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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 auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></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 route-link-active auto-link vp-sidebar-item active" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_8-视图变换管线概览" aria-label="8. 视图变换管线概览"><!--[--><!--[--><!--]--><!--]-->8. 视图变换管线概览<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_9-摄像机视图变换-viewing-camera-transformation" aria-label="9. 摄像机视图变换（Viewing/Camera Transformation)"><!--[--><!--[--><!--]--><!--]-->9. 摄像机视图变换（Viewing/Camera Transformation)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_9-1-定义" aria-label="9.1 定义"><!--[--><!--[--><!--]--><!--]-->9.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_9-2-计算" aria-label="9.2 计算"><!--[--><!--[--><!--]--><!--]-->9.2 计算<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_10-透视投影变换-perspective-projective-transformations" aria-label="10. 透视投影变换(Perspective Projective Transformations)"><!--[--><!--[--><!--]--><!--]-->10. 透视投影变换(Perspective Projective Transformations)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_10-1-定义" aria-label="10.1 定义"><!--[--><!--[--><!--]--><!--]-->10.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_10-2-求解过程" aria-label="10.2 求解过程"><!--[--><!--[--><!--]--><!--]-->10.2 求解过程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_11-正交投影变换-orthogonal-projection-transformations" aria-label="11. 正交投影变换(Orthogonal Projection Transformations)"><!--[--><!--[--><!--]--><!--]-->11. 正交投影变换(Orthogonal Projection Transformations)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_11-1-定义" aria-label="11.1 定义"><!--[--><!--[--><!--]--><!--]-->11.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_11-2-求解过程" aria-label="11.2 求解过程"><!--[--><!--[--><!--]--><!--]-->11.2 求解过程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></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-四" tabindex="-1"><a class="header-anchor" href="#几何变换-四"><span>几何变换(四）</span></a></h1><hr><h2 id="_8-视图变换管线概览" tabindex="-1"><a class="header-anchor" href="#_8-视图变换管线概览"><span>8. 视图变换管线概览</span></a></h2><p>一个物体从独立的物体最终转换为2D图像，期间经过如下坐标变换</p><figure><img src="/assets/transformation_pipeline-YtoCz4fJ.png" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><ul><li>Modeling transformation: 由局部坐标系(Local Space)转换为世界坐标系(World Space)</li><li>Viewing transformation: 由世界坐标系(World Space)转换为摄像机坐标系(View Space)</li><li>Projection transformation: 由摄像机坐标系（View Space)转换为标准设备坐标系NDC(Normalized Device Coordinate)</li><li>Viewport transformation: 由NDC坐标系转换为视口坐标系(Screen coordinate System)</li></ul><h2 id="_9-摄像机视图变换-viewing-camera-transformation" tabindex="-1"><a class="header-anchor" href="#_9-摄像机视图变换-viewing-camera-transformation"><span>9. 摄像机视图变换（Viewing/Camera Transformation)</span></a></h2><h3 id="_9-1-定义" tabindex="-1"><a class="header-anchor" href="#_9-1-定义"><span>9.1 定义</span></a></h3><p>视图变换是将物体的世界坐标系转换到摄像机的坐标系</p><figure><img src="/assets/view_transform-pr23mdSz.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>摄像机坐标系以观察点(eye)为原点，向前方向为-z轴，x轴指向右侧（右手坐标系） 所以</p><mjx-container class="MathJax" jax="CHTML" 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xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">z</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></mtd><mtd><mi></mi><mo>=</mo><mi>l</mi><mi>o</mi><mi>o</mi><mi>k</mi><mi>a</mi><mi>t</mi><mo>−</mo><mi>e</mi><mi>y</mi><mi>e</mi></mtd></mtr><mtr><mtd><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">x</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mover><mrow><mi>u</mi><mi>p</mi></mrow><mo stretchy="false">→</mo></mover></mrow><mo>×</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">z</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr><mtr><mtd><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">y</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></mtd><mtd><mi></mi><mo>=</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">z</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup><mo>×</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">x</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_9-2-计算" tabindex="-1"><a class="header-anchor" href="#_9-2-计算"><span>9.2 计算</span></a></h3><p>根据<a class="route-link" href="/note/graphics/math/transform_02.html#_6-2-3-%E7%BB%84%E5%90%88%E5%90%8E%E7%9A%84%E5%9D%90%E6%A0%87%E8%BD%AC%E6%8D%A2%E7%9F%A9%E9%98%B5">坐标系变换公式</a>，设摄像机在世界坐标系中的位置为<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-c1D452 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>e</mi></math></mjx-assistive-mml></mjx-container>，摄像机的三个轴的单位轴向量为<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>视图变换矩阵为</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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.081em;"><mjx-TeXAtom size="s" 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-c1D456 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-c1D464 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><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:6.667em;vertical-align:-3.083em;" 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:8.828em;"><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-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><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-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D486 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.203em;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-c1D48A TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><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-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 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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-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 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-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-mi class="mjx-b 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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:6.667em;vertical-align:-3.083em;" 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"><msub><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>v</mi><mi>i</mi><mi>e</mi><mi>w</mi></mrow></msub><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><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">e</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">i</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></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 mathvariant="bold-italic">e</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">j</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></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 mathvariant="bold-italic">e</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msup><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">k</mi><mo stretchy="false">→</mo></mover></mrow><mo data-mjx-alternate="1">′</mo></msup></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="_10-透视投影变换-perspective-projective-transformations" tabindex="-1"><a class="header-anchor" href="#_10-透视投影变换-perspective-projective-transformations"><span>10. 透视投影变换(Perspective Projective Transformations)</span></a></h2><h3 id="_10-1-定义" tabindex="-1"><a class="header-anchor" href="#_10-1-定义"><span>10.1 定义</span></a></h3><p>投影变换是将物体从摄像机坐标系转换到标准设备坐标系(NDC)，也就是坐标值在一个<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-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-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:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo><mn>3</mn></msup></math></mjx-assistive-mml></mjx-container>的盒子里(Canonical view volume) 可以将这个过程分为两步，首先构造一个坐标转换，将摄像机的摄影范围构成的视锥体(Frustum)变形成一个立方体(Cuboid)，然后再将这个立方体转化为成标准设备坐标系下(NDC)的标准观察体CCV(Canonical View Volume)</p><figure><img src="/assets/proj_transform-CKWeqHdM.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>NDC坐标系在不同的系统上有不同的约定，在DirectX中，近平面到远平面映射到<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-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-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><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，在OpenGL则是把近平面到远平面映射到<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-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-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><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，但一般都把NDC坐标系设为左手坐标系。</p><h3 id="_10-2-求解过程" tabindex="-1"><a class="header-anchor" href="#_10-2-求解过程"><span>10.2 求解过程</span></a></h3><h4 id="_10-2-1-从视锥体到立方体" tabindex="-1"><a class="header-anchor" href="#_10-2-1-从视锥体到立方体"><span>10.2.1 从视锥体到立方体</span></a></h4><figure><img src="/assets/proj_transform_1-YDn8lTVM.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>设摄像机坐标系中的点<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-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></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-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，转换后的坐标为<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-c28"></mjx-c></mjx-mo><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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><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-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"><mo stretchy="false">(</mo><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，由定义可知<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-c1D45D 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>p</mi></math></mjx-assistive-mml></mjx-container>投影到摄像机近裁剪面上的点为<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-c1D45D 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-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><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-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"><msup><mi>p</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，由相似三角形可知</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:3.795em;"><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-i"><mjx-c class="mjx-c1D466 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><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-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-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><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-c1D466 TEX-I"></mjx-c></mjx-mi></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-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><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-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-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><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-c1D465 TEX-I"></mjx-c></mjx-mi></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-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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>y</mi><mo data-mjx-alternate="1">′</mo></msup></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>y</mi></mrow><mi>z</mi></mfrac></mtd></mtr><mtr><mtd><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>x</mi></mrow><mi>z</mi></mfrac></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>设这个转换矩阵为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ortho</mtext></mrow></msub></math></mjx-assistive-mml></mjx-container>, 可知</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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:5.462em;vertical-align:-2.481em;" 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:1.269em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-top:0.2em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-c1D466 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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:5.462em;vertical-align:-2.481em;" 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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ortho</mtext></mrow></msub><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><mfrac><mrow><mi>n</mi><mi>x</mi></mrow><mi>z</mi></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>n</mi><mi>y</mi></mrow><mi>z</mi></mfrac></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><p>对于表示一个点的其次坐标，所有数值乘以一个同样的数，所表达的意义相同，所以<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-c1D45B TEX-I"></mjx-c></mjx-mi><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-c1D467 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-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><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-c1D467 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B 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-c1D45B 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-msup space="2"><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-mi class="mjx-i"><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-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-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>n</mi><mi>x</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>z</mi><mo>,</mo><mi>n</mi><mi>y</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>z</mi><mo>,</mo><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>n</mi><mi>x</mi><mo>,</mo><mi>n</mi><mi>y</mi><mo>,</mo><msup><mi>z</mi><mo data-mjx-alternate="1">′</mo></msup><mi>z</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，这里先暂时忽略<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，可以得到</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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:5.2em;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:1.172em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><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-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-c1D45B TEX-I"></mjx-c></mjx-mi><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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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: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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ortho</mtext></mrow></msub><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><mi>n</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>n</mi><mi>y</mi></mtd></mtr><mtr><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>由此可以推断出，这个矩阵的第1，2和4行</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:5.2em;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:5.2em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c3F"></mjx-c></mjx-mo><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-c31"></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-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-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:5.2em;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:1.172em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><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-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-c1D45B TEX-I"></mjx-c></mjx-mi><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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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: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"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</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><mi>n</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>n</mi><mi>y</mi></mtd></mtr><mtr><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>考虑特殊情况，近平面上的任意一点<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-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B 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>n</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>经过这个矩阵转换后</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:5.2em;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:5.2em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c3F"></mjx-c></mjx-mo><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-c31"></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-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-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:0.6em;"><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-c1D45B 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:5.2em;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:1.172em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><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-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-c1D45B TEX-I"></mjx-c></mjx-mi><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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c1D45B 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: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"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</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>n</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><mi>n</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>n</mi><mi>y</mi></mtd></mtr><mtr><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>由于近平面上的点经过转换后坐标不变，所以<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-c1D45B 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-c1D45B 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-mo class="mjx-n" space="2"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c1D45B 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-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><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-c1D45B 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>n</mi><mi>x</mi><mo>,</mo><mi>n</mi><mi>y</mi><mo>,</mo><mo>?</mo><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，而<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-c1D45B 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>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>作为其次坐标，所有数值都乘以<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-c1D45B 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>n</mi></math></mjx-assistive-mml></mjx-container>，可以得到<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-c1D45B 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-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c><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-mi class="mjx-i"><mjx-c class="mjx-c1D45B 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-c1D45B 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-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><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-c1D45B 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>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>==</mo><mo stretchy="false">[</mo><mi>n</mi><mi>x</mi><mo>,</mo><mi>n</mi><mi>y</mi><mo>,</mo><msup><mi>n</mi><mn>2</mn></msup><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，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-mrow><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:5.2em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c3F"></mjx-c></mjx-mo><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-c31"></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-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-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:0.6em;"><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-c1D45B 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:5.284em;vertical-align:-2.392em;" 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:1.172em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-bottom:0.2em;"><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-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-c1D45B TEX-I"></mjx-c></mjx-mi><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-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><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-c1D45B 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:5.284em;vertical-align:-2.392em;" 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><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</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>n</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><mi>n</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>n</mi><mi>y</mi></mtd></mtr><mtr><mtd><msup><mi>n</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>同理，远平面上的任意一个点<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-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D453 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>f</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>转换后为<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-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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D453 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><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><msup><mi>y</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><mi>f</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>，可以得到</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:5.2em;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:5.2em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c3F"></mjx-c></mjx-mo><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-c3F"></mjx-c></mjx-mo><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-c31"></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-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-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-c1D453 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:5.302em;vertical-align:-2.401em;" 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:1.04em;"><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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.053em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><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-c1D45B 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:5.302em;vertical-align:-2.401em;" 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><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd><mtd><mo>?</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</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>f</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>f</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>由此可知，第三行的数值和<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-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-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi></math></mjx-assistive-mml></mjx-container>无关，一定是形如<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-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-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D434 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-c1D435 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><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>这样的形式，且</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:5.702em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;margin-left:0.053em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></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" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>A</mi><mi>n</mi><mo>+</mo><mi>B</mi><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>A</mi><mi>f</mi><mo>+</mo><mi>B</mi><mo>=</mo><msup><mi>f</mi><mn>2</mn></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>可以得到</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:4.465em;"><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-c1D434 TEX-I"></mjx-c></mjx-mi><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-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-i"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi><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-c2212"></mjx-c></mjx-mo><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-c1D453 TEX-I"></mjx-c></mjx-mi><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>A</mi></mtd><mtd><mi></mi><mo>=</mo><mi>f</mi><mo>+</mo><mi>n</mi></mtd></mtr><mtr><mtd><mi>B</mi></mtd><mtd><mi></mi><mo>=</mo><mo>−</mo><mi>n</mi><mi>f</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>得到这个转换矩阵为</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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:8.5em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><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-c31"></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-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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ortho</mtext></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mi>n</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mi>f</mi><mo>+</mo><mi>n</mi></mtd><mtd><mo>−</mo><mi>n</mi><mi>f</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h4 id="_10-2-2-从立方体到标准观察体" tabindex="-1"><a class="header-anchor" href="#_10-2-2-从立方体到标准观察体"><span>10.2.2 从立方体到标准观察体</span></a></h4><p>这一步相对简单很多，只要将观察体的中心的<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-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>挪到原点，做一定的放缩，并且将<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>轴反转即可</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:36.404em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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-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:5.778em;" justify="left"><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-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-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-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-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-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-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:5.2em;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-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:10.942em;" justify="left"><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-c32"></mjx-c></mjx-mn><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-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-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-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-c32"></mjx-c></mjx-mn><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-c210E TEX-I"></mjx-c></mjx-mi><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-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-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-c32"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c 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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-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><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>ortho</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub></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><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>1</mn></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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>w</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>h</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo></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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>f</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</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></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><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>w</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>h</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>f</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><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></mtable></math></mjx-assistive-mml></mjx-container><h4 id="_10-2-3-合并后的矩阵" tabindex="-1"><a class="header-anchor" href="#_10-2-3-合并后的矩阵"><span>10.2.3 合并后的矩阵</span></a></h4><p>最终得到合并后的投影矩阵为</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:17.077em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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-msub space="4"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:6.109em;vertical-align:-2.804em;" 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:9.82em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D464 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><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-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-c31"></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-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:6.109em;vertical-align:-2.804em;" 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" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub></mtd><mtd><mi></mi><mo>=</mo><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>ortho</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ortho</mtext></mrow></msub></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><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mi>w</mi></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mi>h</mi></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mi>f</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>f</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mn>2</mn><mi>n</mi><mi>f</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>f</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>在大部分工程项目中，摄像机的输入参数一般是如下几个</p><ul><li>宽高比 <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-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D460 TEX-I"></mjx-c></mjx-mi><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-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-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-c1D464 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-c210E 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>a</mi><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>t</mi><mo>=</mo><mi>w</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mi>h</mi></math></mjx-assistive-mml></mjx-container></li><li>视场角fov，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-n"><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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-c1D463 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><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-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B 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>tan</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mi>o</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>h</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mi>n</mi></math></mjx-assistive-mml></mjx-container></li><li>近裁剪距离 <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-c1D45B 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-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></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-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo></mjx-mrow><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-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B 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>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container></li><li>远裁剪距离 <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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></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-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo></mjx-mrow><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-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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>f</mi><mi>a</mi><mi>r</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>f</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</mo><mi>f</mi></math></mjx-assistive-mml></mjx-container></li></ul><p>将这些值带入上面的公式，可以得到</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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:6.505em;vertical-align:-3.003em;" 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:21.078em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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-c1D463 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-n"><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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-c1D463 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-c31"></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-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:6.505em;vertical-align:-3.003em;" 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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mtext>aspect</mtext><mo>.</mo><mi>tan</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mi>o</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mi>tan</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mi>o</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mfrac><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>+</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mo>−</mo><mn>2</mn><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>.</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><p>一般来说，当针对一个空间点做矩阵转换时，如果<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-c1D474 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-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-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:0.363em;"><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-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-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-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:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><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>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup><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>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></math></mjx-assistive-mml></mjx-container>，那么</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-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></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-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-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-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-c1D458 TEX-I"></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-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-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-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-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><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-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><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-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-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><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-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-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><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-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-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-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-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mi>k</mi><mi mathvariant="bold-italic">M</mi><mo stretchy="false">)</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>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup><mo>=</mo><mi>k</mi><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>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup><mo>=</mo><mo stretchy="false">[</mo><mi>k</mi><msub><mi>v</mi><mi>x</mi></msub><mo>,</mo><mi>k</mi><msub><mi>v</mi><mi>y</mi></msub><mo>,</mo><mi>k</mi><msub><mi>v</mi><mi>z</mi></msub><mo>,</mo><mi>k</mi><msup><mo stretchy="false">]</mo><mi>T</mi></msup><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>1</mn><msup><mo stretchy="false">]</mo><mi>T</mi></msup></math></mjx-assistive-mml></mjx-container><p>可知<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-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 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>k</mi><mi mathvariant="bold-italic">M</mi></math></mjx-assistive-mml></mjx-container>在针对空间点转换矩阵中等同于<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-c1D474 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">M</mi></math></mjx-assistive-mml></mjx-container>，所以为了表达方便，经常将上面得到的这个矩阵乘以-1，得到</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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:6.498em;vertical-align:-2.999em;" 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:19.75em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c70"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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-c1D463 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-n"><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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-c1D463 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-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-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-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:6.498em;vertical-align:-2.999em;" 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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>persp</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>1</mn><mrow><mtext>aspect</mtext><mo>.</mo><mi>tan</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mi>o</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><mrow><mi>tan</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><mi>f</mi><mi>o</mi><mi>v</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mfrac><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>+</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mn>2</mn><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>.</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h4 id="_10-2-4-深度插值" tabindex="-1"><a class="header-anchor" href="#_10-2-4-深度插值"><span>10.2.4 深度插值</span></a></h4><figure><img src="/assets/proj_transform_2-wJzpGoAn.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>设摄像机坐标系中有一个线段<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-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-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-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-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></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-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-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-c32"></mjx-c></mjx-mn></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"><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，在摄像机近平面上的投影为<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-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><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-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><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-c1D45B TEX-I"></mjx-c></mjx-mi><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"><mo stretchy="false">(</mo><msubsup><mi>x</mi><mn>1</mn><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>x</mi><mn>2</mn><mo data-mjx-alternate="1">′</mo></msubsup><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，线段上有一点为<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-c28"></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-c1D467 TEX-I"></mjx-c></mjx-mi><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"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，投影为<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-c28"></mjx-c></mjx-mo><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-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><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"><mo stretchy="false">(</mo><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container> 可知</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:3.663em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-bottom:0.15em;"><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-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 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-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><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-mrow></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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mfrac><mi>x</mi><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mfrac><mo>=</mo><mfrac><mi>z</mi><mi>n</mi></mfrac></mtd></mtr><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mi>z</mi><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>n</mi></mfrac></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>设线段的方程为<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-c1D44E 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" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><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-c1D450 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>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>z</mi><mo>=</mo><mi>c</mi></math></mjx-assistive-mml></mjx-container>，带入可得</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:6.187em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><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-mrow></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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D467 TEX-I"></mjx-c></mjx-mi><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-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-top:0.15em;"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></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-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><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-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>a</mi><mfrac><mrow><mi>z</mi><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mi>n</mi></mfrac><mo>+</mo><mi>b</mi><mi>z</mi><mo>=</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mi>z</mi></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mi>x</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mrow><mi>c</mi><mi>n</mi></mrow></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>当投影点<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>在投影线段上线性移动时，<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-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-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msubsup></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><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mi>x</mi><mn>1</mn><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><mi>t</mi><msubsup><mi>x</mi><mn>2</mn><mo data-mjx-alternate="1">′</mo></msubsup></math></mjx-assistive-mml></mjx-container>，可得</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:16.42em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></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-c1D467 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E 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-mrow><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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mrow><mjx-mo class="mjx-sop"><mjx-c class="mjx-c28 TEX-S1"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 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:0.219em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 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:0.219em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msubsup><mjx-mo class="mjx-sop"><mjx-c class="mjx-c29 TEX-S1"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-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-c28"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msubsup></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-mrow><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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D450 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-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.258em;margin-left:0;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msubsup></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-mrow><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-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D450 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-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><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-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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-mfrac space="3"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><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-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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><mfrac><mn>1</mn><mi>z</mi></mfrac></mtd><mtd><mi></mi><mo>=</mo><mfrac><mi>a</mi><mrow><mi>c</mi><mi>n</mi></mrow></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo stretchy="false">(</mo><mn>1</mn><mo>��</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mi>x</mi><mn>1</mn><mo data-mjx-alternate="1">′</mo></msubsup><mo>+</mo><mi>t</mi><msubsup><mi>x</mi><mn>2</mn><mo data-mjx-alternate="1">′</mo></msubsup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>a</mi><msubsup><mi>x</mi><mn>1</mn><mo data-mjx-alternate="1">′</mo></msubsup></mrow><mrow><mi>c</mi><mi>n</mi></mrow></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>a</mi><msubsup><mi>x</mi><mn>2</mn><mo data-mjx-alternate="1">′</mo></msubsup></mrow><mrow><mi>c</mi><mi>n</mi></mrow></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac><mo stretchy="false">)</mo><mi>t</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>z</mi><mn>1</mn></msub></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><msub><mi>z</mi><mn>2</mn></msub></mfrac><mi>t</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>也就是说，当投影点<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>在投影线段上线性移动时，原线段上的点的深度值的倒数是线性变换的</p><h2 id="_11-正交投影变换-orthogonal-projection-transformations" tabindex="-1"><a class="header-anchor" href="#_11-正交投影变换-orthogonal-projection-transformations"><span>11. 正交投影变换(Orthogonal Projection Transformations)</span></a></h2><h3 id="_11-1-定义" tabindex="-1"><a class="header-anchor" href="#_11-1-定义"><span>11.1 定义</span></a></h3><figure><img src="/assets/proj_transform_3-qj92WLVu.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>正交投影没有“远大近小”的效果</p><h3 id="_11-2-求解过程" tabindex="-1"><a class="header-anchor" href="#_11-2-求解过程"><span>11.2 求解过程</span></a></h3><p>相比透视投影，正交投影只需要将一个立方体形状的摄影范围最终映射到标准设备坐标系(NDC)下的一个标准立方体(Cuboid)即可</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:40.652em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c 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class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><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-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><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-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><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-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><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>ortho</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub></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><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>1</mn></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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo></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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mi>r</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>f</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</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></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><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>l</mi><mo stretchy="false">)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mi>r</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>l</mi><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mtd><mtd><mn>0</mn></mtd><mtd><mo stretchy="false">(</mo><mi>b</mi><mo>+</mo><mi>t</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>f</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>f</mi><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></mtable></math></mjx-assistive-mml></mjx-container><p>同理，在工程中一般标注一个正交投影摄像机的主要参数如下</p><ul><li>近裁剪距离 <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-c1D45B 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-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></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-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo></mjx-mrow><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-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B 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>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container></li><li>远裁剪距离 <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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></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-c class="mjx-c7C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7C"></mjx-c></mjx-mo></mjx-mrow><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-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 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>f</mi><mi>a</mi><mi>r</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>f</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</mo><mi>f</mi></math></mjx-assistive-mml></mjx-container></li><li>左侧裁剪面坐标 <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-c1D459 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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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-c1D459 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>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mo>=</mo><mi>l</mi></math></mjx-assistive-mml></mjx-container></li><li>右侧裁剪面坐标 <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-c1D459 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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><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-c1D459 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>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mo>=</mo><mi>l</mi></math></mjx-assistive-mml></mjx-container></li><li>上侧裁剪面坐标 <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-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-c1D45D TEX-I"></mjx-c></mjx-mi><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-c1D461 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>t</mi><mi>o</mi><mi>p</mi><mo>=</mo><mi>t</mi></math></mjx-assistive-mml></mjx-container></li><li>下侧裁剪面坐标 <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-c1D44F 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-c1D461 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-c1D45A TEX-I"></mjx-c></mjx-mi><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-c1D461 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>b</mi><mi>o</mi><mi>t</mi><mi>t</mi><mi>o</mi><mi>m</mi><mo>=</mo><mi>t</mi></math></mjx-assistive-mml></mjx-container> 带入上面的公式可以得到</li></ul><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-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.077em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c6F"></mjx-c></mjx-mtext><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mtext class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c76"></mjx-c></mjx-mtext></mjx-TeXAtom></mjx-script></mjx-msub><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:6.597em;vertical-align:-3.048em;" 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:18.385em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F 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-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F 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-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D459 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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F 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-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D461 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D461 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-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D461 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-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><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-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F 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-c1D461 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-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><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-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><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-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><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:6.597em;vertical-align:-3.048em;" 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"><msub><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mtext>ortho</mtext><mo stretchy="false">→</mo><mtext>ccv</mtext></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mfrac><mn>2</mn><mrow><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi><mo>−</mo><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mfrac><mrow><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi></mrow><mrow><mi>l</mi><mi>e</mi><mi>f</mi><mi>t</mi><mo>−</mo><mi>r</mi><mi>i</mi><mi>g</mi><mi>h</mi><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>2</mn><mrow><mi>t</mi><mi>o</mi><mi>p</mi><mo>−</mo><mi>b</mi><mi>o</mi><mi>t</mi><mi>t</mi><mi>o</mi><mi>m</mi></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mfrac><mrow><mi>b</mi><mi>o</mi><mi>t</mi><mi>t</mi><mi>o</mi><mi>m</mi><mo>+</mo><mi>t</mi><mi>o</mi><mi>p</mi></mrow><mrow><mi>b</mi><mi>o</mi><mi>t</mi><mi>t</mi><mi>o</mi><mi>m</mi><mo>−</mo><mi>t</mi><mi>o</mi><mi>p</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mfrac><mn>2</mn><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>+</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow><mrow><mi>n</mi><mi>e</mi><mi>a</mi><mi>r</mi><mo>−</mo><mi>f</mi><mi>a</mi><mi>r</mi></mrow></mfrac></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></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>法线变换</span></div><!--]--></a><!----></nav><!--[--><!--]--></main><!--]--></div><!--[--><!----><!--]--><!--]--></div>
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