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Lecture13.html
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</style><title>Lecture13</title>
</head>
<body class='typora-export os-windows typora-export-show-outline typora-export-collapse-outline'><div class='typora-export-content'>
<div class="typora-export-sidebar"><div class="outline-content"><li class="outline-item-wrapper outline-h1"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#games101-lecture-13---ray-tracing-1-whitted-style-ray-tracing">GAMES101 Lecture 13 - Ray Tracing 1 (Whitted-Style Ray Tracing)</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-why-ray-tracing">I. Why Ray Tracing?</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ii-basic-ray-tracing-algorithms">II. Basic Ray-Tracing Algorithms</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ray-casting">Ray Casting</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#recursive-whitted-style-ray-tracing">Recursive (Whitted-Style) Ray Tracing</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h3 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ray-surface-intersection">Ray-Surface Intersection</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#accelerating-ray-surface-intersection">Accelerating Ray-Surface Intersection</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#performance-challenges">Performance Challenges</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#bounding-volumes">Bounding Volumes</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ray-intersection-with-box">Ray-Intersection With Box</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-13---ray-tracing-1-whitted-style-ray-tracing'><span>GAMES101 Lecture 13 - Ray Tracing 1 (Whitted-Style Ray Tracing)</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_13.pdf'><span>GAMES101_Lecture_13.pdf</span></a></p><h2 id='i-why-ray-tracing'><span>I. Why Ray Tracing?</span></h2><ul><li><p><span>Rasterization couldn't handle </span><strong><span>global</span></strong><span> effects well</span></p><ul><li><p><span>(Soft) shadows</span></p></li><li><p><span>Especially when the light sources </span><strong><span>bounce more than once</span></strong></p><ul><li><p><em><span>Glossy reflection</span></em></p></li><li><p><em><span>Indirect Illumination</span></em></p></li></ul></li></ul></li><li><p><span>Rasterization is fast, but quality is </span><strong><span>relatively low</span></strong></p></li><li><p><span>Ray tracing is accurate, but is </span><strong><span>very slow</span></strong></p><ul><li><p><span>Rasterization: </span><em><span>real-time</span></em></p></li><li><p><span>Ray tracing: </span><em><span>offline</span></em></p></li><li><p><span>~10K CPU core hours to render </span><strong><span>one frame</span></strong><span> in production</span></p></li></ul></li></ul><p> </p><h2 id='ii-basic-ray-tracing-algorithms'><span>II. Basic Ray-Tracing Algorithms</span></h2><p><span>Three assumptions:</span></p><ul><li><p><span>Light travels in straight lines </span><em><span>(physically wrong)</span></em></p></li><li><p><span>Light rays do not "collide" with each other if they cross </span><em><span>(physically wrong)</span></em></p></li><li><p><span>Light rays travel from the light sources to the eye (but the physics is invariant under path reversal - </span><em><span>reciprocity</span></em><span>)</span></p></li></ul><p> </p><h3 id='ray-casting'><span>Ray Casting</span></h3><p><img src="../images/Lecture13-img-1.png" alt="img-1" style="zoom:50%;" /></p><ul><li><p><strong><span>Pinhole Camera Model</span></strong></p></li><li><p><strong><span>Point Light Source</span></strong></p></li><li><p><span>Start an </span><em><span>eye ray</span></em><span> at the eye and goes through pixel.</span></p><ul><li><p><span>Collide with the first object, and then</span></p></li><li><p><span>(</span><strong><span>Local</span></strong><span>) Perform </span><strong><span>shading</span></strong><span> calculation to compute color of pixel</span></p></li></ul></li><li><p><span>Directly project to the light source after the eye ray has </span><strong><span>collide</span></strong><span> with an object.</span></p><ul><li><p><em><span>Then the light bounces only </span><strong><span>once</span></strong></em><span>.</span></p></li></ul></li></ul><p> </p><h3 id='recursive-whitted-style-ray-tracing'><span>Recursive (Whitted-Style) Ray Tracing</span></h3><p><em><span>"An improved illumination model for shaded display."</span></em><span> - T. Whitted, CACM 1980</span></p><p><span>Simulates the recursive bouncing of light rays.</span></p><ul><li><p><span>Taking both </span><strong><span>Reflection</span></strong><span> and </span><strong><span>Refraction</span></strong><span> into consideration</span></p></li><li><p><span>Different rays:</span></p><p><img src="../images/Lecture13-img-2.png" alt="img-2" style="zoom:50%;" /></p><ul><li><p><em><span>Primary Ray</span></em><span>: Starts at the eye</span></p></li><li><p><em><span>Secondary Ray</span></em><span>: Generated upon collision of primary ray</span></p></li><li><p><em><span>Shadow Ray</span></em><span>: Directly connected from the intersection to the light source.</span></p></li></ul></li></ul><p> </p><h3 id='ray-surface-intersection'><span>Ray-Surface Intersection</span></h3><ul><li><p><strong><span>Ray Equation</span></strong><span>: A ray is defined by its origin </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-381-TEX-N-28"></use></g><g data-mml-node="mtext" transform="translate(389,0)"><use data-c="1D429" xlink:href="#MJX-381-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(1250.2,0)"><use data-c="2212" xlink:href="#MJX-381-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(2250.4,0)"><g data-mml-node="mtext"><use data-c="1D429" xlink:href="#MJX-381-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(672,413) scale(0.707)"><use data-c="2032" xlink:href="#MJX-381-TEX-V-2032"></use></g></g><g data-mml-node="mo" transform="translate(3166.9,0)"><use data-c="29" xlink:href="#MJX-381-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3778.1,0)"><use data-c="22C5" xlink:href="#MJX-381-TEX-N-22C5"></use></g><g data-mml-node="mtext" transform="translate(4278.3,0)"><use data-c="1D40D" xlink:href="#MJX-381-TEX-B-1D40D"></use></g><g data-mml-node="mo" transform="translate(5456.1,0)"><use data-c="3D" xlink:href="#MJX-381-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(6511.9,0)"><use data-c="28" xlink:href="#MJX-381-TEX-N-28"></use></g><g data-mml-node="mtext" transform="translate(6900.9,0)"><use data-c="1D428" xlink:href="#MJX-381-TEX-B-1D428"></use></g><g data-mml-node="mo" transform="translate(7698.1,0)"><use data-c="2B" xlink:href="#MJX-381-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(8698.3,0)"><use data-c="1D461" xlink:href="#MJX-381-TEX-I-1D461"></use></g><g data-mml-node="mtext" transform="translate(9059.3,0)"><use data-c="1D41D" xlink:href="#MJX-381-TEX-B-1D41D"></use></g><g data-mml-node="mo" transform="translate(9920.6,0)"><use data-c="2212" xlink:href="#MJX-381-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(10920.8,0)"><g data-mml-node="mtext"><use data-c="1D429" xlink:href="#MJX-381-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(672,413) scale(0.707)"><use data-c="2032" xlink:href="#MJX-381-TEX-V-2032"></use></g></g><g data-mml-node="mo" transform="translate(11837.2,0)"><use data-c="29" xlink:href="#MJX-381-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(12448.5,0)"><use data-c="22C5" xlink:href="#MJX-381-TEX-N-22C5"></use></g><g data-mml-node="mtext" transform="translate(12948.7,0)"><use data-c="1D40D" xlink:href="#MJX-381-TEX-B-1D40D"></use></g><g data-mml-node="mo" transform="translate(14126.5,0)"><use data-c="3D" xlink:href="#MJX-381-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(15182.2,0)"><use data-c="30" xlink:href="#MJX-381-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">(</mo><mtext mathvariant="bold">p</mtext><mo>−</mo><msup><mtext mathvariant="bold">p</mtext><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">)</mo><mo>⋅</mo><mtext mathvariant="bold">N</mtext><mo>=</mo><mo stretchy="false">(</mo><mtext mathvariant="bold">o</mtext><mo>+</mo><mi>t</mi><mtext mathvariant="bold">d</mtext><mo>−</mo><msup><mtext mathvariant="bold">p</mtext><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">)</mo><mo>⋅</mo><mtext mathvariant="bold">N</mtext><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>Solving the equation gives</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n262" cid="n262" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg 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data-c="1D429" xlink:href="#MJX-382-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(672,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-382-TEX-V-2032"></use></g></g><g data-mml-node="mo" transform="translate(1527.7,0)"><use data-c="2212" xlink:href="#MJX-382-TEX-N-2212"></use></g><g data-mml-node="mtext" transform="translate(2527.9,0)"><use data-c="1D428" xlink:href="#MJX-382-TEX-B-1D428"></use></g><g data-mml-node="mo" transform="translate(3102.9,0)"><use data-c="29" xlink:href="#MJX-382-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3714.1,0)"><use data-c="22C5" xlink:href="#MJX-382-TEX-N-22C5"></use></g><g data-mml-node="mi" transform="translate(4214.3,0)"><use data-c="1D441" xlink:href="#MJX-382-TEX-I-1D441"></use></g></g><g data-mml-node="mrow" transform="translate(1640.4,-686)"><g data-mml-node="mtext"><use data-c="1D41D" xlink:href="#MJX-382-TEX-B-1D41D"></use></g><g data-mml-node="mo" transform="translate(861.2,0)"><use data-c="22C5" xlink:href="#MJX-382-TEX-N-22C5"></use></g><g data-mml-node="mtext" transform="translate(1361.4,0)"><use data-c="1D40D" xlink:href="#MJX-382-TEX-B-1D40D"></use></g></g><rect width="5302.3" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7236.9,0)"><use data-c="2E" xlink:href="#MJX-382-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mtext mathvariant="bold">p</mtext><mo data-mjx-alternate="1">′</mo></msup><mo>−</mo><mtext mathvariant="bold">o</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mi>N</mi></mrow><mrow><mtext mathvariant="bold">d</mtext><mo>⋅</mo><mtext mathvariant="bold">N</mtext></mrow></mfrac><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><strong><span>Remember to check whether </span><mjx-container class="MathJax" jax="SVG" style="position: 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380 132 331T92 214Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-394-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(777.8,0)"><use data-c="2264" xlink:href="#MJX-394-TEX-N-2264"></use></g><g data-mml-node="mi" transform="translate(1833.6,0)"><use data-c="1D461" xlink:href="#MJX-394-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(2472.3,0)"><use data-c="2264" xlink:href="#MJX-394-TEX-N-2264"></use></g><g data-mml-node="mi" transform="translate(3528.1,0)"><use data-c="221E" xlink:href="#MJX-394-TEX-N-221E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi mathvariant="normal">∞</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex"> 0 \leq t \le \infty</script></strong><span>.</span></p></li><li><p><span>Test if the point is inside the triangle</span></p><p> </p></li></ol></li><li><p><strong><span>Möller Trumbore Algorithm:</span></strong><span> A faster approach by examining the barycentric coordinate.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n269" cid="n269" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="38.434ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 16987.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-383-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 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transform="translate(3270.3,0)"><use data-c="D7" xlink:href="#MJX-389-TEX-N-D7"></use></g><g data-mml-node="msub" transform="translate(4270.6,0)"><g data-mml-node="mtext"><use data-c="1D404" xlink:href="#MJX-389-TEX-B-1D404"></use></g><g data-mml-node="mn" transform="translate(789,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-389-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mtext mathvariant="bold">S</mtext><mn>2</mn></msub><mo>=</mo><mtext mathvariant="bold">S</mtext><mo>×</mo><msub><mtext mathvariant="bold">E</mtext><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container></div></div><p><em><span>See Cramer's rule.</span></em></p></li><li><p><em><span>Inside/outside test: Fire a ray from a given point. Even number of intersections if the point is outside.</span></em></p></li></ul></li></ul><p> </p><h3 id='accelerating-ray-surface-intersection'><span>Accelerating Ray-Surface Intersection</span></h3><h4 id='performance-challenges'><span>Performance Challenges</span></h4><p><span>Simple ray-scene intersection</span></p><ul><li><p><span>Exhaustively test ray-intersection with </span><strong><span>every triangle</span></strong></p></li><li><p><span>Find the closest hit (i.e. minimum </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-395-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-395-TEX-I-1D461"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">t</script><span>)</span></p></li></ul><p> </p><h4 id='bounding-volumes'><span>Bounding Volumes</span></h4><p><span>Build a larger volume for testing the potential intersection quickly.</span></p><ul><li><p><strong><span>Feature</span></strong><span>:</span></p><ul><li><p><span>Object is fully contained in the volume.</span></p></li><li><p><span>If the ray doesn't hit the volume, then it won't hit the object</span></p></li><li><p><span>Test BVol first, then test the object if BVol is hit.</span></p></li></ul></li><li><p><em><span>Can be in any shape as long as it speeds up the test.</span></em></p></li></ul><p> </p><h4 id='ray-intersection-with-box'><span>Ray-Intersection With Box</span></h4><ul><li><p><strong><span>Axis-Aligned Bounding Box (AABB)</span></strong></p><ul><li><p><span>For the 3D box, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="18.102ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 8001.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-396-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 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transform="translate(394,-150) scale(0.707)"><use data-c="6D" xlink:href="#MJX-396-TEX-N-6D"></use><use data-c="69" xlink:href="#MJX-396-TEX-N-69" transform="translate(833,0)"></use><use data-c="6E" xlink:href="#MJX-396-TEX-N-6E" transform="translate(1111,0)"></use></g></g><g data-mml-node="mo" transform="translate(2122.7,0)"><use data-c="7D" xlink:href="#MJX-396-TEX-N-7D"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mtext>enter</mtext></msub><mo>=</mo><mo data-mjx-texclass="OP" movablelimits="true">max</mo><mrow data-mjx-texclass="ORD"><mo fence="false" stretchy="false">{</mo><msub><mi>t</mi><mtext>min</mtext></msub><mo fence="false" stretchy="false">}</mo></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">t_\text{enter} = \max{\{t_\text{min}\}}</script><span>, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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data-mml-node="text"><use data-c="3C" xlink:href="#MJX-398-TEX-N-3C"></use></g><g data-mml-node="text" transform="translate(778,0)"><use data-c="3D" xlink:href="#MJX-398-TEX-N-3D"></use></g></g><g data-mml-node="msub" transform="translate(4128.9,0)"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-398-TEX-I-1D461"></use></g><g data-mml-node="TeXAtom" transform="translate(394,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mtext"><use data-c="65" xlink:href="#MJX-398-TEX-N-65"></use><use data-c="78" xlink:href="#MJX-398-TEX-N-78" transform="translate(444,0)"></use><use data-c="69" xlink:href="#MJX-398-TEX-N-69" transform="translate(972,0)"></use><use data-c="74" xlink:href="#MJX-398-TEX-N-74" transform="translate(1250,0)"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mtext>enter</mtext></mrow></msub><mo><=</mo><msub><mi>t</mi><mrow data-mjx-texclass="ORD"><mtext>exit</mtext></mrow></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">t_{\text{enter}} <= t_{\text{exit}}</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.775ex" height="1.864ex" role="img" focusable="false" viewBox="0 -666 3436.5 823.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-399-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 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257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z"></path><path id="MJX-399-TEX-N-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path id="MJX-399-TEX-N-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path><path id="MJX-399-TEX-N-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 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transform="translate(444,0)"></use><use data-c="69" xlink:href="#MJX-399-TEX-N-69" transform="translate(972,0)"></use><use data-c="74" xlink:href="#MJX-399-TEX-N-74" transform="translate(1250,0)"></use></g></g><g data-mml-node="mo" transform="translate(1880.7,0)"><use data-c="3E" xlink:href="#MJX-399-TEX-N-3E"></use></g><g data-mml-node="mn" transform="translate(2936.5,0)"><use data-c="30" xlink:href="#MJX-399-TEX-N-30"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mtext>exit</mtext></msub><mo>></mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">t_\text{exit} > 0</script><span>, then there is an intersection.</span></p><ul><li><p><span>Equality for BBoxes that contains an axis-aligned plane only</span></p></li></ul></li></ul></li><li><p><strong><span>Why Axis-Aligned?</span></strong></p><ul><li><p><span>General Plane:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n323" cid="n323" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.4ex" height="4.889ex" role="img" focusable="false" viewBox="0 -1469 7248.9 2161" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.566ex;"><defs><path id="MJX-390-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 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749Z"></path><path id="MJX-390-TEX-N-22C5" d="M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z"></path><path id="MJX-390-TEX-B-1D40D" d="M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H171H265Q288 686 297 686T309 684T315 679Q317 676 500 455T684 233V624H576V686H585Q603 683 718 683T851 686H860V624H752V319Q752 15 750 11Q747 4 742 2T718 0H712Q708 0 706 0T700 0T696 1T693 2T690 4T687 7T684 11T679 16T674 23Q671 27 437 311L215 579V62H323V0H314Z"></path><path id="MJX-390-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-390-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(638.8,0)"><use data-c="3D" xlink:href="#MJX-390-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1694.6,0)"><g data-mml-node="mrow" transform="translate(220,710)"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-390-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(389,0)"><g data-mml-node="mtext"><use data-c="1D429" xlink:href="#MJX-390-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(672,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-390-TEX-V-2032"></use></g></g><g data-mml-node="mo" transform="translate(1527.7,0)"><use data-c="2212" xlink:href="#MJX-390-TEX-N-2212"></use></g><g data-mml-node="mtext" transform="translate(2527.9,0)"><use data-c="1D428" xlink:href="#MJX-390-TEX-B-1D428"></use></g><g data-mml-node="mo" transform="translate(3102.9,0)"><use data-c="29" xlink:href="#MJX-390-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3714.1,0)"><use data-c="22C5" xlink:href="#MJX-390-TEX-N-22C5"></use></g><g data-mml-node="mtext" transform="translate(4214.3,0)"><use data-c="1D40D" xlink:href="#MJX-390-TEX-B-1D40D"></use></g></g><g data-mml-node="mrow" transform="translate(1646.4,-686)"><g data-mml-node="mtext"><use data-c="1D41D" xlink:href="#MJX-390-TEX-B-1D41D"></use></g><g data-mml-node="mo" transform="translate(861.2,0)"><use data-c="22C5" xlink:href="#MJX-390-TEX-N-22C5"></use></g><g data-mml-node="mtext" transform="translate(1361.4,0)"><use data-c="1D40D" xlink:href="#MJX-390-TEX-B-1D40D"></use></g></g><rect width="5314.3" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mtext mathvariant="bold">p</mtext><mo data-mjx-alternate="1">′</mo></msup><mo>−</mo><mtext mathvariant="bold">o</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mtext mathvariant="bold">N</mtext></mrow><mrow><mtext mathvariant="bold">d</mtext><mo>⋅</mo><mtext mathvariant="bold">N</mtext></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>3 subtractions, 6 multiplications, 1 division</span></p></li><li><p><span>Slabs perpendicular to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-400-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-400-TEX-I-1D465"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">x</script><span> axis:</span></p><div 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145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path id="MJX-391-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-391-TEX-B-1D429" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path><path id="MJX-391-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path><path id="MJX-391-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-391-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-391-TEX-B-1D428" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path id="MJX-391-TEX-B-1D41D" d="M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-391-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(638.8,0)"><use data-c="3D" xlink:href="#MJX-391-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1694.6,0)"><g data-mml-node="mrow" transform="translate(220,714.8)"><g data-mml-node="msubsup"><g data-mml-node="mtext"><use data-c="1D429" xlink:href="#MJX-391-TEX-B-1D429"></use></g><g data-mml-node="mo" transform="translate(672,363) scale(0.707)"><use data-c="2032" xlink:href="#MJX-391-TEX-V-2032"></use></g><g data-mml-node="mi" transform="translate(672,-247) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-391-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(1348.7,0)"><use data-c="2212" xlink:href="#MJX-391-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2348.9,0)"><g data-mml-node="mtext"><use data-c="1D428" xlink:href="#MJX-391-TEX-B-1D428"></use></g><g data-mml-node="mi" transform="translate(608,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-391-TEX-I-1D465"></use></g></g></g><g data-mml-node="msub" transform="translate(1362.5,-686)"><g data-mml-node="mtext"><use data-c="1D41D" xlink:href="#MJX-391-TEX-B-1D41D"></use></g><g data-mml-node="mi" transform="translate(672,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-391-TEX-I-1D465"></use></g></g><rect width="3611.4" height="60" x="120" y="220"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>t</mi><mo>=</mo><mfrac><mrow><msubsup><mtext mathvariant="bold">p</mtext><mi>x</mi><mo data-mjx-alternate="1">′</mo></msubsup><mo>−</mo><msub><mtext mathvariant="bold">o</mtext><mi>x</mi></msub></mrow><msub><mtext mathvariant="bold">d</mtext><mi>x</mi></msub></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>1 subtraction, 1 division</span></p></li></ul></li></ul></div></div>
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