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class="vp-sidebar-children"><!--[--><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/turbolink" aria-label="TurboLink" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->TurboLink<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading active collapsible">学习笔记 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学相关 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">图形学 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item active collapsible">数学基础 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_4-仿射变换" aria-label="4. 仿射变换"><!--[--><!--[--><!--]--><!--]-->4. 仿射变换<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_4-1-齐次坐标-homogeneous-coordinates" aria-label="4.1 齐次坐标(Homogeneous Coordinates)"><!--[--><!--[--><!--]--><!--]-->4.1 齐次坐标(Homogeneous Coordinates)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_4-2-齐次矩阵表达" aria-label="4.2 齐次矩阵表达"><!--[--><!--[--><!--]--><!--]-->4.2 齐次矩阵表达<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-平移变换" aria-label="5. 平移变换"><!--[--><!--[--><!--]--><!--]-->5. 平移变换<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_5-1-定义" aria-label="5.1 定义"><!--[--><!--[--><!--]--><!--]-->5.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_6-组合变换" aria-label="6. 组合变换"><!--[--><!--[--><!--]--><!--]-->6. 组合变换<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_6-1-定义" aria-label="6.1 定义"><!--[--><!--[--><!--]--><!--]-->6.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_6-2-坐标系变换" aria-label="6.2 坐标系变换"><!--[--><!--[--><!--]--><!--]-->6.2 坐标系变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="几何变换-二" tabindex="-1"><a class="header-anchor" href="#几何变换-二"><span>几何变换（二）</span></a></h1><hr><h2 id="_4-仿射变换" tabindex="-1"><a class="header-anchor" href="#_4-仿射变换"><span>4. 仿射变换</span></a></h2><h3 id="_4-1-齐次坐标-homogeneous-coordinates" tabindex="-1"><a class="header-anchor" href="#_4-1-齐次坐标-homogeneous-coordinates"><span>4.1 齐次坐标(Homogeneous Coordinates)</span></a></h3><h4 id="_4-1-1-定义" tabindex="-1"><a class="header-anchor" href="#_4-1-1-定义"><span>4.1.1 定义</span></a></h4><p>齐次坐标就是将一个原本是n维的向量或者一个n维的点用一个n+1维坐标来表示，例如二维点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>,用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>表示，它等同于笛卡尔坐标系中的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>w</mi><mo>,</mo><mi>y</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>w</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c21D4"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo stretchy="false">⇔</mo><mo stretchy="false">[</mo><mfrac><mi>x</mi><mi>w</mi></mfrac><mo>,</mo><mfrac><mi>y</mi><mi>w</mi></mfrac><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container><h4 id="_4-1-2-齐次的含义" tabindex="-1"><a class="header-anchor" href="#_4-1-2-齐次的含义"><span>4.1.2 齐次的含义</span></a></h4><p>笛卡尔坐标<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，等同于齐次坐标<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>k</mi><mi>x</mi><mo>,</mo><mi>k</mi><mi>y</mi><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，所以称作该坐标是"齐次"，Homogeneous一词有“同质”的含义，译作同质坐标可能更准确一些</p><h4 id="_4-1-3-点和矢量" tabindex="-1"><a class="header-anchor" href="#_4-1-3-点和矢量"><span>4.1.3 点和矢量</span></a></h4><p>在图形学中，一般用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>表示一个二维矢量，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>表示一个二维点。 这样可以保证矢量的“平移不变性”，也就是一个矢量经过平移变换，仍然保持和原先一致。</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:3.848em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.572em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.572em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>t</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><msub><mi>t</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>两个矢量的和或者差</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:15.69em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cB1"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo>±</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><mo>,</mo><msub><mi>u</mi><mi>y</mi></msub><mo>,</mo><msub><mi>u</mi><mi>z</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo><mo>±</mo><mo stretchy="false">[</mo><msub><mi>v</mi><mi>x</mi></msub><mo>,</mo><msub><mi>v</mi><mi>y</mi></msub><mo>,</mo><msub><mi>v</mi><mi>z</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>u</mi><mi>x</mi></msub><mo>±</mo><msub><mi>v</mi><mi>x</mi></msub><mo>,</mo><msub><mi>u</mi><mi>y</mi></msub><mo>±</mo><msub><mi>v</mi><mi>y</mi></msub><mo>,</mo><msub><mi>u</mi><mi>z</mi></msub><mo>±</mo><msub><mi>v</mi><mi>z</mi></msub><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>仍然是一个矢量，而两个点的差也是矢量，点和矢量的和或者差也是矢量</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:10.599em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mo>?</mo></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mi>p</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>其次坐标表示的两个点的和也是有意义的，表示两个点的中点</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:15.132em;" align="x_1,y_1,1"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt" align="x_1,y_1,1"><mtr><mtd><mo>+</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>2</mn></msub><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>,</mo><mn>2</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mo>,</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_4-2-齐次矩阵表达" tabindex="-1"><a class="header-anchor" href="#_4-2-齐次矩阵表达"><span>4.2 齐次矩阵表达</span></a></h3><h4 id="_4-2-1-线性变换无法表示图形学中所有的几何变换-因此需要定义一种变换为" tabindex="-1"><a class="header-anchor" href="#_4-2-1-线性变换无法表示图形学中所有的几何变换-因此需要定义一种变换为"><span>4.2.1 线性变换无法表示图形学中所有的几何变换，因此需要定义一种变换为</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D70F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.261em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D483 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>α</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>τ</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><p>用矩阵表达为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:20.335em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.315em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D497 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.261em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D483 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.62em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.572em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mrow space="3"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.916em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.845em;vertical-align:-1.672em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:right;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.827em;vertical-align:-1.663em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.849em;" justify="right"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.827em;vertical-align:-1.663em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left right" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>α</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">v</mi><mo stretchy="false">→</mo></mover></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>b</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>常用4X4齐次矩阵来做这种运算</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mrow><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.245em;vertical-align:-2.372em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:8.537em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.245em;vertical-align:-2.372em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.572em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.227em;vertical-align:-2.363em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.849em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.227em;vertical-align:-2.363em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd><mtd><msub><mi>b</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd><mtd><msub><mi>b</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd><mtd><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h2 id="_5-平移变换" tabindex="-1"><a class="header-anchor" href="#_5-平移变换"><span>5. 平移变换</span></a></h2><h3 id="_5-1-定义" tabindex="-1"><a class="header-anchor" href="#_5-1-定义"><span>5.1 定义</span></a></h3><p>空间中的某一点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">u</mi></math></mjx-assistive-mml></mjx-container>，移动到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-TeXAtom space="3" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.261em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D483 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">u</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，那么用齐次矩阵表示这个变换为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.245em;vertical-align:-2.372em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:5.416em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.245em;vertical-align:-2.372em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>b</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>b</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><msub><mi>b</mi><mi>z</mi></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h2 id="_6-组合变换" tabindex="-1"><a class="header-anchor" href="#_6-组合变换"><span>6. 组合变换</span></a></h2><h3 id="_6-1-定义" tabindex="-1"><a class="header-anchor" href="#_6-1-定义"><span>6.1 定义</span></a></h3><p>图形学中将所有变换统一使用4X4的齐次坐标组合</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>R</mi><mo>=</mo><msub><mi>R</mi><mi>n</mi></msub><mo>⋯</mo><msub><mi>R</mi><mn>3</mn></msub><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><p>这样，一系列变换可以组合到一个矩阵中完成</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>⋯</mo><mo stretchy="false">(</mo><msub><mi>R</mi><mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>R</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>R</mi><mn>1</mn></msub><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>⋯</mo><msub><mi>R</mi><mn>3</mn></msub><msub><mi>R</mi><mn>2</mn></msub><msub><mi>R</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><h3 id="_6-2-坐标系变换" tabindex="-1"><a class="header-anchor" href="#_6-2-坐标系变换"><span>6.2 坐标系变换</span></a></h3><p>一个物体在不同坐标系中的变换，比如常用的局部坐标系到世界坐标系的变换</p><figure><img src="/assets/transform_axis-DnM_BPj4.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>设新坐标系x'y'z'的原点在原坐标系的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.514em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D476 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">O</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>o</mi></msub><mo>,</mo><msub><mi>y</mi><mi>o</mi></msub><mo>,</mo><msub><mi>z</mi><mi>o</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>处，新坐标系的单位坐标矢量在原坐标系中的矢量为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.341em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48A TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.374em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48B TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.441em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D48C TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><msup><mi mathvariant="bold-italic">i</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msup><mi mathvariant="bold-italic">j</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow><mo>,</mo><mrow data-mjx-texclass="ORD"><mover><msup><mi mathvariant="bold-italic">k</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，那么将物体的坐标从原先坐标系xyz转换到新坐标系x'y'z'的转换可以分为两部分</p><h4 id="_6-2-1-平移矩阵" tabindex="-1"><a class="header-anchor" href="#_6-2-1-平移矩阵"><span>6.2.1 平移矩阵</span></a></h4><p>构建平移矩阵，目的是将物体在原坐标系中的坐标做一个偏移，这个偏移可以视作将物体和新坐标系一起移动，将新坐标系的原点<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D476 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">O</mi></math></mjx-assistive-mml></mjx-container>挪到原坐标系的原点处，可以得到</p><figure><img src="/assets/transform_axis_move-Ck2kYwSA.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.287em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>−</mo><msub><mi>z</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h4 id="_6-2-2-构建旋转矩阵" tabindex="-1"><a class="header-anchor" href="#_6-2-2-构建旋转矩阵"><span>6.2.2 构建旋转矩阵</span></a></h4><p>接下来要构建一个旋转矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math></mjx-assistive-mml></mjx-container>，目的是将片以后的物体坐标和新坐标系旋转到和原坐标系一致，也就是经过这个矩阵，可以将新坐标系中的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container>轴旋转到原坐标系的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container>轴，也就是:</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.378em;vertical-align:-2.439em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.832em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.378em;vertical-align:-2.439em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:0.5em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.2em;vertical-align:-2.35em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd></mtr><mtr><mtd><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd></mtr><mtr><mtd><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>同理，也需要将<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container>轴旋转到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></mjx-assistive-mml></mjx-container>轴，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup></math></mjx-assistive-mml></mjx-container>轴旋转到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></mjx-assistive-mml></mjx-container>轴</p><figure><img src="/assets/transform_axis_roate-Bm4Ra9LF.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>这个矩阵并不好直接写出，但是它的逆矩阵就比较容易，也就是能够将写出</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.378em;vertical-align:-2.439em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.24em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.378em;vertical-align:-2.439em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>这个矩阵满足</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:13.526em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msup><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><msup><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>由于这是一个正交矩阵，所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:5.653em;vertical-align:-2.576em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.392em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd 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class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi mathvariant="bold-italic">R</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h4 id="_6-2-3-组合后的坐标转换矩阵" tabindex="-1"><a class="header-anchor" href="#_6-2-3-组合后的坐标转换矩阵"><span>6.2.3 组合后的坐标转换矩阵</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" 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class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:6.697em;vertical-align:-3.098em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:8.954em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 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style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-base></mjx-mover></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.328em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 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rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">T</mi></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>y</mi><mn>0</mn></msub><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>z</mi><mn>0</mn></msub><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>y</mi><mn>0</mn></msub><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>z</mi><mn>0</mn></msub><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>y</mi><mn>0</mn></msub><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><msub><mi>z</mi><mn>0</mn></msub><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>O</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msup><mi>i</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>O</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msup><mi>j</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi>O</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><msup><mi>k</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">→</mo></mover></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>验证</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:13.311em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msubsup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.378em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D479 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D47B TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">T</mi><mo stretchy="false">[</mo><msubsup><mi>i</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>i</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>i</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">T</mi><mo stretchy="false">[</mo><msubsup><mi>j</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>j</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>j</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">T</mi><mo stretchy="false">[</mo><msubsup><mi>k</mi><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>k</mi><mi>y</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><msubsup><mi>k</mi><mi>z</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">T</mi><mo stretchy="false">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div 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