex{标量部}{scalar part}{quaternion四元数}，
$x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$称为\keyindex{虚部}{imaginary part}{quaternion四元数}、\keyindex{纯部}{pure part}{quaternion四元数}或\keyindex{向量部}{vector part}{quaternion四元数}。
\end{definition}
\begin{definition}
当$c=0$且$xyz\neq 0$时，${\bm q}$称为\keyindex{向量四元数}{vector quaternion}{quaternion四元数}。
\end{definition}
\begin{definition}
当$x=y=z=0$时，${\bm q}$称为\keyindex{标量四元数}{scalar quaternion}{quaternion四元数}。
其中，当$c=x=y=z=0$时，${\bm q}$称为\keyindex{零四元数}{zero quaternion}{quaternion四元数}，记作$0$。
\end{definition}
\begin{notation}
在实际书写时，我们做如下约定：
\begin{itemize}
\item $c,x,y,z$之一等于$0$时，略写相应项；
\item $x,y,z$之一等于$1$时，相应项简写为$\mathbf{i,j}$或$\mathbf{k}$；
\item 也可写作${\bm q}=c+{\bm u}$，其中$c$为标量，${\bm u}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$为向量；
\item 所有四元数构成的集合记作$\mathbb{H}$。
\end{itemize}
\end{notation}
\begin{remark}
因为实数域$\mathbb{R}$、向量空间$\mathbb{R}^3$分别
与$\mathbb{H}$的子集\keyindex{同构}{isomorphic}{}，
所以即便在这种记法下
实数与标量四元数、三维向量与向量四元数记号一样，
其逻辑也是自洽的。
\end{remark}
\subsection{四元数的运算}\label{sub:四元数的运算}
分别记四元数为
\begin{align}
{\bm q}{\bm q}' & =(c+{\bm u})(c'+{\bm u}')\nonumber \\
& =(cc'-{\bm u}\cdot{\bm u}')+({\bm u}\times{\bm u}'+\langle c{\bm u}'\rangle+\langle c'{\bm u}\rangle)\, ,
{\bm q} & =c+{\bm u}=c+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\, , \\
{\bm q}_1 & =c_1+{\bm u}_1=c_1+x_1\mathbf{i}+y_1\mathbf{j}+z_1\mathbf{k}\, , \\
{\bm q}_2 & =c_2+{\bm u}_2=c_2+x_2\mathbf{i}+y_2\mathbf{j}+z_2\mathbf{k}\, ,
\end{align}
其中
\begin{align*}
{\bm u}\cdot{\bm u}' & =xx'+yy'+zz'\, , \\
\langle c{\bm u}\rangle & =cx{\rm i}+cy{\rm j}+cz{\rm k}\, , \\
{\bm u}\times{\bm u}' & =(yz'-zy'){\rm i}+(zx'-xz'){\rm j}+(xy'-yx'){\rm k}\,
\end{align*}
分别为\keyindex{内积}{inner product}{}、数乘、\keyindex{叉积}{cross product}{}。
其中$c,x,y,z,c_1,x_1,y_1,z_1,c_2,x_2,y_2,z_2\in\mathbb{R}$，${\bm u},{\bm u}_1,{\bm u}_2\in\mathbb{R}_3$。
\begin{definition}
四元数\keyindex{加法}{addition}{}为
\begin{align}
{\bm q}_1+{\bm q}_2 & =(c_1+c_2)+({\bm u}_1+{\bm u}_2)\nonumber \\
& =(c_1+c_2)+(x_1+x_2)\mathbf{i}+(y_1+y_2)\mathbf{j}+(z_1+z_2)\mathbf{k}\, .
\end{align}
\end{definition}
\begin{definition}
基元$\mathbf{i},\mathbf{j},\mathbf{k}$的乘法为
\begin{align}
1\mathbf{i} & =\mathbf{i}1=\mathbf{i}, & 1\mathbf{j} & =\mathbf{j}1=\mathbf{j}, & 1\mathbf{k} & =\mathbf{k}1=\mathbf{k}\, , \\
\mathbf{i}^2 & =-1, & \mathbf{j}^2 & =-1, & \mathbf{k}^2 & =-1\, , \\
\mathbf{ij} & =\mathbf{k}, & \mathbf{jk} & =\mathbf{i}, & \mathbf{ki} & =\mathbf{j}\, , \\
\mathbf{ji} & =-\mathbf{k}, & \mathbf{kj} & =-\mathbf{i}, & \mathbf{ik} & =-\mathbf{j}\, .
\end{align}
\end{definition}
\begin{definition}
对于$\lambda\in\mathbb{R}$，四元数的\keyindex{数乘}{scalar multiplication}{}为
\begin{align}
\lambda{\bm q} & =\lambda c+\lambda {\bm u}\nonumber \\
& =\lambda c+(\lambda x)\mathbf{i}+(\lambda y)\mathbf{j}+(\lambda z)\mathbf{k}\, .
\end{align}
\end{definition}
8 changes: 5 additions & 3 deletions structure.tex
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\newcommand{\ud}{\mathop{\mathrm{{}d}}\mathopen{}}
\newcommand{\intff}[2]{\mathopen{[}#1\,;#2\mathclose{]}}
\renewcommand{\qedsymbol}{$\blacksquare$}
\newtheorem{notation}{Notation}[chapter]
\newtheorem{notation}{记号}[section]

% Boxed/framed environments
\newtheoremstyle{ocrenumbox}% Theorem style name
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\newtheorem{exerciseT}{Exercise}[chapter]
\theoremstyle{blacknumex}
\newtheorem{exampleT}{Example}[chapter]
\newtheorem{proveT}{证明}[section]
\newenvironment{prove}{\begin{proveT}}{\hfill{\tiny\ensuremath{\qedhere\blacksquare}}\end{proveT}}
\theoremstyle{blacknumbox}
\newtheorem{vocabulary}{Vocabulary}[chapter]
\newtheorem{definitionT}{定义}[section]
\newtheorem{corollaryT}[dummy]{Corollary}
\newtheorem{corollaryT}[dummy]{推论}
\theoremstyle{ocrenum}
\newtheorem{proposition}[dummy]{Proposition}
\newtheorem{proposition}[dummy]{定理}

%----------------------------------------------------------------------------------------
% DEFINITION OF COLORED BOXES
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