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    <title>Javier Amador</title>
    <subtitle>Blog and portfolio of Javier Amador — technical and environment artist. Notes on environment art, shaders, and real-time worlds.</subtitle>
    <link href="https://javieramador.dev/feed.xml" rel="self"/>
    <link href="https://javieramador.dev/"/>
    <updated>2026-07-13T00:00:00Z</updated>
    <id>https://javieramador.dev/</id>
    <author>
        <name>Javier Amador</name>
    </author>
    <entry>
        <title>Shaders, live: GLSL in the browser (and HLSL alongside)</title>
        <link href="https://javieramador.dev/posts/post-6/"/>
        <updated>2026-07-13T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-6/</id>
        <summary>A mockup post for the live shader system — a running WebGL demo, its GLSL source, and an HLSL snippet shown side-by-side.</summary>
        <content type="html">&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note to self:&lt;/strong&gt; mockup post to test the &lt;code&gt;{% shader %}&lt;/code&gt; embed and
&lt;code&gt;glsl&lt;/code&gt; / &lt;code&gt;hlsl&lt;/code&gt; highlighting. Safe to delete before launch.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Half of technical-art writing is &amp;quot;here&#39;s the code, here&#39;s what it &lt;em&gt;does&lt;/em&gt;.&amp;quot; A static
screenshot loses the motion; a wall of code loses the intuition. So on this site a shader
can &lt;strong&gt;run right in the page&lt;/strong&gt;, with its source shown directly beneath it.&lt;/p&gt;
&lt;h2&gt;A live one — edit it yourself&lt;/h2&gt;
&lt;p&gt;The code below the canvas &lt;em&gt;is&lt;/em&gt; the shader driving it, and it&#39;s &lt;strong&gt;editable&lt;/strong&gt;: change a number,
tweak the colours, and the canvas recompiles as you type. Try bumping the &lt;code&gt;0.4&lt;/code&gt; in the ramp
or swapping the &lt;code&gt;green&lt;/code&gt;/&lt;code&gt;amber&lt;/code&gt; values. Broke it? The last working version keeps running and
the compile error shows over the canvas; hit &lt;strong&gt;revert&lt;/strong&gt; to restore the original.&lt;/p&gt;
&lt;figure class=&quot;shader is-editable&quot; data-shader=&quot;void mainImage(out vec4 O, in vec2 I)&amp;#10;{&amp;#10;    // Centre the coords and keep them square regardless of canvas size.&amp;#10;    vec2 uv = (2.0 * I - iResolution.xy) / iResolution.y;&amp;#10;&amp;#10;    // Domain warping: fold the space back on itself a few times.&amp;#10;    float t = iTime * 0.15;&amp;#10;    for (float i = 1.0; i &amp;lt; 6.0; i++) {&amp;#10;        uv.x += 0.32 / i * sin(i * 2.4 * uv.y + t * 2.0 + i);&amp;#10;        uv.y += 0.30 / i * cos(i * 1.9 * uv.x + t * 1.7 + i);&amp;#10;    }&amp;#10;&amp;#10;    // Ramp between a mossy green and a warm amber.&amp;#10;    float v = 0.5 + 0.5 * sin(uv.x * 2.0 + uv.y * 2.0 + iTime * 0.4);&amp;#10;    vec3 green = vec3(0.30, 0.48, 0.32);&amp;#10;    vec3 amber = vec3(0.93, 0.78, 0.42);&amp;#10;    vec3 col = mix(green, amber, v) * (0.7 + 0.3 * v);&amp;#10;&amp;#10;    O = vec4(col, 1.0);&amp;#10;}&quot; style=&quot;--shader-h:300px&quot;&gt;&lt;div class=&quot;shader__stage&quot;&gt;&lt;canvas class=&quot;shader__canvas&quot;&gt;&lt;/canvas&gt;&lt;div class=&quot;shader__bar&quot;&gt;&lt;button type=&quot;button&quot; class=&quot;shader__btn&quot; data-act=&quot;toggle&quot; aria-label=&quot;Play or pause&quot;&gt;❚❚&lt;/button&gt;&lt;button type=&quot;button&quot; class=&quot;shader__btn&quot; data-act=&quot;reset&quot; aria-label=&quot;Restart&quot;&gt;↺&lt;/button&gt;&lt;span class=&quot;shader__clock&quot; aria-hidden=&quot;true&quot;&gt;0.0s&lt;/span&gt;&lt;/div&gt;&lt;p class=&quot;shader__err&quot; role=&quot;alert&quot; hidden=&quot;&quot;&gt;&lt;/p&gt;&lt;/div&gt;&lt;figcaption class=&quot;shader__caption&quot;&gt;Domain-warped flow — five cheap sine folds over a green→amber ramp. Editable: type to recompile.&lt;/figcaption&gt;&lt;div class=&quot;shader__code shader__code--edit&quot;&gt;&lt;div class=&quot;shader__codehead&quot;&gt;&lt;span class=&quot;shader__file&quot;&gt;fragment.glsl · live&lt;/span&gt;&lt;button type=&quot;button&quot; class=&quot;shader__revert&quot; data-act=&quot;revert&quot;&gt;revert&lt;/button&gt;&lt;/div&gt;&lt;div class=&quot;shader__editor&quot;&gt;&lt;pre class=&quot;language-glsl&quot; aria-hidden=&quot;true&quot;&gt;&lt;code class=&quot;language-glsl&quot;&gt;&lt;span class=&quot;token keyword&quot;&gt;void&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;mainImage&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token keyword&quot;&gt;out&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec4&lt;/span&gt; O&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt; I&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&amp;#10;&lt;span class=&quot;token punctuation&quot;&gt;{&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token comment&quot;&gt;// Centre the coords and keep them square regardless of canvas size.&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt; uv &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;2.0&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; I &lt;span class=&quot;token operator&quot;&gt;-&lt;/span&gt; iResolution&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xy&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;/&lt;/span&gt; iResolution&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;y&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&amp;#10;    &lt;span class=&quot;token comment&quot;&gt;// Domain warping: fold the space back on itself a few times.&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; t &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; iTime &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;for&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; i &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt; i &lt;span class=&quot;token operator&quot;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;6.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt; i&lt;span class=&quot;token operator&quot;&gt;++&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;{&lt;/span&gt;&amp;#10;        uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;x &lt;span class=&quot;token operator&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.32&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;/&lt;/span&gt; i &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;i &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;2.4&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;y &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; t &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;2.0&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; i&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;        uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;y &lt;span class=&quot;token operator&quot;&gt;+=&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.30&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;/&lt;/span&gt; i &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;i &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;1.9&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;x &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; t &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;1.7&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; i&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token punctuation&quot;&gt;}&lt;/span&gt;&amp;#10;&amp;#10;    &lt;span class=&quot;token comment&quot;&gt;// Ramp between a mossy green and a warm amber.&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; v &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.5&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.5&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;x &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;2.0&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; uv&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;y &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;2.0&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; iTime &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.4&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt; green &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.30&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.48&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.32&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt; amber &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.93&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.78&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.42&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt; col &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;mix&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;green&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; amber&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; v&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.7&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.3&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; v&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&amp;#10;    O &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec4&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;col&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&lt;span class=&quot;token punctuation&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;textarea class=&quot;shader__input&quot; spellcheck=&quot;false&quot; autocapitalize=&quot;off&quot; autocomplete=&quot;off&quot; autocorrect=&quot;off&quot; aria-label=&quot;Editable GLSL source&quot;&gt;void mainImage(out vec4 O, in vec2 I)&amp;#10;{&amp;#10;    // Centre the coords and keep them square regardless of canvas size.&amp;#10;    vec2 uv = (2.0 * I - iResolution.xy) / iResolution.y;&amp;#10;&amp;#10;    // Domain warping: fold the space back on itself a few times.&amp;#10;    float t = iTime * 0.15;&amp;#10;    for (float i = 1.0; i &amp;lt; 6.0; i++) {&amp;#10;        uv.x += 0.32 / i * sin(i * 2.4 * uv.y + t * 2.0 + i);&amp;#10;        uv.y += 0.30 / i * cos(i * 1.9 * uv.x + t * 1.7 + i);&amp;#10;    }&amp;#10;&amp;#10;    // Ramp between a mossy green and a warm amber.&amp;#10;    float v = 0.5 + 0.5 * sin(uv.x * 2.0 + uv.y * 2.0 + iTime * 0.4);&amp;#10;    vec3 green = vec3(0.30, 0.48, 0.32);&amp;#10;    vec3 amber = vec3(0.93, 0.78, 0.42);&amp;#10;    vec3 col = mix(green, amber, v) * (0.7 + 0.3 * v);&amp;#10;&amp;#10;    O = vec4(col, 1.0);&amp;#10;}&lt;/textarea&gt;&lt;/div&gt;&lt;/div&gt;&lt;/figure&gt;
&lt;p&gt;The embed follows the &lt;strong&gt;Shadertoy convention&lt;/strong&gt;, so anything you prototype there drops
straight in: write &lt;code&gt;mainImage(out vec4, in vec2)&lt;/code&gt; and you get &lt;code&gt;iResolution&lt;/code&gt;, &lt;code&gt;iTime&lt;/code&gt;, and
&lt;code&gt;iMouse&lt;/code&gt; for free.&lt;/p&gt;
&lt;h2&gt;It reads the pointer, too&lt;/h2&gt;
&lt;p&gt;&lt;code&gt;iMouse.xy&lt;/code&gt; is the cursor in pixels — hover the canvas below and the glow follows you.&lt;/p&gt;
&lt;figure class=&quot;shader&quot; data-shader=&quot;void mainImage(out vec4 O, in vec2 I)&amp;#10;{&amp;#10;    vec2 uv = I / iResolution.xy;&amp;#10;    vec2 m  = iMouse.xy / iResolution.xy;&amp;#10;&amp;#10;    // Fall off from the cursor; drift to centre before it&#39;s ever touched.&amp;#10;    if (iMouse.xy == vec2(0.0)) m = vec2(0.5);&amp;#10;    float d = distance(uv, m);&amp;#10;    float glow = smoothstep(0.45, 0.0, d);&amp;#10;&amp;#10;    vec3 col = mix(vec3(0.11, 0.12, 0.10), vec3(0.47, 0.68, 0.51), glow);&amp;#10;    O = vec4(col, 1.0);&amp;#10;}&quot; style=&quot;--shader-h:220px&quot;&gt;&lt;div class=&quot;shader__stage&quot;&gt;&lt;canvas class=&quot;shader__canvas&quot;&gt;&lt;/canvas&gt;&lt;div class=&quot;shader__bar&quot;&gt;&lt;button type=&quot;button&quot; class=&quot;shader__btn&quot; data-act=&quot;toggle&quot; aria-label=&quot;Play or pause&quot;&gt;❚❚&lt;/button&gt;&lt;button type=&quot;button&quot; class=&quot;shader__btn&quot; data-act=&quot;reset&quot; aria-label=&quot;Restart&quot;&gt;↺&lt;/button&gt;&lt;span class=&quot;shader__clock&quot; aria-hidden=&quot;true&quot;&gt;0.0s&lt;/span&gt;&lt;/div&gt;&lt;p class=&quot;shader__err&quot; role=&quot;alert&quot; hidden=&quot;&quot;&gt;&lt;/p&gt;&lt;/div&gt;&lt;figcaption class=&quot;shader__caption&quot;&gt;Move your cursor over me — a soft light tracking iMouse.&lt;/figcaption&gt;&lt;div class=&quot;shader__code&quot;&gt;&lt;span class=&quot;shader__file&quot;&gt;fragment.glsl&lt;/span&gt;&lt;pre class=&quot;language-glsl&quot;&gt;&lt;code class=&quot;language-glsl&quot;&gt;&lt;span class=&quot;token keyword&quot;&gt;void&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;mainImage&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token keyword&quot;&gt;out&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec4&lt;/span&gt; O&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;in&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt; I&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&amp;#10;&lt;span class=&quot;token punctuation&quot;&gt;{&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt; uv &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; I &lt;span class=&quot;token operator&quot;&gt;/&lt;/span&gt; iResolution&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xy&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt; m  &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; iMouse&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xy &lt;span class=&quot;token operator&quot;&gt;/&lt;/span&gt; iResolution&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xy&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&amp;#10;    &lt;span class=&quot;token comment&quot;&gt;// Fall off from the cursor; drift to centre before it&#39;s ever touched.&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;if&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;iMouse&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xy &lt;span class=&quot;token operator&quot;&gt;==&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt; m &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec2&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; d &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;distance&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;uv&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; m&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; glow &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;smoothstep&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.45&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; d&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&amp;#10;    &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt; col &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;mix&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.11&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.12&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.10&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec3&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;0.47&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.68&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;0.51&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; glow&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;    O &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;vec4&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;col&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token number&quot;&gt;1.0&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&amp;#10;&lt;span class=&quot;token punctuation&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/figure&gt;
&lt;h2&gt;The HLSL side&lt;/h2&gt;
&lt;p&gt;WebGL only speaks GLSL, so HLSL can&#39;t &lt;em&gt;run&lt;/em&gt; here — but it still gets first-class
highlighting, which is what you want when the source of truth is an Unreal &lt;strong&gt;Custom&lt;/strong&gt; node
or a Unity surface function. Here&#39;s a rim-light term I reach for constantly:&lt;/p&gt;
&lt;pre class=&quot;language-hlsl&quot;&gt;&lt;code class=&quot;language-hlsl&quot;&gt;&lt;span class=&quot;token comment&quot;&gt;// Unreal &quot;Custom&quot; node — Fresnel rim term.&lt;/span&gt;
&lt;span class=&quot;token comment&quot;&gt;// In:  N (float3 world normal), V (float3 view dir), power, scale, tint&lt;/span&gt;
&lt;span class=&quot;token comment&quot;&gt;// Out: float3 additive rim colour&lt;/span&gt;
&lt;span class=&quot;token keyword&quot;&gt;float3&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;RimLight&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token keyword&quot;&gt;float3&lt;/span&gt; N&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;float3&lt;/span&gt; V&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; power&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; scale&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token keyword&quot;&gt;float3&lt;/span&gt; tint&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;
&lt;span class=&quot;token punctuation&quot;&gt;{&lt;/span&gt;
    &lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; fresnel &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;pow&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token function&quot;&gt;saturate&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token number&quot;&gt;1.0&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;-&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;dot&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;token function&quot;&gt;normalize&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;N&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;normalize&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;V&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;,&lt;/span&gt; power&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;
    &lt;span class=&quot;token keyword&quot;&gt;return&lt;/span&gt; tint &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; &lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;fresnel &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; scale&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;
&lt;span class=&quot;token punctuation&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;When a preview actually helps, I port the effect to a GLSL &lt;code&gt;mainImage&lt;/code&gt; and drop it in a
live canvas next to the HLSL — same maths, running proof. The HLSL stays the reference; the
GLSL is the demo.&lt;/p&gt;
&lt;p&gt;That&#39;s the whole system: &lt;strong&gt;GLSL runs, HLSL is shown, and the source is never a screenshot.&lt;/strong&gt;&lt;/p&gt;
</content>
    </entry>
    <entry>
        <title>Multi-tag test: shading a stylised waterfall</title>
        <link href="https://javieramador.dev/posts/post-5/"/>
        <updated>2026-07-13T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-5/</id>
        <summary>A mockup post that lives under two topics at once — used to test multi-tag filtering. Safe to delete.</summary>
        <content type="html">&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note to self:&lt;/strong&gt; throwaway post to test posts that carry more than one category
(&lt;code&gt;categories: [environment, shaders]&lt;/code&gt;). It should appear under both the &lt;strong&gt;environment&lt;/strong&gt;
and &lt;strong&gt;shaders&lt;/strong&gt; filters on &lt;code&gt;/blog/&lt;/code&gt;. Delete before launch.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;A waterfall is a nice test case because it sits squarely between two disciplines: the
&lt;strong&gt;environment art&lt;/strong&gt; of the rock shelf and surrounding scatter, and the &lt;strong&gt;shader&lt;/strong&gt; work that
sells the moving water. This post is tagged with both.&lt;/p&gt;
&lt;h2&gt;The environment side&lt;/h2&gt;
&lt;p&gt;The silhouette does most of the work. A modular rock kit gives the ledge its stepped
profile, and a light scatter of wet pebbles and moss breaks up the hard edges where the
water meets stone.&lt;/p&gt;
&lt;h2&gt;The shader side&lt;/h2&gt;
&lt;p&gt;The water itself is two scrolling normal maps at different speeds, plus a foam mask driven
by depth so the white water gathers where the surface meets geometry. Nothing fancy — just
enough motion to read as &amp;quot;falling.&amp;quot;&lt;/p&gt;
&lt;h2&gt;Why tag it twice&lt;/h2&gt;
&lt;p&gt;Some work genuinely belongs to more than one topic. Tagging this under both
&lt;strong&gt;environment&lt;/strong&gt; and &lt;strong&gt;shaders&lt;/strong&gt; means it surfaces whichever lens a reader is browsing
through — and it&#39;s a good check that the filter matches on &lt;em&gt;any&lt;/em&gt; of a post&#39;s tags, not just
the first.&lt;/p&gt;
</content>
    </entry>
    <entry>
        <title>Math test: the rendering equation, briefly</title>
        <link href="https://javieramador.dev/posts/post-4/"/>
        <updated>2026-07-13T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-4/</id>
        <summary>A KaTeX-heavy mockup post used to test inline and block math rendering. Safe to delete.</summary>
        <content type="html">&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note to self:&lt;/strong&gt; mockup post to test KaTeX (inline + block, fractions, integrals, Greek,
matrices). Delete before launch.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Almost everything a technical artist touches in lighting traces back to one equation. It
looks intimidating written out, but each piece has a plain-language job. Here it is.&lt;/p&gt;
&lt;h2&gt;The equation&lt;/h2&gt;
&lt;p&gt;The outgoing radiance &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L_o&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; from a point &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;x&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; in direction &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;omega_o&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the emitted light
plus everything reflected from the hemisphere &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;Omega&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;Ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; above the surface:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Ω&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L_o(x, &#92;omega_o) = L_e(x, &#92;omega_o) + &#92;int_{&#92;Omega} f_r(x, &#92;omega_i, &#92;omega_o)&#92;, L_i(x, &#92;omega_i)&#92;,(&#92;omega_i &#92;cdot n)&#92;, d&#92;omega_i
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;e&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.2719em;vertical-align:-0.9119em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol large-op&quot; style=&quot;margin-right:0.4445em;position:relative;top:-0.0011em;&quot;&gt;∫&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:-0.4336em;&quot;&gt;&lt;span style=&quot;top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;Ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9119em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Reading it left to right: take the emitted light &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L_e&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;e&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, then add up — that&#39;s the integral
&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Ω&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;int_&#92;Omega&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1608em;vertical-align:-0.3558em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;margin-right:0.1945em;position:relative;top:-0.0006em;&quot;&gt;∫&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1225em;&quot;&gt;&lt;span style=&quot;top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;Ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3558em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — the contribution of every incoming direction &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;omega_i&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, weighted by how the
surface reflects it and how grazing the angle is.&lt;/p&gt;
&lt;h2&gt;The pieces&lt;/h2&gt;
&lt;ul&gt;
&lt;li&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;L_i(x, &#92;omega_i)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;L&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — the light &lt;em&gt;arriving&lt;/em&gt; from direction &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;omega_i&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo separator=&quot;true&quot;&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f_r(x, &#92;omega_i, &#92;omega_o)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mpunct&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;o&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — the &lt;strong&gt;BRDF&lt;/strong&gt;, the surface&#39;s rule for turning incoming light
into outgoing light.&lt;/li&gt;
&lt;li&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;omega_i &#92;cdot n&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5945em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;⋅&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; — the cosine term, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;cos&#92;theta_i&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8444em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3117em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;. Light hitting head-on counts fully;
light at a grazing angle barely counts at all.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;That cosine is why a floor looks bright underfoot and dim toward the horizon.&lt;/p&gt;
&lt;h2&gt;A concrete BRDF&lt;/h2&gt;
&lt;p&gt;For a simple diffuse surface with albedo &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;rho&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ρ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, the BRDF is just a constant:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f_r = &#92;frac{&#92;rho}{&#92;pi}
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1514em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0278em;&quot;&gt;r&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.7936em;vertical-align:-0.686em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.1076em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ρ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;pi&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; in the denominator is the part everyone forgets — it&#39;s what keeps energy conserved
so a surface never reflects more light than it receives.&lt;/p&gt;
&lt;p&gt;Specular is where it gets interesting. The Fresnel term, using Schlick&#39;s approximation, is:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F(&#92;theta) = F_0 + (1 - F_0)&#92;,(1 - &#92;cos&#92;theta)^5
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1141em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8641em;&quot;&gt;&lt;span style=&quot;top:-3.113em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;5&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;where &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F_0&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the reflectance at normal incidence. For most dielectrics &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mn&gt;0.04&lt;/mn&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;F_0 &#92;approx 0.04&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8333em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1389em;&quot;&gt;F&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3011em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6444em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;0.04&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;,
which is why even matte plastic gets a bright rim at grazing angles.&lt;/p&gt;
&lt;h2&gt;Why the integral is hard&lt;/h2&gt;
&lt;p&gt;We can&#39;t evaluate &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Ω&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;int_&#92;Omega&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1608em;vertical-align:-0.3558em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol small-op&quot; style=&quot;margin-right:0.1945em;position:relative;top:-0.0006em;&quot;&gt;∫&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.1225em;&quot;&gt;&lt;span style=&quot;top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;Ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3558em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; exactly in real time, so we approximate it. Monte Carlo
integration samples &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;N&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; random directions and averages:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;Ω&lt;/mi&gt;&lt;/msub&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;int_{&#92;Omega} f(&#92;omega)&#92;, d&#92;omega &#92;approx &#92;frac{1}{N} &#92;sum_{k=1}^{N} &#92;frac{f(&#92;omega_k)}{p(&#92;omega_k)}
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.2719em;vertical-align:-0.9119em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;&lt;span class=&quot;mop op-symbol large-op&quot; style=&quot;margin-right:0.4445em;position:relative;top:-0.0011em;&quot;&gt;∫&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:-0.4336em;&quot;&gt;&lt;span style=&quot;top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;Ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.9119em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;≈&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:3.1304em;vertical-align:-1.3021em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.3214em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.686em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop op-limits&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.8283em;&quot;&gt;&lt;span style=&quot;top:-1.8479em;margin-left:0em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;mrel mtight&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span&gt;&lt;span class=&quot;mop op-symbol large-op&quot;&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-4.3em;margin-left:0em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3.05em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.109em;&quot;&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.3021em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mopen nulldelimiter&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mfrac&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.427em;&quot;&gt;&lt;span style=&quot;top:-2.314em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.23em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;frac-line&quot; style=&quot;border-bottom-width:0.04em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-3.677em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.936em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose nulldelimiter&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;p(&#92;omega_k)&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot; style=&quot;margin-right:0.0315em;&quot;&gt;k&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the probability of picking each sample — importance sampling just means
choosing &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;p&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.625em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;p&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; to match where &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;f&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.1076em;&quot;&gt;f&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is large, so you spend samples where they matter.&lt;/p&gt;
&lt;h2&gt;A tiny bit of linear algebra&lt;/h2&gt;
&lt;p&gt;Transforming a normal isn&#39;t the same as transforming a position. If &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;M&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;M&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is your model
matrix, normals must use the inverse-transpose:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo mathvariant=&quot;normal&quot; lspace=&quot;0em&quot; rspace=&quot;0em&quot;&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi mathvariant=&quot;sans-serif&quot;&gt;T&lt;/mi&gt;&lt;/msup&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;n&amp;#x27; = (M^{-1})^{&#92;mathsf{T}}&#92;, n
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8019em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8019em;&quot;&gt;&lt;span style=&quot;top:-3.113em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;′&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1.1491em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.109em;&quot;&gt;M&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8641em;&quot;&gt;&lt;span style=&quot;top:-3.113em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mord mtight&quot;&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.8991em;&quot;&gt;&lt;span style=&quot;top:-3.113em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mtight&quot;&gt;&lt;span class=&quot;mord mathsf mtight&quot;&gt;T&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;For a rotation like&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo fence=&quot;true&quot;&gt;[&lt;/mo&gt;&lt;mtable rowspacing=&quot;0.16em&quot; columnalign=&quot;center center&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mstyle scriptlevel=&quot;0&quot; displaystyle=&quot;false&quot;&gt;&lt;mrow&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mstyle&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mstyle scriptlevel=&quot;0&quot; displaystyle=&quot;false&quot;&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mstyle&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mstyle scriptlevel=&quot;0&quot; displaystyle=&quot;false&quot;&gt;&lt;mrow&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mstyle&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mstyle scriptlevel=&quot;0&quot; displaystyle=&quot;false&quot;&gt;&lt;mrow&gt;&lt;mi&gt;cos&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/mstyle&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo fence=&quot;true&quot;&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R = &#92;begin{bmatrix} &#92;cos&#92;theta &amp;amp; -&#92;sin&#92;theta &#92;&#92; &#92;sin&#92;theta &amp;amp; &#92;cos&#92;theta &#92;end{bmatrix}
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:2.4em;vertical-align:-0.95em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;minner&quot;&gt;&lt;span class=&quot;mopen delimcenter&quot; style=&quot;top:0em;&quot;&gt;&lt;span class=&quot;delimsizing size3&quot;&gt;[&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mtable&quot;&gt;&lt;span class=&quot;col-align-c&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.45em;&quot;&gt;&lt;span style=&quot;top:-3.61em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.41em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.95em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;arraycolsep&quot; style=&quot;width:0.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;arraycolsep&quot; style=&quot;width:0.5em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;col-align-c&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:1.45em;&quot;&gt;&lt;span style=&quot;top:-3.61em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord&quot;&gt;−&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style=&quot;top:-2.41em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:3em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mop&quot;&gt;cos&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0278em;&quot;&gt;θ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.95em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;mclose delimcenter&quot; style=&quot;top:0em;&quot;&gt;&lt;span class=&quot;delimsizing size3&quot;&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;the inverse-transpose equals &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;R&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0077em;&quot;&gt;R&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; itself, which is why pure rotations never distort your
normals — but non-uniform scale does, and that is the usual culprit behind broken lighting.&lt;/p&gt;
&lt;h2&gt;Wrapping up&lt;/h2&gt;
&lt;p&gt;You rarely type these equations into a shader directly, but knowing what each term &lt;em&gt;does&lt;/em&gt;
turns &amp;quot;the lighting looks wrong&amp;quot; into a diagnosis. The cosine, the &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;pi&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;π&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, the Fresnel rim —
each one is a knob with a physical meaning.&lt;/p&gt;
&lt;p&gt;If the math above rendered as crisp symbols rather than raw dollar signs, KaTeX is working.&lt;/p&gt;
</content>
    </entry>
    <entry>
        <title>Scroll test: building a modular rock kit</title>
        <link href="https://javieramador.dev/posts/post-3/"/>
        <updated>2026-07-13T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-3/</id>
        <summary>A deliberately long post used to test the back-to-top button and long-form prose styling. Safe to delete.</summary>
        <content type="html">&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note to self:&lt;/strong&gt; this is a throwaway post to test the back-to-top button and how a long
article reads. Delete it before launch.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;p&gt;Modular kits are the backbone of any large environment. Instead of sculpting a whole cliff
face as one giant mesh, you build a small set of reusable pieces and assemble them like
LEGO. Done well, a kit of eight rocks can dress an entire canyon and still tile cleanly at
every seam. This post walks through how I approach one.&lt;/p&gt;
&lt;h2&gt;Why modular&lt;/h2&gt;
&lt;p&gt;The temptation early on is to sculpt everything bespoke. It looks great in isolation, but it
does not scale: memory balloons, iteration slows to a crawl, and the moment a level designer
wants the cliff two metres taller you are back in ZBrush for an afternoon.&lt;/p&gt;
&lt;p&gt;A kit flips that trade. You spend more time up front on a handful of pieces, and in return
you get:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Fast blockouts&lt;/strong&gt; — designers grey-box with the real assets from day one.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Consistent silhouette&lt;/strong&gt; — every rock shares the same material and wear language.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Cheap memory&lt;/strong&gt; — a few textures and meshes, instanced hundreds of times.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Painless changes&lt;/strong&gt; — retexture once, and the whole canyon updates.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;The base set&lt;/h2&gt;
&lt;p&gt;I usually start with the smallest set that still reads as varied. Eight shapes is my
comfort zone:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;A large hero boulder with a strong directional silhouette.&lt;/li&gt;
&lt;li&gt;Two medium blockers that hide seams between larger pieces.&lt;/li&gt;
&lt;li&gt;Three small scatter rocks for breaking up flat ground.&lt;/li&gt;
&lt;li&gt;A wide, flat &amp;quot;shelf&amp;quot; piece for ledges and steps.&lt;/li&gt;
&lt;li&gt;A tall spire for verticality and reading distance.&lt;/li&gt;
&lt;/ol&gt;
&lt;p&gt;The trick is that none of these should look finished on their own. A modular piece that
looks perfect in isolation is usually too distinctive to repeat.&lt;/p&gt;
&lt;h3&gt;A note on scale&lt;/h3&gt;
&lt;p&gt;Model everything to real-world scale from the start. If your engine treats one unit as one
metre, a hero boulder around three metres tall feels right next to a human character. Getting
this wrong early means refitting every texture density later.&lt;/p&gt;
&lt;h2&gt;Sculpting for tiling&lt;/h2&gt;
&lt;p&gt;The seams are where modular kits live or die. Two habits keep them invisible:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Keep the &lt;strong&gt;contact edges flat and consistent&lt;/strong&gt; in height, so pieces butt together without
gaps or intersections poking through.&lt;/li&gt;
&lt;li&gt;Push the interesting detail toward the &lt;strong&gt;centre of each piece&lt;/strong&gt;, away from the seam, so the
eye is drawn to the silhouette rather than the join.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Here is the rough density I aim for when I retopologise:&lt;/p&gt;
&lt;pre&gt;&lt;code&gt;hero boulder     ~4,000 tris
medium blocker   ~1,500 tris
small scatter     ~600 tris
shelf             ~1,200 tris
spire            ~2,000 tris
&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Those are starting points, not rules. A rock read from thirty metres away can shed half its
triangles and nobody will notice.&lt;/p&gt;
&lt;h2&gt;Texturing the set&lt;/h2&gt;
&lt;p&gt;I bake a single trim-friendly material and drive variation with vertex colours rather than
unique textures per rock. A three-channel mask goes a long way:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Red&lt;/strong&gt; — moss and lichen in the crevices.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Green&lt;/strong&gt; — sand and dust accumulation on upward faces.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Blue&lt;/strong&gt; — wet, darkened rock near the base.&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Blending those in the shader means one material serves a mossy forest floor and a dry desert
mesa, just by changing the mask weights per instance.&lt;/p&gt;
&lt;h3&gt;Wear and edges&lt;/h3&gt;
&lt;p&gt;Edge wear is the cheapest way to sell &amp;quot;rock&amp;quot;. A curvature map driving a lighter, chipped
colour along convex edges reads as chunks that have broken away over time. Keep it subtle —
the moment every edge is bright, the illusion collapses into a cartoon.&lt;/p&gt;
&lt;h2&gt;Placing the kit&lt;/h2&gt;
&lt;p&gt;With the pieces built, dressing a scene becomes a composition problem rather than a modelling
one. A few habits that help:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Rotate and scale aggressively.&lt;/strong&gt; The same boulder at 0.8x and rotated 140 degrees reads as
a different rock. Non-uniform scale within reason is your friend.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Bury the seams.&lt;/strong&gt; Sink scatter rocks slightly into the ground and let smaller pieces hide
the joins between big ones.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Vary the spacing.&lt;/strong&gt; Nature clumps. Tight clusters with open gaps look far more believable
than an even scatter.&lt;/li&gt;
&lt;/ul&gt;
&lt;blockquote&gt;
&lt;p&gt;The goal is that a player never counts eight rocks. They see a canyon.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2&gt;Lighting and final read&lt;/h2&gt;
&lt;p&gt;None of this matters if the lighting flattens it. I always dress and light in tandem: a
grazing key light rakes across the silhouettes and makes the sculpted detail earn its
triangles. Ambient occlusion in the crevices ties the pieces to the ground and hides the
last of the seams.&lt;/p&gt;
&lt;h2&gt;A quick checklist&lt;/h2&gt;
&lt;p&gt;Before I call a kit done, I run through this:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;Do the pieces tile at every rotation without gaps?&lt;/li&gt;
&lt;li&gt;Does the silhouette read from the intended viewing distance?&lt;/li&gt;
&lt;li&gt;Is the triangle budget justified by what the camera actually sees?&lt;/li&gt;
&lt;li&gt;Can I redress the kit into a &lt;em&gt;different&lt;/em&gt; biome with only mask changes?&lt;/li&gt;
&lt;li&gt;Would a level designer be able to build with it without asking me anything?&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;If the answer to all five is yes, the kit is ready to ship — and I never have to open ZBrush
just because someone wanted the cliff a little taller.&lt;/p&gt;
&lt;h2&gt;Wrapping up&lt;/h2&gt;
&lt;p&gt;Modular kits are less about modelling skill and more about restraint and planning. Build the
smallest set that reads as varied, keep the seams boring, and let placement and lighting do
the heavy lifting. Your future self — and every designer on the team — will thank you.&lt;/p&gt;
&lt;p&gt;Scroll back up with the little arrow in the corner. That is what it is here to test.&lt;/p&gt;
</content>
    </entry>
    <entry>
        <title>A tiny shader for wind</title>
        <link href="https://javieramador.dev/posts/post-2/"/>
        <updated>2026-06-24T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-2/</id>
        <summary>A minimal vertex shader that adds believable wind to foliage without much setup.</summary>
        <content type="html">&lt;p&gt;Wind is one of those effects that sells a scene instantly. Here&#39;s the smallest version
I keep reaching for.&lt;/p&gt;
&lt;h2&gt;The idea&lt;/h2&gt;
&lt;p&gt;Offset each vertex by a sine wave, scaled by how high up the vertex sits — so the trunk
stays put and the tips move most.&lt;/p&gt;
&lt;pre class=&quot;language-glsl&quot;&gt;&lt;code class=&quot;language-glsl&quot;&gt;&lt;span class=&quot;token keyword&quot;&gt;float&lt;/span&gt; sway &lt;span class=&quot;token operator&quot;&gt;=&lt;/span&gt; &lt;span class=&quot;token function&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;token punctuation&quot;&gt;(&lt;/span&gt;time &lt;span class=&quot;token operator&quot;&gt;+&lt;/span&gt; worldPos&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;x&lt;span class=&quot;token punctuation&quot;&gt;)&lt;/span&gt; &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; heightMask&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;
position&lt;span class=&quot;token punctuation&quot;&gt;.&lt;/span&gt;xz &lt;span class=&quot;token operator&quot;&gt;+=&lt;/span&gt; sway &lt;span class=&quot;token operator&quot;&gt;*&lt;/span&gt; strength&lt;span class=&quot;token punctuation&quot;&gt;;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;
&lt;p&gt;Written out, the horizontal offset for a vertex is &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;o = A &#92;sin(&#92;omega t + &#92;phi)&#92;, m_h&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;o&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ϕ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;,
where &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;m_h&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.5806em;vertical-align:-0.15em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is the height mask and &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;A&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.6833em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;omega&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;&#92;phi&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.8889em;vertical-align:-0.1944em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ϕ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; are amplitude, frequency and phase:&lt;/p&gt;
&lt;p class=&quot;katex-block &quot;&gt;&lt;span class=&quot;katex-display&quot;&gt;&lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;sin&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mtext&gt; &lt;/mtext&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;annotation encoding=&quot;application/x-tex&quot;&gt;o = A &#92;sin(&#92;omega t + &#92;phi)&#92;, m_h
&lt;/annotation&gt;&lt;/semantics&gt;&lt;/math&gt;&lt;/span&gt;&lt;span class=&quot;katex-html&quot; aria-hidden=&quot;true&quot;&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:0.4306em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;o&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mrel&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2778em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;A&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mop&quot;&gt;sin&lt;/span&gt;&lt;span class=&quot;mopen&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot; style=&quot;margin-right:0.0359em;&quot;&gt;ω&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;t&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mbin&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.2222em;&quot;&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;base&quot;&gt;&lt;span class=&quot;strut&quot; style=&quot;height:1em;vertical-align:-0.25em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;ϕ&lt;/span&gt;&lt;span class=&quot;mclose&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;mspace&quot; style=&quot;margin-right:0.1667em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;mord&quot;&gt;&lt;span class=&quot;mord mathnormal&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;msupsub&quot;&gt;&lt;span class=&quot;vlist-t vlist-t2&quot;&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.3361em;&quot;&gt;&lt;span style=&quot;top:-2.55em;margin-left:0em;margin-right:0.05em;&quot;&gt;&lt;span class=&quot;pstrut&quot; style=&quot;height:2.7em;&quot;&gt;&lt;/span&gt;&lt;span class=&quot;sizing reset-size6 size3 mtight&quot;&gt;&lt;span class=&quot;mord mathnormal mtight&quot;&gt;h&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-s&quot;&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class=&quot;vlist-r&quot;&gt;&lt;span class=&quot;vlist&quot; style=&quot;height:0.15em;&quot;&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;Add a second wave at a different frequency and you get something that reads as natural
motion rather than a metronome.&lt;/p&gt;
</content>
    </entry>
    <entry>
        <title>Building a stylised forest floor</title>
        <link href="https://javieramador.dev/posts/post-1/"/>
        <updated>2026-06-02T00:00:00Z</updated>
        <id>https://javieramador.dev/posts/post-1/</id>
        <summary>Layered materials and a scatter workflow for a believable, stylised forest floor.</summary>
        <content type="html">&lt;p&gt;A quick breakdown of how I approach a stylised forest floor — from the base material
layers to the scatter pass that ties everything together.&lt;/p&gt;
&lt;h2&gt;Starting with the ground material&lt;/h2&gt;
&lt;p&gt;The base is three blended layers: soil, moss, and scattered leaf litter. Keeping each
layer independent means I can tune coverage without repainting.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Soil&lt;/strong&gt; — the darkest value, does most of the shadow work.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Moss&lt;/strong&gt; — a cooler mid-tone that reads as &amp;quot;damp&amp;quot;.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Leaf litter&lt;/strong&gt; — warm accents, used sparingly.&lt;/li&gt;
&lt;/ul&gt;
&lt;blockquote&gt;
&lt;p&gt;The trick with stylised work is restraint: let a few strong shapes carry the read.&lt;/p&gt;
&lt;/blockquote&gt;
&lt;h2&gt;The scatter pass&lt;/h2&gt;
&lt;p&gt;Once the material reads well up close, I scatter meshes across it — small rocks, ferns,
and fallen branches — using vertex colours to mask density.&lt;/p&gt;
&lt;p&gt;That&#39;s the whole loop: material first, scatter second, lighting last.&lt;/p&gt;
</content>
    </entry>
</feed>
