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Lecture14.html
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</style><title>Lecture14</title>
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<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-14---ray-tracing-2-acceleration-and-radiometry">GAMES101 Lecture 14 - Ray Tracing 2 (Acceleration and Radiometry)</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-spatial-partitions">I. Spatial Partitions</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#trivial-partitions">Trivial Partitions</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="#uniform-spatial-partitions-grid">Uniform Spatial Partitions (Grid)</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#tree-shaped-partitions">Tree-Shaped Partitions</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="#oct-tree">Oct-Tree</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="#bsp-tree-binary-space-partitioning">BSP-Tree (Binary Space Partitioning)</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="#kd-tree">KD-Tree</a></div><ul class="outline-children"></ul></li></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="#general-problems">General Problems</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ii-object-partitions">II. Object Partitions</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="#bounding-volume-hierarchy-bvh">Bounding Volume Hierarchy (BVH)</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="#spatial-vs-object-partitions">Spatial vs Object Partitions</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#iii-basic-radiometry">III. Basic Radiometry</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="#radiant-energy-and-flux-power">Radiant Energy and Flux (Power)</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="#radiant-intensity">Radiant Intensity</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="#angles-solid-angles-and-direction-vectors">Angles, Solid Angles and Direction Vectors</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="#differential-solid-angle">Differential Solid Angle</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="#isotropic-point-source">Isotropic Point Source</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="#irradiance">Irradiance</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="#correction-irradiance-falloff">Correction: Irradiance Falloff</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#radiance">Radiance</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="#incident-radiance">Incident Radiance</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="#exiting-radiance">Exiting Radiance</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="#irradiance-vs-radiance">Irradiance vs. Radiance</a></div><ul class="outline-children"></ul></li></ul></li></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#appendix-a-surface-area-heuristics-sah">Appendix A: Surface Area Heuristics (SAH)</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="#idea-behind-the-sah-cost-model">Idea behind the SAH Cost Model</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="#the-bucket-algorithm">The Bucket Algorithm</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="#pros">Pros</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="#cons">Cons</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-14---ray-tracing-2-acceleration-and-radiometry'><span>GAMES101 Lecture 14 - Ray Tracing 2 (Acceleration and Radiometry)</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_14.pdf'><span>GAMES101_Lecture_14.pdf</span></a></p><h2 id='i-spatial-partitions'><span>I. Spatial Partitions</span></h2><h3 id='trivial-partitions'><span>Trivial Partitions</span></h3><h4 id='uniform-spatial-partitions-grid'><span>Uniform Spatial Partitions (Grid)</span></h4><ul><li><p><strong><span>Preprocessing - Build Acceleration Grid</span></strong></p><ul><li><p><span>Find bounding box</span></p></li><li><p><span>Create grid</span></p></li><li><p><span>Store each object in overlapping cells</span></p></li></ul></li><li><p><strong><span>Grid Resolution Heuristic</span></strong></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="21.663ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 9575 910" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-203-TEX-N-23" d="M56 347Q56 360 70 367H313L355 524Q394 676 401 686Q406 694 416 694Q434 694 436 676Q436 672 396 522Q355 374 355 369L354 367H543L585 524Q626 679 630 685Q636 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style="zoom:33%;" /></p><h4 id='oct-tree'><span>Oct-Tree</span></h4><p><em><span>Recursively divide the current octant into </span><strong><span>8 sub-octants</span></strong><span> until the current working space is empty or the number of objects contained reaches certain minimum.</span></em></p><ul><li><p><strong><span>Cons</span></strong><span>:</span></p><ul><li><p><span>Too many branches</span></p></li></ul></li></ul><p> </p><h4 id='bsp-tree-binary-space-partitioning'><span>BSP-Tree (Binary Space Partitioning)</span></h4><p><em><span>Recursively divide the space using a hyperplane.</span></em></p><p> </p><h4 id='kd-tree'><span>KD-Tree</span></h4><p><img src="../images/Lecture14-img-2.png" alt="img-2" style="zoom: 33%;" /></p><p><em><span>Recursively divide the space along </span><strong><span>alternating</span></strong><span> axes (</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.493ex" height="1.62ex" 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287Z"></path><path id="MJX-205-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></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-205-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="2192" xlink:href="#MJX-205-TEX-N-2192"></use></g><g data-mml-node="mi" transform="translate(2127.6,0)"><use data-c="1D466" xlink:href="#MJX-205-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(2895.3,0)"><use data-c="2192" xlink:href="#MJX-205-TEX-N-2192"></use></g><g data-mml-node="mi" transform="translate(4173.1,0)"><use data-c="1D467" xlink:href="#MJX-205-TEX-I-1D467"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo accent="false" stretchy="false">→</mo><mi>y</mi><mo accent="false" stretchy="false">→</mo><mi>z</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">x \to y \to z</script><span> loop) using a hyperplane, creating a binary tree structure.</span></em></p><p><em><span>Separations don't have to be </span><strong><span>even</span></strong></em><span>.</span></p><p> </p><p><strong><span>Data Structure:</span></strong></p><ul><li><p><strong><span>Internal Nodes</span></strong><span>:</span></p><ul><li><p><span>Store:</span></p><ul><li><p><em><span>Split axis</span></em></p></li><li><p><em><span>Split position</span></em></p></li><li><p><em><span>Children</span></em></p></li><li><p><em><strong><span>No objects are stored</span></strong></em></p></li></ul></li></ul></li><li><p><strong><span>Leaf Nodes</span></strong><span>:</span></p><ul><li><p><span>Store:</span></p><ul><li><p><em><span>List of objects contained</span></em></p></li></ul></li></ul></li></ul><p> </p><p><strong><span>Traversing a KD-Tree</span></strong><span>: If the ray has intersected with the current node, recursively check all its child nodes.</span></p><ul><li><p><span>If the current node is a leaf node, test intersection with all contained objects.</span></p></li></ul><p> </p><h3 id='general-problems'><span>General Problems</span></h3><ul><li><p><em><span>Inside a partition, there may be an triangle which passes through this partition but has none of its vertices inside this partition.</span></em></p></li><li><p><em><span>An object may be contained inside multiple partitions, leading to </span><strong><span>memory inefficiency</span></strong></em><span>.</span></p><ul><li><p><span>Ideally we want each object stored in a </span><strong><span>single</span></strong><span> node only.</span></p></li></ul></li></ul><p> </p><h2 id='ii-object-partitions'><span>II. Object Partitions</span></h2><h3 id='bounding-volume-hierarchy-bvh'><span>Bounding Volume Hierarchy (BVH)</span></h3><p><strong><span>Summary</span></strong><span>: </span></p><ul><li><p><span>Find bounding box</span></p></li><li><p><span>Recursively split set of objects into two subsets</span></p></li><li><p><strong><span>Recompute</span></strong><span> the bounding box of the subsets</span></p></li><li><p><span>Stop when necessary</span></p></li><li><p><span>Store objects in each leaf node</span></p></li></ul><p> </p><p><strong><span>How to subdivide a node</span></strong><span>? </span><em><span>Make the split as separated and evenly-sized as possible.</span></em></p><ul><li><p><span>Choose a dimension to split</span></p></li><li><p><strong><span>Heuristic #1</span></strong><span>: Always choose the </span><strong><span>longest</span></strong><span> axis in the current node</span></p><ul><li><p><span>To make the partitions more evenly-spaced</span></p></li></ul></li><li><p><strong><span>Heuristic #2</span></strong><span>: Split node at location of </span><strong><span>median</span></strong><span> object</span></p><ul><li><p><span>To make the tree </span><strong><span>balanced</span></strong><span>, resulting in less depth of the hierarchy</span></p></li><li><p><span>This can be done in </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.844ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2141 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-207-TEX-I-1D442" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path id="MJX-207-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-207-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-207-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D442" xlink:href="#MJX-207-TEX-I-1D442"></use></g><g data-mml-node="mo" transform="translate(763,0)"><use data-c="28" xlink:href="#MJX-207-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1152,0)"><use data-c="1D45B" xlink:href="#MJX-207-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(1752,0)"><use data-c="29" xlink:href="#MJX-207-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">O(n)</script><span> on average using randomized algorithms, and strictly in </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.844ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2141 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-207-TEX-I-1D442" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path id="MJX-207-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-207-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-207-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D442" xlink:href="#MJX-207-TEX-I-1D442"></use></g><g data-mml-node="mo" transform="translate(763,0)"><use data-c="28" xlink:href="#MJX-207-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1152,0)"><use data-c="1D45B" xlink:href="#MJX-207-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(1752,0)"><use data-c="29" xlink:href="#MJX-207-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">O(n)</script><span> using a type of deterministic algorithm at the cost of a significantly higher constant factor.</span></p></li></ul></li></ul><p><em><span>Heuristics are of great research interest.</span></em></p><p> </p><p><strong><span>Termination criteria</span></strong><span>?</span></p><ul><li><p><span>Heuristic: Stop when the number of elements contained in the current node reaches certain minimum.</span></p></li></ul><p> </p><p><strong><span>Data Structure:</span></strong></p><ul><li><p><strong><span>Internal Nodes</span></strong><span>:</span></p><ul><li><p><span>Store:</span></p><ul><li><p><span>Bounding box</span></p></li><li><p><em><span>Children</span></em></p></li><li><p><em><strong><span>No objects are stored</span></strong></em></p></li></ul></li></ul></li><li><p><strong><span>Leaf Nodes</span></strong><span>:</span></p><ul><li><p><span>Store:</span></p><ul><li><p><span>Bounding box</span></p></li><li><p><em><span>List of objects contained</span></em></p></li></ul></li></ul></li></ul><p><em><span>All objects are in subtrees.</span></em></p><p> </p><p><strong><span>Traversing a BVH</span></strong><span>:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="pseudocode"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="pseudocode"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.5191px; left: 7.98608px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Intersect</span><span class="cm-bracket">(</span><span class="cm-variable">Ray</span> <span class="cm-variable">ray</span>, <span class="cm-variable">BVH</span> <span class="cm-variable">node</span><span class="cm-bracket">)</span> <span class="cm-bracket">{</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">if</span> <span class="cm-bracket">(</span><span class="cm-variable">ray</span> <span class="cm-variable">misses</span> <span class="cm-variable">node</span>.<span class="cm-variable">bbox</span><span class="cm-bracket">)</span> <span class="cm-keyword">return</span><span class="cm-bracket">;</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">if</span> <span class="cm-bracket">(</span><span class="cm-variable">node</span> <span class="cm-keyword">is</span> <span class="cm-variable">a</span> <span class="cm-variable">leaf</span> <span class="cm-variable">node</span><span class="cm-bracket">)</span> <span class="cm-bracket">{</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">test</span> <span class="cm-variable">intersection</span> <span class="cm-variable">with</span> <span class="cm-keyword">all</span> <span class="cm-variable">objects</span><span class="cm-bracket">;</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">return</span> <span class="cm-variable">the</span> <span class="cm-variable">closest</span> <span class="cm-variable">intersection</span><span class="cm-bracket">;</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-bracket">}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">hit1</span> <span class="cm-operator">=</span> <span class="cm-variable">Intersect</span><span class="cm-bracket">(</span><span class="cm-variable">ray</span>, <span class="cm-variable">node</span>.<span class="cm-variable">child1</span><span class="cm-bracket">);</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-variable">hit2</span> <span class="cm-operator">=</span> <span class="cm-variable">Intersect</span><span class="cm-bracket">(</span><span class="cm-variable">ray</span>, <span class="cm-variable">node</span>.<span class="cm-variable">child2</span><span class="cm-bracket">);</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" class="cm-tab-wrap-hack" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-tab" role="presentation" cm-text=" "> </span><span class="cm-keyword">return</span> <span class="cm-variable">the</span> <span class="cm-variable">closer</span> <span class="cm-keyword">of</span> <span class="cm-variable">hit1</span>, <span class="cm-variable">hit2</span><span class="cm-bracket">;</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-bracket">}</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 299px;"></div><div class="CodeMirror-gutters" style="display: none; height: 299px;"></div></div></div></pre><p><img src="../images/Lecture14-img-3.png" alt="img-3" style="zoom:50%;" /></p><p> </p><p> </p><h3 id='spatial-vs-object-partitions'><span>Spatial vs Object Partitions</span></h3><ul><li><p><strong><span>Spatial Partition</span></strong></p><ul><li><p><span>Partition space into </span><strong><span>non-overlapping</span></strong><span> regions</span></p></li><li><p><span>An object can be contained in multiple regions</span></p></li></ul></li><li><p><strong><span>Object Partition</span></strong></p><ul><li><p><span>Partition set of objects into </span><strong><span>disjoint subsets</span></strong></p></li><li><p><span>Bounding boxes for each set may overlap in space</span></p></li></ul></li></ul><p> </p><h2 id='iii-basic-radiometry'><span>III. Basic Radiometry</span></h2><p><strong><span>Motivations and stuff to learn</span></strong><span>: </span><em><span>Describe the light in a precise manner</span></em><span>.</span></p><ul><li><p><span>Measure system and units for illumination</span></p></li><li><p><strong><span>Accurately measure</span></strong><span> the spatial properties of light</span></p><ul><li><p><span>Radiant Flux</span></p></li><li><p><span>Intensity</span></p></li><li><p><span>Irradiance</span></p></li><li><p><span>Radiance</span></p></li></ul></li><li><p><span>Perform lighting calculations </span><strong><span>in a physically correct manner</span></strong><span>.</span></p></li><li><p><em><span>Still based on Geometric Optics</span></em><span>.</span></p></li></ul><p> </p><h3 id='radiant-energy-and-flux-power'><span>Radiant Energy and Flux (Power)</span></h3><p><em><strong><span>Definition</span></strong></em><span>: </span><strong><span>Radiant energy</span></strong><span> is the energy of electromagnetic radiation. 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transform="translate(1264,0)"></use><use data-c="6C" xlink:href="#MJX-185-TEX-N-6C" transform="translate(1820,0)"></use><use data-c="65" xlink:href="#MJX-185-TEX-N-65" transform="translate(2098,0)"></use></g></g><g data-mml-node="mo" transform="translate(6153,0)"><use data-c="5D" xlink:href="#MJX-185-TEX-N-5D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>Q</mi><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mo stretchy="false">[</mo><mrow><mtext>J </mtext><mrow data-mjx-texclass="ORD"><mo>=</mo></mrow><mtext> Joule</mtext></mrow><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container></div></div><p><em><strong><span>Definition</span></strong></em><span>: </span><strong><span>Radiant flux</span></strong><span> (power) is the energy emitted, reflected, transmitted or received, </span><strong><span>per unit time.</span></strong></p><div contenteditable="false" 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Intensity</span></h3><p><img src="../images/Lecture14-img-4.png" alt="img-4" style="zoom:50%;" /></p><p><em><strong><span>Definition</span></strong></em><span>: The </span><strong><span>radiant (luminous) intensity</span></strong><span> is the power </span><strong><span>per unit solid angle</span></strong><span> emitted by a point light source.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n566" cid="n566" 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="42.492ex" height="5.428ex" role="img" focusable="false" viewBox="0 -1449.5 18781.7 2399" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.148ex;"><defs><path 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data-mjx-texclass="CLOSE">]</mo></mrow></mrow><mtext> </mtext><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mtext>lm</mtext><mtext>sr</mtext></mfrac><mo>=</mo><mtext>cd</mtext><mo>=</mo><mtext>candela</mtext><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow><mtext> </mtext></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span><em><span>candela</span></em><span> is one of the seven SI base units.</span></p><p> </p><h3 id='angles-solid-angles-and-direction-vectors'><span>Angles, Solid Angles and Direction Vectors</span></h3><p><em><strong><span>Definition</span></strong></em><span>: </span><strong><span>Angle</span></strong><span> is the ratio of subtended arc length on a circle to the radius</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n571" cid="n571" mdtype="math_block" data-math-tag-before="0" 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316T57 280Q55 278 41 278H27Q21 284 21 287Z"></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="1D703" xlink:href="#MJX-188-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(746.8,0)"><use data-c="3D" xlink:href="#MJX-188-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1802.6,0)"><g data-mml-node="mi" transform="translate(296.5,676)"><use data-c="1D459" xlink:href="#MJX-188-TEX-I-1D459"></use></g><g data-mml-node="mi" transform="translate(220,-686)"><use data-c="1D45F" xlink:href="#MJX-188-TEX-I-1D45F"></use></g><rect width="651" 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>θ</mi><mo>=</mo><mfrac><mi>l</mi><mi>r</mi></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>A circle has </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.421ex" height="1.532ex" role="img" focusable="false" viewBox="0 -666 1070 677" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-208-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-208-TEX-I-1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></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="32" xlink:href="#MJX-208-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D70B" xlink:href="#MJX-208-TEX-I-1D70B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">2\pi</script><span> </span><strong><span>radians</span></strong><span>.</span></p><p><em><strong><span>Definition</span></strong></em><span>: </span><strong><span>Solid angle</span></strong><span> is the ratio of subtended area on a sphere to the radius </span><em><span>squared</span></em></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n574" cid="n574" 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="7.654ex" height="4.803ex" role="img" focusable="false" viewBox="0 -1392 3383.1 2122.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.654ex;"><defs><path id="MJX-189-TEX-N-3A9" d="M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 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mathvariant="normal">Ω</mi><mo>=</mo><mfrac><mi>A</mi><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>A sphere has </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.421ex" height="1.557ex" role="img" focusable="false" viewBox="0 -677 1070 688" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-209-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path id="MJX-209-TEX-I-1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></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="34" xlink:href="#MJX-209-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D70B" xlink:href="#MJX-209-TEX-I-1D70B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>π</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">4\pi</script><span> </span><strong><span>steradians</span></strong><span>.</span></p><p><em><span>The area, when calculated, must be that of a part of the shell, or that projected to the shell.</span></em></p><p> </p><p><strong><span>Direction Vector</span></strong><span>: </span><strong><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.407ex" height="1.027ex" role="img" focusable="false" viewBox="0 -443 622 454" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-210-TEX-I-1D714" d="M495 384Q495 406 514 424T555 443Q574 443 589 425T604 364Q604 334 592 278T555 155T483 38T377 -11Q297 -11 267 66Q266 68 260 61Q201 -11 125 -11Q15 -11 15 139Q15 230 56 325T123 434Q135 441 147 436Q160 429 160 418Q160 406 140 379T94 306T62 208Q61 202 61 187Q61 124 85 100T143 76Q201 76 245 129L253 137V156Q258 297 317 297Q348 297 348 261Q348 243 338 213T318 158L308 135Q309 133 310 129T318 115T334 97T358 83T393 76Q456 76 501 148T546 274Q546 305 533 325T508 357T495 384Z"></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="1D714" xlink:href="#MJX-210-TEX-I-1D714"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega</script></strong><span> will be used to denote a </span><strong><span>direction vector</span></strong><span> of unit length.</span></p><p><img src="../images\Lecture14-img-5.png" alt="img-5" style="zoom: 33%;" /></p><p> </p><h3 id='differential-solid-angle'><span>Differential Solid Angle</span></h3><p><img src="../images/Lecture14-img-6.png" alt="img-6" style="zoom: 33%;" /></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n583" cid="n583" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" 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stretchy="false">(</mo><mtext mathvariant="bold">x</mtext><mo stretchy="false">)</mo><mo>≡</mo><mfrac><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mtext mathvariant="bold">x</mtext><mo stretchy="false">)</mo></mrow><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>A</mi></mrow></mfrac><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mtext>W</mtext><msup><mtext>m</mtext><mn>2</mn></msup></mfrac><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow><mtext> </mtext><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mtext>lm</mtext><msup><mtext>m</mtext><mn>2</mn></msup></mfrac><mo>=</mo><mtext>lux</mtext><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-239-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g 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data-c="2061" xlink:href="#MJX-213-TEX-N-2061"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1504.7,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-213-TEX-I-1D703"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\cos{\theta}</script><span> operation as adjusting the plane such that it sits perpendicular to the incident light.</span></em></p><p> </p><h4 id='correction-irradiance-falloff'><span>Correction: Irradiance Falloff</span></h4><p><img src="../images/Lecture14-img-9.png" alt="img-9" style="zoom:33%;" /></p><p> </p><h3 id='radiance'><span>Radiance</span></h3><p><img src="../images/Lecture14-img-10.png" alt="img-10" style="zoom:33%;" /></p><p><span>Radiance is the fundamental field quantity 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stretchy="false">)</mo><mo>≡</mo><mfrac><mrow><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>ω</mi></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>A</mi></mrow><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></mrow></mfrac><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mrow><mtext> </mtext><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mtext>W</mtext><msup><mtext>sr m</mtext><mn>2</mn></msup></mfrac><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow><mtext> </mtext><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mfrac><mtext>cd</mtext><msup><mtext>m</mtext><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mtext>lm</mtext><msup><mtext>sr m</mtext><mn>2</mn></msup></mfrac><mo>=</mo><mtext>nit</mtext><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow><mtext> </mtext></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><em><span>where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.465ex" height="1.62ex" role="img" focusable="false" viewBox="0 -705 1973.7 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-213-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-213-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-213-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-213-TEX-N-2061" d=""></path><path id="MJX-213-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></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="63" xlink:href="#MJX-213-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-213-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-213-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(1338,0)"><use data-c="2061" xlink:href="#MJX-213-TEX-N-2061"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1504.7,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-213-TEX-I-1D703"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\cos{\theta}</script><span> accounts for projected surface area.</span></em></p><ul><li><p><span>Irradiance per solid angle</span></p></li><li><p><span>Intensity per </span><strong><span>projected</span></strong><span> unit area</span></p></li></ul><p> </p><ul><li><p><span>Why dividing by </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.465ex" height="1.62ex" role="img" focusable="false" viewBox="0 -705 1973.7 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-238-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-238-TEX-N-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path id="MJX-238-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-238-TEX-N-2061" d=""></path><path id="MJX-238-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></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="63" xlink:href="#MJX-238-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-238-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-238-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(1338,0)"><use data-c="2061" xlink:href="#MJX-238-TEX-N-2061"></use></g><g data-mml-node="mi" transform="translate(1504.7,0)"><use data-c="1D703" xlink:href="#MJX-238-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\cos \theta</script><span>?</span></p><ul><li><p><span>To recover the true radiance from that angle (regardless of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.618ex" role="img" focusable="false" viewBox="0 -705 469 715" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-239-TEX-I-1D703" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></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="1D703" xlink:href="#MJX-239-TEX-I-1D703"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\theta</script><span> which considers how much energy is actually received)</span></p></li></ul></li></ul><p> </p><h4 id='incident-radiance'><span>Incident Radiance</span></h4><p><strong><span>Incident radiance</span></strong><span> is the irradiance per unit solid angle arriving at the surface. </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n623" cid="n623" 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="18.492ex" height="4.88ex" role="img" focusable="false" viewBox="0 -1460 8173.6 2157" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.577ex;"><defs><path id="MJX-196-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 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xlink:href="#MJX-196-TEX-I-1D703"></use></g></g></g><rect width="3518.3" 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>L</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>E</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo stretchy="false">)</mo></mrow><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>ω</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>The light arriving at the surface along a given ray.</span></p><p> </p><h4 id='exiting-radiance'><span>Exiting Radiance</span></h4><p><strong><span>Exiting surface</span></strong><span> radiance is the intensity per unit 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transform="translate(1060,0)"><use data-c="28" xlink:href="#MJX-197-TEX-N-28"></use></g><g data-mml-node="mtext" transform="translate(1449,0)"><use data-c="70" xlink:href="#MJX-197-TEX-N-70"></use></g><g data-mml-node="mo" transform="translate(2005,0)"><use data-c="2C" xlink:href="#MJX-197-TEX-N-2C"></use></g><g data-mml-node="mi" transform="translate(2449.7,0)"><use data-c="1D714" xlink:href="#MJX-197-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(3071.7,0)"><use data-c="29" xlink:href="#MJX-197-TEX-N-29"></use></g></g><g data-mml-node="mrow" transform="translate(227.2,-686)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-197-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D434" xlink:href="#MJX-197-TEX-I-1D434"></use></g><g data-mml-node="mi" transform="translate(1472.7,0)"><use data-c="63" xlink:href="#MJX-197-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-197-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-197-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(2810.7,0)"><use data-c="2061" xlink:href="#MJX-197-TEX-N-2061"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2977.3,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-197-TEX-I-1D703"></use></g></g></g><rect width="3660.7" 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>L</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≡</mo><mfrac><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>I</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>A</mi><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><em><span>Hint: For the two formulas for incident/exiting radiance, considering taking them has differentials and multiply </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.972ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3081.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-214-TEX-I-1D43F" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path id="MJX-214-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-214-TEX-N-70" d="M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo stretchy="false">(</mo><mtext>p</mtext><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">L(\text{p}, \omega)</script><span> by the denominator.</span></em></p><p> </p><h4 id='irradiance-vs-radiance'><span>Irradiance vs. Radiance</span></h4><p><img src="../images/Lecture14-img-12.png" alt="img-12" style="zoom:33%;" /></p><ul><li><p><strong><span>Irradiance</span></strong><span>: Total power received by area </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.955ex" height="1.645ex" role="img" focusable="false" viewBox="0 -716 1306 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-165-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 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href='https://pbr-book.org/3ed-2018/Primitives_and_Intersection_Acceleration/Bounding_Volume_Hierarchies#sec:sah'><span>Bounding Volume Hierarchies (pbr-book.org)</span></a></p><p><span>The two primitive partitioning approaches described in the BVH section can work well for some distributions of primitives, but they often choose partitions that perform poorly in practice, leading to more nodes of the tree being visited by rays and hence unnecessary inefficient ray-primitive intersection computations at rendering time.</span></p><p><span>Most of the best algorithms (as of 2018) for building acceleration structures for ray-tracing are based on the "surface area heuristic" (SAH), which provides a </span><strong><span>well-grounded cost model</span></strong><span> for answer questions like</span></p><ul><li><p><span>"which of a number of partitions of primitives will lead to a better BVH for ray-primitive intersection tests?", or</span></p></li><li><p><span>"which of a number of possible positions to split space in a spatial subdivision scheme will lead to a better acceleration structure?"</span></p></li></ul><p> </p><h3 id='idea-behind-the-sah-cost-model'><span>Idea behind the SAH Cost Model</span></h3><p><span>The SAH model estimates the </span><strong><span>computational cost</span></strong><span> of performing ray intersection tests, including the time spent on:</span></p><ul><li><p><strong><span>Traversing</span></strong><span> nodes of the tree</span></p></li><li><p><span>Ray-primitive </span><strong><span>intersection tests</span></strong><span> for a particular partitioning of primitives</span></p></li></ul><p><span>By assuming the current working node is a leaf node regardless of the number of primitives it contains, we know that any ray that passes through this node will be tested against all of the overlapping primitives and will incur a cost of </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block 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146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-200-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-200-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 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data-mml-node="math"><g data-mml-node="munderover"><g data-mml-node="mo"><use data-c="2211" xlink:href="#MJX-200-TEX-LO-2211"></use></g><g data-mml-node="TeXAtom" transform="translate(148.2,-1087.9) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-200-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(345,0)"><use data-c="3D" xlink:href="#MJX-200-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1123,0)"><use data-c="31" xlink:href="#MJX-200-TEX-N-31"></use></g></g><g data-mml-node="mi" transform="translate(408,1150) scale(0.707)"><use data-c="1D441" xlink:href="#MJX-200-TEX-I-1D441"></use></g></g><g data-mml-node="msub" transform="translate(1610.7,0)"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-200-TEX-I-1D461"></use></g><g data-mml-node="mtext" transform="translate(394,-150) scale(0.707)"><use data-c="69" xlink:href="#MJX-200-TEX-N-69"></use><use data-c="73" xlink:href="#MJX-200-TEX-N-73" transform="translate(278,0)"></use><use data-c="65" xlink:href="#MJX-200-TEX-N-65" transform="translate(672,0)"></use><use data-c="63" xlink:href="#MJX-200-TEX-N-63" transform="translate(1116,0)"></use><use data-c="74" xlink:href="#MJX-200-TEX-N-74" transform="translate(1560,0)"></use></g></g><g data-mml-node="mo" transform="translate(3432.8,0)"><use data-c="28" xlink:href="#MJX-200-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(3821.8,0)"><use data-c="1D456" xlink:href="#MJX-200-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4166.8,0)"><use data-c="29" xlink:href="#MJX-200-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>t</mi><mtext>isect</mtext></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.009ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 888 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-234-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></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="1D441" xlink:href="#MJX-234-TEX-I-1D441"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">N</script><span> is the number of objects</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.123ex" height="1.773ex" role="img" focusable="false" viewBox="0 -626 1822.2 783.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-220-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><path id="MJX-220-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-220-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 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xlink:href="#MJX-220-TEX-N-74" transform="translate(1560,0)"></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>isect</mtext></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">t_\text{isect}</script><span> is the time to compute a ray-object intersection with the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-219-TEX-I-1D456" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></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="1D456" xlink:href="#MJX-219-TEX-I-1D456"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">i</script><span>th primitive</span></p><ul><li><p><span>We here assume that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.123ex" height="1.773ex" role="img" focusable="false" viewBox="0 -626 1822.2 783.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-220-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><path id="MJX-220-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-220-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-220-TEX-N-65" d="M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z"></path><path id="MJX-220-TEX-N-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path id="MJX-220-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-220-TEX-I-1D461"></use></g><g data-mml-node="mtext" transform="translate(394,-150) scale(0.707)"><use data-c="69" xlink:href="#MJX-220-TEX-N-69"></use><use data-c="73" xlink:href="#MJX-220-TEX-N-73" 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stretchy="false">)</mo><mo>=</mo><msub><mi>t</mi><mtext>trav</mtext></msub><mo>+</mo><msub><mi>p</mi><mi>A</mi></msub><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>N</mi><mi>A</mi></msub></mrow></munderover><msub><mi>t</mi><mtext>isect</mtext></msub><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mi>B</mi></msub><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>N</mi><mi>B</mi></msub></mrow></munderover><msub><mi>t</mi><mtext>isect</mtext></msub><mo stretchy="false">(</mo><msub><mi>b</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>where</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="3.899ex" height="1.773ex" role="img" focusable="false" viewBox="0 -626 1723.2 783.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-221-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><path id="MJX-221-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-221-TEX-N-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path id="MJX-221-TEX-N-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path id="MJX-221-TEX-N-76" d="M338 431Q344 429 422 429Q479 429 503 431H508V385H497Q439 381 423 345Q421 341 356 172T288 -2Q283 -11 263 -11Q244 -11 239 -2Q99 359 98 364Q93 378 82 381T43 385H19V431H25L33 430Q41 430 53 430T79 430T104 429T122 428Q217 428 232 431H240V385H226Q187 384 184 370Q184 366 235 234L286 102L377 341V349Q377 363 367 372T349 383T335 385H331V431H338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-221-TEX-I-1D461"></use></g><g data-mml-node="mtext" transform="translate(394,-150) scale(0.707)"><use data-c="74" xlink:href="#MJX-221-TEX-N-74"></use><use data-c="72" xlink:href="#MJX-221-TEX-N-72" transform="translate(389,0)"></use><use data-c="61" xlink:href="#MJX-221-TEX-N-61" transform="translate(781,0)"></use><use data-c="76" xlink:href="#MJX-221-TEX-N-76" transform="translate(1281,0)"></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>trav</mtext></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">t_\text{trav}</script><span> is the time it takes to traverse the interior node and determine which of the children the ray passes through</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.526ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 1116.3 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-222-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-222-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-222-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-152.7) scale(0.707)"><use data-c="1D434" xlink:href="#MJX-222-TEX-I-1D434"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>A</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_A</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.54ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 1122.7 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-223-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path id="MJX-223-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D45D" xlink:href="#MJX-223-TEX-I-1D45D"></use></g><g data-mml-node="mi" transform="translate(536,-150) scale(0.707)"><use data-c="1D435" xlink:href="#MJX-223-TEX-I-1D435"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>B</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">p_B</script><span> are the probabilities that the ray passes through each of the child nodes (assuming binary subdivision)</span></p><ul><li><p><span>They can be computed using ideas from geometric probability.</span></p><p><span>For a convex volume </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-227-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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="1D434" xlink:href="#MJX-227-TEX-I-1D434"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A</script><span> contained in another convex volume </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-226-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></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="1D435" xlink:href="#MJX-226-TEX-I-1D435"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">B</script><span>, the conditional probability that a </span><strong><span>uniformly distributed random ray</span></strong><span> passing through </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 759 683" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-226-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></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="1D435" xlink:href="#MJX-226-TEX-I-1D435"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">B</script><span> will also pass through </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-227-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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="1D434" xlink:href="#MJX-227-TEX-I-1D434"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">A</script><span> is the ratio of their surface areas, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.449ex" height="1.345ex" role="img" focusable="false" viewBox="0 -442 1082.3 594.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.345ex;"><defs><path id="MJX-228-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 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xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mi>B</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">N_B</script><span> are the number of primitives contained in two child nodes</span></p></li></ul><p> </p><p> </p><p><span>The algorithm then optimize the partitioning with the goal of minimizing the total cost.</span></p><ul><li><p><span>Typically, a </span><strong><span>greedy</span></strong><span> algorithm is used that minimizes the cost for each single node of the hiearchy.</span></p></li></ul><p> </p><h3 id='the-bucket-algorithm'><span>The Bucket Algorithm</span></h3><p><span>The bucket algorithm follows a very simple pattern.</span></p><ul><li><p><span>If the number of primitives in the current node is less than a threshold, do nothing</span></p></li><li><p><span>Else:</span></p><ul><li><p><span>Choose the axis to split. </span><em><span>May split the longest axis.</span></em></p></li><li><p><span>Split the range on that axis into 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transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D441" xlink:href="#MJX-234-TEX-I-1D441"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">N</script><span> equally-sized regions.</span></p></li><li><p><span>Compute the cost if we split the node at those </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.906ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 2610.4 765" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-235-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-235-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-235-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></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="1D441" xlink:href="#MJX-235-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(1110.2,0)"><use data-c="2212" xlink:href="#MJX-235-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2110.4,0)"><use data-c="31" xlink:href="#MJX-235-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>−</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">N-1</script><span> planes.</span></p></li><li><p><span>Which plane is the best to patition? Choose that and recursively build BVH.</span></p></li></ul></li></ul><p><span>In the book, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.289ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 3221.6 765" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-236-TEX-I-1D441" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path><path id="MJX-236-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-236-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-236-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></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="1D441" xlink:href="#MJX-236-TEX-I-1D441"></use></g><g data-mml-node="mo" transform="translate(1165.8,0)"><use data-c="3D" xlink:href="#MJX-236-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2221.6,0)"><use data-c="31" xlink:href="#MJX-236-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-236-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>12</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">N=12</script><span>.</span></p><p> </p><h3 id='pros'><span>Pros</span></h3><ul><li><p><span>Probably the </span><strong><span>most efficient</span></strong><span> BVH split method</span></p></li></ul><p> </p><h3 id='cons'><span>Cons</span></h3><ul><li><p><span>Many passes are taken over the scene primitives to compute the SAH costs at all of the levels of the tree</span></p></li><li><p><span>Hard to </span><strong><span>parallelize</span></strong></p></li></ul></div></div>
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