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style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">图形学 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <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/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 auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item active collapsible">光照模型 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/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 route-link-active auto-link vp-sidebar-item active" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_1-概述" aria-label="1. 概述"><!--[--><!--[--><!--]--><!--]-->1. 概述<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-法线分布函数" aria-label="2. 法线分布函数"><!--[--><!--[--><!--]--><!--]-->2. 法线分布函数<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_2-1-定义" aria-label="2.1 定义"><!--[--><!--[--><!--]--><!--]-->2.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-2-微表面的面积" aria-label="2.2 微表面的面积"><!--[--><!--[--><!--]--><!--]-->2.2 微表面的面积<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-几何衰减因子" aria-label="3. 几何衰减因子"><!--[--><!--[--><!--]--><!--]-->3. 几何衰减因子<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-1-定义" aria-label="3.1 定义"><!--[--><!--[--><!--]--><!--]-->3.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_4-菲涅尔方程" aria-label="4. 菲涅尔方程"><!--[--><!--[--><!--]--><!--]-->4. 菲涅尔方程<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_4-1-定义" aria-label="4.1 定义"><!--[--><!--[--><!--]--><!--]-->4.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-cook-torrance模型" aria-label="5.Cook-Torrance模型"><!--[--><!--[--><!--]--><!--]-->5.Cook-Torrance模型<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_5-1-定义" aria-label="5.1 定义"><!--[--><!--[--><!--]--><!--]-->5.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-2-推导过程" aria-label="5.2 推导过程"><!--[--><!--[--><!--]--><!--]-->5.2 推导过程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></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="微平面理论-一" tabindex="-1"><a class="header-anchor" href="#微平面理论-一"><span>微平面理论(一)</span></a></h1><hr><h2 id="_1-概述" tabindex="-1"><a class="header-anchor" href="#_1-概述"><span>1. 概述</span></a></h2><p>微表面理论（Microfacet Theory）认为我们看到的表面上的一点是由很多朝向各异且光学平的微小表面组成。当光线从<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:0.232em;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-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>l</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>方向照射到这点，而我们在<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:0.27em;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-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>方向观察时，由于光学平面只会将光线<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:0.232em;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-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>l</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>反射到关于法线对称的<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.27em;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-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>方向，而<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:0.232em;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-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>l</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<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:0.27em;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-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>已经确定，所以只有法线朝向正好是<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:0.232em;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-i"><mjx-c class="mjx-c1D459 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>l</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>和<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:0.27em;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-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的半角向量<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:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的微表面才会将光线发射到<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:0.27em;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-i"><mjx-c class="mjx-c1D463 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>v</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>方向，从而被我们看见。</p><figure><img src="/assets/microfacet_1-B65RAyZw.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h2 id="_2-法线分布函数" tabindex="-1"><a class="header-anchor" href="#_2-法线分布函数"><span>2. 法线分布函数</span></a></h2><h3 id="_2-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-1-定义"><span>2.1 定义</span></a></h3><p>法线分布函数（Normal Distribution Function，简写为NDF）用来描述微表面法线的概率分布，可以这样理解：向NDF输入一个朝向h，NDF会返回法线是<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:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的微表面数占微表面总数的比例，需要注意的是<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-c1D437 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-c210E 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>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>是描述微表面<strong>数量</strong>的概率分布函数</p><h3 id="_2-2-微表面的面积" tabindex="-1"><a class="header-anchor" href="#_2-2-微表面的面积"><span>2.2 微表面的面积</span></a></h3><p>设一个小平面，面积为<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 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>d</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container>，此小平面由更小的微表面组成，其中法线为<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:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的微表面的概率为<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-c1D437 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-c210E 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>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，这些微表面的面积总和为<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-c1D437 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-c210E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 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>D</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>d</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container>，考虑到法线范围，定义微表面的法线为<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:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，范围在立体角<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>之内的所有微表面的表面积为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.26em;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-i"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>d</mi><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>h</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub><mi>d</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container><h2 id="_3-几何衰减因子" tabindex="-1"><a class="header-anchor" href="#_3-几何衰减因子"><span>3. 几何衰减因子</span></a></h2><h3 id="_3-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-1-定义"><span>3.1 定义</span></a></h3><p>由于微表面之间的互相遮挡(Shadowing)，并不是所有微表面能够接收到光线，还有一部分微表面反射的光线会被阻挡(Masking)，还有一部分光线会在微表面之间互相反射. Shadowing和Masking效应用几何衰减因子(Geometrical Attenuation Factor)来表示，用<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-c1D43A 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-c1D459 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-c1D463 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>G</mi><mo stretchy="false">(</mo><mi>l</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>表示</p><figure><img src="/assets/microfacet_2-CsaQTCgV.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h2 id="_4-菲涅尔方程" tabindex="-1"><a class="header-anchor" href="#_4-菲涅尔方程"><span>4. 菲涅尔方程</span></a></h2><h3 id="_4-1-定义" tabindex="-1"><a class="header-anchor" href="#_4-1-定义"><span>4.1 定义</span></a></h3><p>光学平面会将一部分光线反射，一部分光线折射，其中反射光线的比例用菲尼尔方程(Fresnel Equations)计算，用<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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>表示，因为菲尼尔函数一般只和光线和法线之间的夹角相关，有时候也会用<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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>表示</p><h2 id="_5-cook-torrance模型" tabindex="-1"><a class="header-anchor" href="#_5-cook-torrance模型"><span>5.Cook-Torrance模型</span></a></h2><h3 id="_5-1-定义" tabindex="-1"><a class="header-anchor" href="#_5-1-定义"><span>5.1 定义</span></a></h3><p>Cook-Torrance模型用来描述反射平面的的BRDF模型中的高光(Specular)部分，公式为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-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-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;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-i"><mjx-c class="mjx-c1D45B TEX-I"></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-msub space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.3em;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-i"><mjx-c class="mjx-c1D45B TEX-I"></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-msub space="3"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo>⋅</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></math></mjx-assistive-mml></mjx-container><h3 id="_5-2-推导过程" tabindex="-1"><a class="header-anchor" href="#_5-2-推导过程"><span>5.2 推导过程</span></a></h3><p>设一个小平面，面积为<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 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>d</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container>，法线方向<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:0.3em;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-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>，入射光方向<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>，微分变量为<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>，观察方向<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>，微分变量为<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container></p><figure><img src="/assets/microfacet_3-BvKDQL98.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>根据上面的推导，这个小平面上所有法线为<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:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></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><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container>的微平面的面积为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>d</mi><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub><mi>d</mi><mi>A</mi></math></mjx-assistive-mml></mjx-container><p>由于其他微平面接受的入射光通量不会反射到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>方向上，所以不用考虑，所以在<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>方向上有效入射光通量为</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:16.132em;"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></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-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></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-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-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>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>h</mi></msub></mtd><mtd><mi></mi><mo>=</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>A</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub><mi>d</mi><mi>A</mi><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>出射光通量为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>o</mi></msub><mo>=</mo><mi>d</mi><msub><mi>L</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>A</mi><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><p>根据菲尼尔公式，出射光通量和入射光通量的比例可以用如下公式表示</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>o</mi></msub><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><p>联立以上公式可得</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:21.056em;"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-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-mstyle space="4"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-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-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><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-mstyle space="4"><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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-n"><mjx-c class="mjx-c3A6"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><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-mstyle space="4"><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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.556em;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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><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>d</mi><msub><mi>L</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>o</mi></msub></mrow><mrow><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>A</mi><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi mathvariant="normal">Φ</mi><mi>h</mi></msub></mrow><mrow><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>d</mi><mi>A</mi><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></mrow><mrow><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>根据BRDF的定义</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:18.331em;"><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-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-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-mstyle space="4"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><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-mstyle space="4"><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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-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-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mstyle space="3"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><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>f</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>L</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>ω</mi><mi>i</mi></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>⋅</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></mrow><mrow><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><p>下面考察<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>的关系，设小平面上的一个微平面，接受的入射光线方向为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>, 法线为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>，出射光线方向为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>，和单位圆的交点分别是<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-c1D43C 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-c1D444 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-c1D445 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>I</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi></math></mjx-assistive-mml></mjx-container></p><figure><img src="/assets/microfacet_4-De12LVzJ.svg" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>立体角<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>和立体角<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>之间的约束关系在于，保持<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>不变，让法线<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>在立体角<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>范围之内移动，则<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>需要保证是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container>的半角法线，它的活动范围<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>比较小，具体可以计算一下</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-mstyle><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></mrow><mrow><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>A</mi><mi>h</mi></msub></mrow><mrow><mi>d</mi><msub><mi>A</mi><mi>r</mi></msub></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>连接<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-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 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>I</mi><mi>R</mi></math></mjx-assistive-mml></mjx-container>相交法线<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container>于交点<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-c1D443 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>，由于<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-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 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>I</mi><mi>R</mi><mo>=</mo><mn>2</mn><mi>I</mi><mi>P</mi></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-mstyle><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2033"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c34"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>A</mi><mi>r</mi></msub></mrow><mrow><mi>d</mi><msup><mi>A</mi><mo data-mjx-alternate="1">″</mo></msup></mrow></mfrac></mstyle><mo>=</mo><mn>4</mn></math></mjx-assistive-mml></mjx-container><p>由于<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-c1D442 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 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-n" space="4"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>P</mi><mo>=</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><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-mstyle><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2033"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><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-mrow><mjx-msup><mjx-mi class="mjx-n"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.289em;"><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-c2061"></mjx-c></mjx-mo><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-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msup><mi>A</mi><mo data-mjx-alternate="1">′</mo></msup></mrow><mrow><mi>d</mi><msup><mi>A</mi><mo data-mjx-alternate="1">″</mo></msup></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>由于<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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 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-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><msub><mi>A</mi><mi>h</mi></msub><mo>=</mo><mi>d</mi><msup><mi>A</mi><mo data-mjx-alternate="1">′</mo></msup><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，联立以上公式，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-mstyle><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-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><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-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>d</mi><msub><mi>ω</mi><mi>h</mi></msub></mrow><mrow><mi>d</mi><msub><mi>ω</mi><mi>o</mi></msub></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>4</mn><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>带入上面的BRDF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-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-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-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>将几何衰减因子G考虑在内，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-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><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-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c210E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 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-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.311em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D714 TEX-I"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-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-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c63"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c73"></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-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>h</mi></msub><mo stretchy="false">)</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>i</mi></msub><mo>,</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>ω</mi><mo stretchy="false">→</mo></mover></mrow><mi>o</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE">⁡</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></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 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