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Lecture17.html
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</style><title>Lecture17</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-17---materials-and-appearances">GAMES101 Lecture 17 - Materials and Appearances</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-materials-and-appearances">I. Materials and Appearances</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="#diffuselambertian-material">Diffuse/Lambertian Material</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="#glossy-material">Glossy Material</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="#ideal-reflectiverefractive-material">Ideal Reflective/Refractive Material</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="#perfect-specular-reflection">Perfect Specular Reflection</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="#specular-refraction">Specular Refraction</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="#snells-law">Snell's Law</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="#law-of-refraction">Law of Refraction</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="#fresnel-reflectionterm">Fresnel Reflection/Term</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#fresnel-term">Fresnel Term</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h5 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#formulae">Formulae</a></div><ul class="outline-children"></ul></li></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="#microfacet-material">Microfacet Material</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="#microfacet-theory">Microfacet Theory</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="#isotropicanisotropic-materials-brdfs">Isotropic/Anisotropic Materials (BRDFs)</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-further-on-brdfs">II. Further on BRDFs</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="#properties-of-brdfs">Properties of BRDFs</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="#measuring-brdfs">Measuring BRDFs</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="#general-approach">General Approach</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="#improving-efficiency">Improving Efficiency</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="#challenges">Challenges</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#representing-measured-brdfs">Representing Measured BRDFs</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="#tabular-representation">Tabular Representation</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-microfacet-models">Appendix A: Microfacet Models</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="#introduction">Introduction</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="#oren-nayar-diffuse-reflection">Oren-Nayar Diffuse Reflection</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="#microfacet-distribution-functions">Microfacet Distribution Functions</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="#beckmann-distribution">Beckmann Distribution</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="#trowbridge-reitz-distribution">Trowbridge-Reitz Distribution</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="#masking-and-shadowing">Masking and Shadowing</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-torrance-sparrow-model">The Torrance-Sparrow Model</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-17---materials-and-appearances'><span>GAMES101 Lecture 17 - Materials and Appearances</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_17.pdf'><span>GAMES101_Lecture_17.pdf</span></a></p><h2 id='i-materials-and-appearances'><span>I. Materials and Appearances</span></h2><p><img src="../images/Lecture17-img-27.png" referrerpolicy="no-referrer" alt="img-27"></p><p><span>Textures and appearances are closely related:</span></p><ul><li><p><span>Under different lighting conditions textures appears to be different.</span></p></li></ul><p><span>Some of the features from natural materials:</span></p><ul><li><p><strong><span>Water</span></strong></p></li><li><p><strong><span>Scattering</span></strong></p></li><li><p><strong><span>Hair/Fur</span></strong></p></li><li><p><strong><span>Clothes</span></strong></p></li><li><p><strong><span>Subsurface Scattering</span></strong><span> (SSS)</span></p></li><li><p><span>...</span></p></li></ul><p> </p><p><img src="../images/Lecture17-img-1.png" referrerpolicy="no-referrer" alt="img-1"></p><p><span>The term </span><strong><span>material</span></strong><span> is equivalent to </span><strong><span>BSDF</span></strong><span>.</span></p><p><strong><span>Bidirectional Scattering Distribution Function, BSDF</span></strong></p><ul><li><p><span>The generalization of BRDF and BTDF (</span><strong><span>Bidirectional Transmittance Distribution Function</span></strong><span>), which takes both refraction and reflection into consideration.</span></p></li></ul><p> </p><h3 id='diffuselambertian-material'><span>Diffuse/Lambertian Material</span></h3><p><img src="../images/Lecture17-img-2.png" alt="img-2" style="zoom: 25%;" /></p><p><img src="../images/Lecture17-img-3.png" alt="img-3" style="zoom: 33%;" /></p><p align="center">From [Mitsuba render, Wenzel Jakob, 2010</p><p><span>Light is </span><strong><span>equally</span></strong><span> reflected in each output direction.</span></p><p><img src="../images/Lecture17-img-28.png" alt="img-28" style="zoom:50%;" /></p><p><span>Suppose the incident lighting is </span><strong><span>uniform in radiance</span></strong><span>, and without self-emission we have:</span></p><div contenteditable="false" spellcheck="false" 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data-mml-node="msub" transform="translate(556,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-447-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-447-TEX-I-1D456"></use></g></g></g><g data-mml-node="mo" transform="translate(1782.7,0)"><use data-c="3D" xlink:href="#MJX-447-TEX-N-3D"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(2838.5,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-447-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D434" xlink:href="#MJX-447-TEX-I-1D434"></use></g></g><g data-mml-node="mo" transform="translate(4422.3,0)"><use data-c="3D" xlink:href="#MJX-447-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(5478.1,0)"><use data-c="73" xlink:href="#MJX-447-TEX-N-73"></use><use data-c="69" xlink:href="#MJX-447-TEX-N-69" transform="translate(394,0)"></use><use data-c="6E" xlink:href="#MJX-447-TEX-N-6E" transform="translate(672,0)"></use></g><g data-mml-node="mo" transform="translate(6706.1,0)"><use data-c="2061" xlink:href="#MJX-447-TEX-N-2061"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6872.7,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-447-TEX-I-1D703"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(7508.4,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-447-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D703" xlink:href="#MJX-447-TEX-I-1D703"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(8700.1,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-447-TEX-N-64"></use></g></g><g data-mml-node="mi" transform="translate(556,0)"><use data-c="1D719" xlink:href="#MJX-447-TEX-I-1D719"></use></g></g></g><g data-mml-node="mtext" transform="translate(12607.1,0)"><use data-c="29" xlink:href="#MJX-447-TEX-N-29"></use></g></g></g></g><g data-mml-node="mtr" transform="translate(0,-2557.4)"><g data-mml-node="mtd" transform="translate(2932.9,0)"></g><g data-mml-node="mtd" transform="translate(2932.9,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-447-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1333.6,0)"><use data-c="1D70B" xlink:href="#MJX-447-TEX-I-1D70B"></use></g><g data-mml-node="msub" transform="translate(1903.6,0)"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-447-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-447-TEX-I-1D45F"></use></g></g><g data-mml-node="msub" transform="translate(2795.5,0)"><g data-mml-node="mi"><use data-c="1D43F" xlink:href="#MJX-447-TEX-I-1D43F"></use></g><g data-mml-node="mi" transform="translate(714,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-447-TEX-I-1D456"></use></g></g></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>L</mi><mi>o</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup></mrow></msub><msub><mi>f</mi><mi>r</mi></msub><msub><mi>L</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>f</mi><mi>r</mi></msub><msub><mi>L</mi><mi>i</mi></msub><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup></mrow></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>i</mi></msub></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mrow><mtext>(since </mtext><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow><mo>=</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>A</mi></mrow><mo>=</mo><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><mi>θ</mi></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>θ</mi></mrow><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>ϕ</mi></mrow></mrow><mtext>)</mtext></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>π</mi><msub><mi>f</mi><mi>r</mi></msub><msub><mi>L</mi><mi>i</mi></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>If the material absorbs no light, then </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.587ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3795.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-558-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-558-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-558-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-558-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-558-TEX-N-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path id="MJX-558-TEX-I-1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 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style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="20.055ex" height="4.106ex" role="img" focusable="false" viewBox="0 -1118 8864.2 1815" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.577ex;"><defs><path id="MJX-448-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-448-TEX-I-1D45F" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-448-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-448-TEX-I-1D70C" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 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-16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-448-TEX-N-2264" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path id="MJX-448-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="msub"><g data-mml-node="mi"><use data-c="1D453" xlink:href="#MJX-448-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(523,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-448-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(1169.7,0)"><use data-c="3D" xlink:href="#MJX-448-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2225.5,0)"><g data-mml-node="mi" transform="translate(246.5,676)"><use data-c="1D70C" xlink:href="#MJX-448-TEX-I-1D70C"></use></g><g data-mml-node="mi" transform="translate(220,-686)"><use data-c="1D70B" xlink:href="#MJX-448-TEX-I-1D70B"></use></g><rect width="770" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3235.5,0)"><use data-c="2C" xlink:href="#MJX-448-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(3513.5,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mn" transform="translate(4680.1,0)"><use data-c="30" xlink:href="#MJX-448-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(5457.9,0)"><use data-c="2264" xlink:href="#MJX-448-TEX-N-2264"></use></g><g data-mml-node="mi" transform="translate(6513.7,0)"><use data-c="1D70C" xlink:href="#MJX-448-TEX-I-1D70C"></use></g><g data-mml-node="mo" transform="translate(7308.5,0)"><use data-c="2264" xlink:href="#MJX-448-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(8364.2,0)"><use data-c="31" xlink:href="#MJX-448-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>f</mi><mi>r</mi></msub><mo>=</mo><mfrac><mi>ρ</mi><mi>π</mi></mfrac><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><mn>0</mn><mo>≤</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>in which </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.17ex" height="1.489ex" role="img" focusable="false" viewBox="0 -442 517 658" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-559-TEX-I-1D70C" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></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="1D70C" xlink:href="#MJX-559-TEX-I-1D70C"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\rho</script><span> is called </span><strong><span>albedo</span></strong><span>, or color.</span></p><p> </p><h3 id='glossy-material'><span>Glossy Material</span></h3><p><img src="../images/Lecture17-img-4.png" alt="img-4" style="zoom: 25%;" /></p><p><img src="../images/Lecture17-img-5.png" alt="img-5" style="zoom:25%;" /></p><p align="center">From [Mitsuba render, Wenzel Jakob, 2010</p><p> </p><h3 id='ideal-reflectiverefractive-material'><span>Ideal Reflective/Refractive Material</span></h3><p><img src="../images/Lecture17-img-6.png" alt="img-6" style="zoom: 25%;" /></p><p><img src="../images/Lecture17-img-7.png" alt="img-17" style="zoom:25%;" /></p><p align="center">From [Mitsuba render, Wenzel Jakob, 2010</p><ul><li><p><span>Part of the spectrum is absorbed by the underlying material.</span></p></li></ul><p> </p><h3 id='perfect-specular-reflection'><span>Perfect Specular Reflection</span></h3><p><img src="../images/Lecture17-img-9.png" alt="img-9" style="zoom:50%;" 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display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>ω</mi><mi>o</mi></msub><mo>=</mo><mo>−</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>⋅</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mover><mi>n</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><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-605-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 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\phi</script><span> are obtained from the local coordinate system.</span></p><p><strong><span>BRDFs</span></strong><span> for the perfect specular reflection are difficult to write.</span></p><ul><li><p><span>Related to the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.005ex" height="1.645ex" role="img" focusable="false" viewBox="0 -717 444 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-562-TEX-I-1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 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xlink:href="#MJX-452-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(6017.5,0)"><use data-c="2212" xlink:href="#MJX-452-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(7017.7,0)"><g data-mml-node="mi"><use data-c="63" xlink:href="#MJX-452-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-452-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-452-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mn" transform="translate(1371,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-452-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8792.3,0)"><use data-c="2061" xlink:href="#MJX-452-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(8958.9,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-452-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-452-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(9754.9,0)"><use data-c="29" xlink:href="#MJX-452-TEX-N-29"></use></g></g><g data-mml-node="mo" transform="translate(0,214.2)"><use data-c="221A" xlink:href="#MJX-452-TEX-S4-221A"></use></g><rect width="10143.9" height="60" x="1020" y="1904.2"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -5190 1 9880"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:total_internal_reflection" transform="translate(0,-2904.2)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-750)"><g data-mml-node="mtext"><use data-c="28" xlink:href="#MJX-452-TEX-N-28"></use><use data-c="31" xlink:href="#MJX-452-TEX-N-31" transform="translate(389,0)"></use><use data-c="29" xlink:href="#MJX-452-TEX-N-29" transform="translate(889,0)"></use></g></g></g></g></svg></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><msub><mi>η</mi><mi>i</mi></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>i</mi></msub></mrow></mtd><mtd><mi></mi><mo>=</mo><msub><mi>η</mi><mi>t</mi></msub><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>t</mi></msub></mrow></mtd></mtr><mtr><mtd><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow data-mjx-texclass="ORD"><msub><mi>θ</mi><mi>t</mi></msub></mrow></mtd><mtd><mi></mi><mo>=</mo><msqrt><mn>1</mn><mo>−</mo><msup><mi>sin</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>t</mi></msub></msqrt></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msqrt><mn>1</mn><mo>−</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><msub><mi>η</mi><mi>i</mi></msub><msub><mi>η</mi><mi>t</mi></msub></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><msup><mi>sin</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>i</mi></msub></msqrt></mtd></mtr><mlabeledtr><mtd><mtext>(1)</mtext></mtd><mtd></mtd><mtd><mi></mi><mo>=</mo><msqrt><mn>1</mn><mo>−</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><msub><mi>η</mi><mi>i</mi></msub><msub><mi>η</mi><mi>t</mi></msub></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></msqrt></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><em><strong><span>Definition</span></strong></em><span>: </span><strong><span>Total internal reflection</span></strong><span>: When light is moving from a more optically dense medium to a less optically dense medium, i.e., </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n101" cid="n101" 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="7.034ex" height="4.57ex" role="img" focusable="false" viewBox="0 -1118 3108.8 2020" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.041ex;"><defs><path id="MJX-453-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-453-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><path id="MJX-453-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-453-TEX-N-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path id="MJX-453-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="mfrac"><g data-mml-node="msub" transform="translate(225.7,676)"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-453-TEX-I-1D702"></use></g><g data-mml-node="mi" transform="translate(530,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-453-TEX-I-1D456"></use></g></g><g data-mml-node="msub" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-453-TEX-I-1D702"></use></g><g data-mml-node="mi" transform="translate(530,-150) scale(0.707)"><use data-c="1D461" xlink:href="#MJX-453-TEX-I-1D461"></use></g></g><rect width="1035.3" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1553,0)"><use data-c="3E" xlink:href="#MJX-453-TEX-N-3E"></use></g><g data-mml-node="mn" transform="translate(2608.8,0)"><use data-c="31" xlink:href="#MJX-453-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>η</mi><mi>i</mi></msub><msub><mi>η</mi><mi>t</mi></msub></mfrac><mo>></mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div><p><span>then light incident on boundary from large enough angle will not exit the medium. The critical angle can be computed from equation </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-565-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"><a href="#mjx-eqn%3Atotal_internal_reflection"><g data-mml-node="mrow" class=""><rect data-hitbox="true" fill="none" stroke="none" pointer-events="all" width="500" height="666" y="0"></rect><g data-mml-node="mtext"><use data-c="31" xlink:href="#MJX-565-TEX-N-31"></use></g></g></a></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow href="#mjx-eqn%3Atotal_internal_reflection" class="MathJax_ref"><mtext>1</mtext></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\ref{total_internal_reflection}</script><span> by substituting </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.396ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3710.8 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-566-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><path id="MJX-566-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-566-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-566-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><path id="MJX-566-TEX-N-2F" d="M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z"></path><path id="MJX-566-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="msub"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-566-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D461" xlink:href="#MJX-566-TEX-I-1D461"></use></g></g><g data-mml-node="mo" transform="translate(1085,0)"><use data-c="3D" xlink:href="#MJX-566-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(2140.8,0)"><use data-c="1D70B" xlink:href="#MJX-566-TEX-I-1D70B"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2710.8,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-566-TEX-N-2F"></use></g></g><g data-mml-node="mn" transform="translate(3210.8,0)"><use data-c="32" xlink:href="#MJX-566-TEX-N-32"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mi>t</mi></msub><mo>=</mo><mi>π</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">\theta_t = \pi / 2</script><span> into the equation.</span></p><p> </p><ul><li><p><strong><span>Snell's Window/Circle</span></strong></p><p><img src="../images/Lecture17-img-30.png" alt="img-30" style="zoom:50%;" /></p></li></ul><p> </p><p> </p><h3 id='fresnel-reflectionterm'><span>Fresnel Reflection/Term</span></h3><p><span>Reflectance depends on </span><strong><span>incident angle</span></strong><span> (and </span><strong><span>polarization</span></strong><span> of light).</span></p><p><img src="../images/Lecture17-img-11.png" referrerpolicy="no-referrer" alt="img-11"></p><p align="center">This example: reflectance increases with grazing angle [Lafortune et al. 1997]</p><h4 id='fresnel-term'><span>Fresnel Term</span></h4><ul><li><p><strong><span>Polarization</span></strong><span>: The component of the electric field parallel to the incidence plane is termed </span><em><span>p-like</span></em><span> (parallel) and the component perpendicular to this plane is termed </span><em><span>s-like</span></em><span> (from </span><em><span>senkrecht</span></em><span>, German for perpendicular). (Wikipedia)</span></p></li></ul><p><strong><span>Dielectric</span></strong><span>, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.033ex" height="1.995ex" role="img" focusable="false" viewBox="0 -666 3108.6 882" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.489ex;"><defs><path id="MJX-567-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-567-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 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44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z"></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="1D702" xlink:href="#MJX-567-TEX-I-1D702"></use></g><g data-mml-node="mo" transform="translate(774.8,0)"><use data-c="3D" xlink:href="#MJX-567-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1830.6,0)"><use data-c="31" xlink:href="#MJX-567-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-567-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-567-TEX-N-35" transform="translate(778,0)"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi><mo>=</mo><mn>1.5</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta = 1.5</script><span>:</span></p><p><img src="../images/Lecture17-img-12.png" alt="img-12" style="zoom:33%;" /></p><p> </p><p><strong><span>Conductor</span></strong><span>:</span></p><p><img src="../images/Lecture17-img-13.png" alt="img-13" style="zoom:33%;" /></p><ul><li><p><strong><span>Conductors</span></strong><span> have negative indices of refraction.</span></p></li></ul><p> </p><h5 id='formulae'><span>Formulae</span></h5><p><strong><span>Accurate</span></strong><span>: </span><strong><span>polarization</span></strong><span> taken into consideration</span></p><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 600 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-568-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></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="1D45B" xlink:href="#MJX-568-TEX-I-1D45B"></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 related to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.124ex" height="1.489ex" role="img" focusable="false" viewBox="0 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xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\eta</script><span>, the intrinsic property of the material.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n130" cid="n130" 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="64.151ex" height="12.529ex" role="img" focusable="false" viewBox="0 -3118 28354.6 5538" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -5.475ex;"><defs><path id="MJX-454-TEX-I-1D445" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 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id='microfacet-theory'><span>Microfacet Theory</span></h4><p><img src="../images/Lecture17-img-15.png" referrerpolicy="no-referrer" alt="img-15"></p><p><span>Rough surface:</span></p><ul><li><p><span>Macroscale: flat & rough</span></p></li><li><p><span>Microscale: bumpy & </span><strong><span>specular</span></strong></p></li></ul><p> </p><p><strong><span>Microfacet</span></strong><span>: individual elements of surface act like </span><em><span>mirrors</span></em></p><p><strong><span>Key</span></strong><span>: The </span><strong><span>distribution</span></strong><span> of their </span><strong><span>normals</span></strong><span>. Each microfacet has its own normal.</span></p><ul><li><p><span>Concentrated <==> Glossy</span></p><p><img src="../images/Lecture17-img-16.png" alt="img-16" style="zoom: 67%;" /></p><p><img src="../images/Lecture17-img-17.png" alt="img-17" style="zoom: 50%;" /></p></li><li><p><span>Spread <==> Diffuse</span></p><p><img src="../images/Lecture17-img-18.png" alt="img-18" style="zoom:67%;" /></p><p><img src="../images/Lecture17-img-19.png" alt="img-19" style="zoom:50%;" /></p></li></ul><p><img src="../images/Lecture17-img-20.png" alt="img-20" style="zoom:50%;" /></p><p><strong><span>Microfacet BRDF</span></strong><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n162" cid="n162" 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" 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xlink:href="#MJX-459-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(500,0)"><use data-c="28" xlink:href="#MJX-459-TEX-N-28"></use></g><g data-mml-node="mtext" transform="translate(889,0)"><use data-c="1D427" xlink:href="#MJX-459-TEX-B-1D427"></use></g><g data-mml-node="mo" transform="translate(1528,0)"><use data-c="2C" xlink:href="#MJX-459-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1972.7,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-459-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-459-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(2921.6,0)"><use data-c="29" xlink:href="#MJX-459-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3310.6,0)"><use data-c="28" xlink:href="#MJX-459-TEX-N-28"></use></g><g data-mml-node="mtext" transform="translate(3699.6,0)"><use data-c="1D427" xlink:href="#MJX-459-TEX-B-1D427"></use></g><g data-mml-node="mo" transform="translate(4338.6,0)"><use data-c="2C" xlink:href="#MJX-459-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(4783.3,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-459-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-459-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(5831.2,0)"><use data-c="29" xlink:href="#MJX-459-TEX-N-29"></use></g></g><rect width="11169.9" 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>f</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mtext>F</mtext><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><mtext mathvariant="bold">h</mtext><mo stretchy="false">)</mo><mtext mathvariant="bold">G</mtext><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo>,</mo><mtext mathvariant="bold">h</mtext><mo stretchy="false">)</mo><mtext mathvariant="bold">D</mtext><mo stretchy="false">(</mo><mtext mathvariant="bold">h</mtext><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mo stretchy="false">(</mo><mtext mathvariant="bold">n</mtext><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext mathvariant="bold">n</mtext><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>In which:</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.477ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 653 680" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-570-TEX-N-46" d="M128 619Q121 626 117 628T101 631T58 634H25V680H582V676Q584 670 596 560T610 444V440H570V444Q563 493 561 501Q555 538 543 563T516 601T477 622T431 631T374 633H334H286Q252 633 244 631T233 621Q232 619 232 490V363H284Q287 363 303 363T327 364T349 367T372 373T389 385Q407 403 410 459V480H450V200H410V221Q407 276 389 296Q381 303 371 307T348 313T327 316T303 317T284 317H232V189L233 61Q240 54 245 52T270 48T333 46H360V0H348Q324 3 182 3Q51 3 36 0H25V46H58Q100 47 109 49T128 61V619Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="46" xlink:href="#MJX-570-TEX-N-46"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math></mjx-assistive-mml></mjx-container><script type="math/tex">\text{F}</script><span> is the Fresnel term</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.045ex" height="1.6ex" role="img" focusable="false" viewBox="0 -697 904 707" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-571-TEX-B-1D406" d="M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="1D406" xlink:href="#MJX-571-TEX-B-1D406"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">G</mtext></math></mjx-assistive-mml></mjx-container><script type="math/tex">\textbf{G}</script><span> is the shadowing-masking term</span></p><ul><li><p><em><span>Microfacets may </span><strong><span>block</span></strong><span> each other</span></em><span>.</span></p></li><li><p><em><span>Happens often when lights are near the </span><strong><span>grazing angle</span></strong></em><span>.</span></p></li></ul></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.995ex" height="1.552ex" role="img" focusable="false" viewBox="0 -686 882 686" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-572-TEX-B-1D403" d="M39 624V686H270H310H408Q500 686 545 680T638 649Q768 584 805 438Q817 388 817 338Q817 171 702 75Q628 17 515 2Q504 1 270 0H39V62H147V624H39ZM655 337Q655 370 655 390T650 442T639 494T616 540T580 580T526 607T451 623Q443 624 368 624H298V62H377H387H407Q445 62 472 65T540 83T606 129Q629 156 640 195T653 262T655 337Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="1D403" xlink:href="#MJX-572-TEX-B-1D403"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">D</mtext></math></mjx-assistive-mml></mjx-container><script type="math/tex">\textbf{D}</script><span> is the distribution of normals</span></p><ul><li><p><em><span>How many normals are there that match the direction of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.446ex" height="1.57ex" role="img" focusable="false" viewBox="0 -694 639 694" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: 0px;"><defs><path id="MJX-573-TEX-B-1D421" d="M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="1D421" xlink:href="#MJX-573-TEX-B-1D421"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">h</mtext></math></mjx-assistive-mml></mjx-container><script type="math/tex">\textbf{h}</script></em><span>?</span></p></li></ul></li></ul><p> </p><h3 id='isotropicanisotropic-materials-brdfs'><span>Isotropic/Anisotropic Materials (BRDFs)</span></h3><p><img src="../images/Lecture17-img-21.png" referrerpolicy="no-referrer" alt="img-21"></p><p><strong><span>Key</span></strong><span>: </span><strong><span>Directionality</span></strong><span> of the underlying surface.</span></p><p><img src="../images/Lecture17-img-22.png" alt="img-22" style="zoom:50%;" /></p><p> </p><p><img src="../images/Lecture17-img-23.png" alt="img-23" style="zoom:50%;" /></p><p><strong><span>Anisotropic</span></strong><span>: Reflection depends on </span><strong><span>azimuthal angle</span></strong><span> </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" 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stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo>;</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>≠</mo><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><span>Results from </span><strong><span>oriented microstructure</span></strong><span> of the surface, e.g.</span></p><ul><li><p><span>Brushed Metal</span></p></li><li><p><span>Nylon</span></p></li><li><p><span>Velvet</span></p></li></ul></li></ul><p> </p><h2 id='ii-further-on-brdfs'><span>II. 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xlink:href="#MJX-464-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">∀</mi><msub><mi>ω</mi><mi>r</mi></msub><mo>,</mo><mstyle scriptlevel="0"><mspace width="1em"></mspace></mstyle><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup></mrow></msub><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>i</mi></msub><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow><mo>≤</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div></li><li><p><strong><span>Isotropic/Anisotropic</span></strong></p><ul><li><p><span>If isotropic: </span><mjx-container 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transform="translate(11317.6,0)"><use data-c="2C" xlink:href="#MJX-576-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(11762.3,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-576-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D45F" xlink:href="#MJX-576-TEX-I-1D45F"></use></g></g><g data-mml-node="mo" transform="translate(12982.4,0)"><use data-c="2212" xlink:href="#MJX-576-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(13982.6,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-576-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-576-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(14905.6,0)"><use data-c="29" xlink:href="#MJX-576-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo>;</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">f_r (\theta_i, \phi_i; \theta_r, \phi_r) = f_r (\theta_i, \theta_r, \phi_r - \phi_i)</script><span>, which essentially means that </span><strong><span>the dimension of BRDF is reduced by 1.</span></strong></p><p><span>Then from reciprocity we have</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" 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data-c="1D456" xlink:href="#MJX-465-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3421.3,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-465-TEX-N-7C"></use></g></g><g data-mml-node="mo" transform="translate(24942.3,0)"><use data-c="29" xlink:href="#MJX-465-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"><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>r</mi></msub><mo>,</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>ϕ</mi><mi>r</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div></li></ul><p><img src="../images/Lecture17-img-26.png" alt="img-26" style="zoom:50%;" /></p></li></ul><p> </p><h3 id='measuring-brdfs'><span>Measuring BRDFs</span></h3><p><span>Target:</span></p><ul><li><p><span>Avoid need to develop/derive models</span></p><ul><li><p><span>Automatically includes all of the scattering effects present</span></p></li></ul></li><li><p><span>Can accurately render with real-world materials</span></p><ul><li><p><span>Useful for product design, special effects, ...</span></p></li></ul></li></ul><p> </p><h4 id='general-approach'><span>General Approach</span></h4><p><img src="../images/Lecture17-img-31.png" alt="img-31" style="zoom: 33%;" /></p><ul><li><p><span>For each outgoing direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1047.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-634-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><path id="MJX-634-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D714" xlink:href="#MJX-634-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-634-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_o</script></p><ul><li><p><span>move light to illuminate surface with a thin beam from </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1047.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-634-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><path id="MJX-634-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D714" xlink:href="#MJX-634-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-634-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_o</script></p></li><li><p><span>For each incoming direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 949 600.8" 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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="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-635-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-635-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_i</script></p><ul><li><p><span>move sensor to be at direction </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" 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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="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-635-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-635-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_i</script><span> from surface</span></p></li><li><p><span>measure incident radiance</span></p></li></ul></li></ul></li></ul><p> </p><p><strong><span>Gonioreflectometer</span></strong><span>:</span></p><p><img src="../images/Lecture17-img-32.png" alt="img-32" style="zoom: 50%;" /></p><p align="center">Spherical gantry at UCSD</p><p> </p><h4 id='improving-efficiency'><span>Improving Efficiency</span></h4><ul><li><p><span>Isotropic surfaces reduce dimensionality from 4D to 3D</span></p></li><li><p><span>Reciprocity reduces # of measurements by half</span></p></li><li><p><span>Clever optical systems</span></p></li></ul><p> </p><h4 id='challenges'><span>Challenges</span></h4><ul><li><p><span>Accurate measurements at grazing angles</span></p><ul><li><p><span>Important due to Fresnel effects</span></p></li></ul></li><li><p><span>Measuring with dens enough sampling to capture high frequency specularities</span></p></li><li><p><span>Retro-reflection</span></p></li><li><p><span>Spatially-varying reflectance</span></p></li><li><p><span>...</span></p></li></ul><p> </p><h4 id='representing-measured-brdfs'><span>Representing Measured BRDFs</span></h4><p><span>Desirable qualities:</span></p><ul><li><p><span>Compact representation</span></p></li><li><p><span>Accurate representation of measured data</span></p></li><li><p><span>Efficient evaluation for arbitrary pairs of directions</span></p></li><li><p><span>Good distributions available for importance sampling</span></p></li></ul><p> </p><h4 id='tabular-representation'><span>Tabular Representation</span></h4><p><strong><span>MERL BRDF Database</span></strong><span> [Matusik et al. 2004], </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="13.45ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 5944.9 688" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-581-TEX-N-39" d="M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 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xlink:href="#MJX-582-TEX-N-7C"></use></g><g data-mml-node="msub" transform="translate(278,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-582-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-582-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(1423.2,0)"><use data-c="2212" xlink:href="#MJX-582-TEX-N-2212"></use></g><g data-mml-node="msub" transform="translate(2423.4,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-582-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-582-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(3445.3,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-582-TEX-N-7C"></use></g></g><g data-mml-node="mo" transform="translate(6692.6,0)"><use data-c="29" xlink:href="#MJX-582-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>o</mi></msub><mo>,</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>o</mi></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">(\theta_i, \theta_o, \abs{\phi_i - \phi_o})</script></p><ul><li><p><strong><span>Better</span></strong><span>: reparameterize angles to better match specularities</span></p></li><li><p><span>Generally need to resample measured values to table</span></p></li><li><p><strong><span>Very high</span></strong><span> storage requirements</span></p></li></ul><p> </p><h2 id='appendix-a-microfacet-models'><span>Appendix A: Microfacet Models</span></h2><p><span>Reference: </span><a href='https://www.pbr-book.org/3ed-2018/Reflection_Models/Microfacet_Models'><span>Microfacet Models (pbr-book.org)</span></a></p><h3 id='introduction'><span>Introduction</span></h3><p><span>Many geometric-optics-based approaches to modeling surface reflection and transmission are based on the idea that rough surfaces can be modeled as a collection of small </span><em><span>microfacets</span></em><span>. They are often modeled as heightfields, where the </span><strong><span>distribution</span></strong><span> of facet orientations is described statistically.</span></p><p><img src="../images/Lecture17-img-33.png" alt="img-33" style="zoom: 67%;" /></p><ul><li><p><strong><span>Microsurface</span></strong><span> is used to describe microfacet surfaces</span></p></li><li><p><strong><span>Macrosurface</span></strong><span> is used to describe the underlying smooth surface (as represented by a </span><code>Shape</code><span>, or other </span><code>Object</code><span> in our framework for homework).</span></p></li></ul><p> </p><p><span>The microfacet-based BRDF models work by statistically modeling the scattering of light from a large collection of microfacets.</span></p><ul><li><p><span>If we assume that the differential 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" 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number of microfacets are illuminated, </span></p></li><li><p><span>and it is the </span><strong><span>aggregate behavior</span></strong><span> of these microfacets that determines the observed scattering.</span></p></li></ul><p> </p><p><span>The two main components of microfacet models are:</span></p><ul><li><p><span>A representation of the </span><strong><span>distribution</span></strong><span> of facets, and</span></p></li><li><p><span>A </span><strong><span>BRDF</span></strong><span> for </span><strong><span>individual microfacets</span></strong><span>.</span></p><ul><li><p><em><span>Perfect Mirror Reflection</span></em></p><ul><li><p><span>most commonly used</span></p></li></ul></li><li><p><em><span>Specular Transmission</span></em></p><ul><li><p><span>useful for modeling many translucent materials</span></p></li></ul></li><li><p><em><span>The Oren-Nayar Model</span></em></p><ul><li><p><span>treats microfacets as Lambertian reflectors</span></p></li></ul></li></ul></li></ul><p> </p><p><span>Three important geometric effects to consider with Microfacet Reflection Models:</span></p><p><img src="../images/Lecture17-img-34.png" alt="img-34" style="zoom:50%;" /></p><ul><li><p><strong><span>Masking</span></strong></p><ul><li><p><span>Microfacet is occluded by another facet</span></p></li></ul></li><li><p><strong><span>Shadowing</span></strong></p><ul><li><p><span>Microfacet may lie in the shadow of a neighboring microfacet</span></p></li></ul></li><li><p><strong><span>Interreflection</span></strong></p><ul><li><p><span>Cause a microfacet to reflect more light than predicted by the amount of direct illumination and the low-level microfacet BRDF</span></p></li></ul></li></ul><p> </p><h3 id='oren-nayar-diffuse-reflection'><span>Oren-Nayar Diffuse Reflection</span></h3><p><strong><span>Idea:</span></strong><span> Real-world objects do not exhibit perfect Lambertian reflection.</span></p><ul><li><p><span>Rough surfaces generally appear brighter as the illumination direction approaches the viewing direction.</span></p></li><li><p><span>Describe rough surfaces by </span><strong><span>V-shaped</span></strong><span> microfacets, which is</span></p><ul><li><p><span>Described by a spherical Gaussian distribution with a single parameter </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.292ex" height="1ex" role="img" focusable="false" viewBox="0 -431 571 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-592-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></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="1D70E" xlink:href="#MJX-592-TEX-I-1D70E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\sigma</script><span>, the standard deviation of the microfacet orientation angle</span></p></li><li><p><strong><span>Interreflections</span></strong><span>: Only consider the neighboring microfacet</span></p></li></ul></li></ul><p> </p><p><strong><span>Approximation:</span></strong></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n384" cid="n384" 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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xlink:href="#MJX-466-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1756.6,0)"><g data-mml-node="mi"><use data-c="1D719" xlink:href="#MJX-466-TEX-I-1D719"></use></g><g data-mml-node="mi" transform="translate(629,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-466-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(2778.6,0)"><use data-c="29" xlink:href="#MJX-466-TEX-N-29"></use></g></g><g data-mml-node="mo" transform="translate(17631.4,0)"><use data-c="29" xlink:href="#MJX-466-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(18187,0)"><use data-c="73" xlink:href="#MJX-466-TEX-N-73"></use><use data-c="69" xlink:href="#MJX-466-TEX-N-69" transform="translate(394,0)"></use><use data-c="6E" xlink:href="#MJX-466-TEX-N-6E" transform="translate(672,0)"></use></g><g data-mml-node="mo" transform="translate(19415,0)"><use data-c="2061" xlink:href="#MJX-466-TEX-N-2061"></use></g><g data-mml-node="mi" transform="translate(19581.7,0)"><use data-c="1D6FC" xlink:href="#MJX-466-TEX-I-1D6FC"></use></g><g data-mml-node="mi" transform="translate(20388.4,0)"><use data-c="74" xlink:href="#MJX-466-TEX-N-74"></use><use data-c="61" xlink:href="#MJX-466-TEX-N-61" transform="translate(389,0)"></use><use data-c="6E" xlink:href="#MJX-466-TEX-N-6E" transform="translate(889,0)"></use></g><g data-mml-node="mo" transform="translate(21833.4,0)"><use data-c="2061" xlink:href="#MJX-466-TEX-N-2061"></use></g><g data-mml-node="mi" transform="translate(22000,0)"><use data-c="1D6FD" xlink:href="#MJX-466-TEX-I-1D6FD"></use></g><g data-mml-node="mo" transform="translate(22566,0)"><use data-c="29" xlink:href="#MJX-466-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"><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>R</mi><mi>π</mi></mfrac><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo data-mjx-texclass="OP" movablelimits="true">max</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mrow><mo data-mjx-texclass="OPEN">(</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>ϕ</mi><mi>o</mi></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">)</mo><mi>sin</mi><mo data-mjx-texclass="NONE"></mo><mi>α</mi><mi>tan</mi><mo data-mjx-texclass="NONE"></mo><mi>β</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>where if </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.292ex" height="1ex" role="img" focusable="false" viewBox="0 -431 571 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path 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transform="translate(833,0)"></use><use data-c="6E" xlink:href="#MJX-470-TEX-N-6E" transform="translate(1111,0)"></use></g><g data-mml-node="mo" transform="translate(3566.6,0)"><use data-c="28" xlink:href="#MJX-470-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(3955.6,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-470-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-470-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(4751.5,0)"><use data-c="2C" xlink:href="#MJX-470-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(5196.2,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-470-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-470-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(6091.1,0)"><use data-c="29" xlink:href="#MJX-470-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"><mi>β</mi><mo>=</mo><mo data-mjx-texclass="OP" movablelimits="true">min</mo><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo>,</mo><msub><mi>θ</mi><mi>o</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p> </p><h3 id='microfacet-distribution-functions'><span>Microfacet Distribution Functions</span></h3><p><span>One important characteristics of a microfacet surface is represented by the distribution function </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.15ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2718.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-619-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path id="MJX-619-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-619-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><path id="MJX-619-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path id="MJX-619-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="1D437" xlink:href="#MJX-619-TEX-I-1D437"></use></g><g data-mml-node="mo" transform="translate(828,0)"><use data-c="28" xlink:href="#MJX-619-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(1217,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-619-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-619-TEX-I-210E"></use></g></g><g data-mml-node="mo" transform="translate(2329.3,0)"><use data-c="29" xlink:href="#MJX-619-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>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">D(\omega_h)</script><span>, which gives the differential area of microfacets with the surface normal </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.517ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1112.3 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-638-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><path id="MJX-638-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D714" xlink:href="#MJX-638-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-638-TEX-I-210E"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_h</script><span>. In </span><code>pbrt</code><span>, microfacet distribution functions are defined in the same BSDF coordinate system as BxDFs. As such, </span></p><ul><li><p><span>a perfectly smooth surface could be describe by a delta distribution that was non-zero only when </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.517ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1112.3 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-638-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><path id="MJX-638-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D714" xlink:href="#MJX-638-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-638-TEX-I-210E"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_h</script><span> was equal to the surface normal: </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="24.38ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10775.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-589-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z"></path><path id="MJX-589-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-589-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><path id="MJX-589-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path id="MJX-589-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><path id="MJX-589-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-589-TEX-I-1D6FF" d="M195 609Q195 656 227 686T302 717Q319 716 351 709T407 697T433 690Q451 682 451 662Q451 644 438 628T403 612Q382 612 348 641T288 671T249 657T235 628Q235 584 334 463Q401 379 401 292Q401 169 340 80T205 -10H198Q127 -10 83 36T36 153Q36 286 151 382Q191 413 252 434Q252 435 245 449T230 481T214 521T201 566T195 609ZM112 130Q112 83 136 55T204 27Q233 27 256 51T291 111T309 178T316 232Q316 267 309 298T295 344T269 400L259 396Q215 381 183 342T137 256T118 179T112 130Z"></path><path id="MJX-589-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-589-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-589-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 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xlink:href="#MJX-589-TEX-I-210E"></use></g></g><g data-mml-node="mo" transform="translate(2329.3,0)"><use data-c="29" xlink:href="#MJX-589-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2996.1,0)"><use data-c="3D" xlink:href="#MJX-589-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(4051.8,0)"><use data-c="1D6FF" xlink:href="#MJX-589-TEX-I-1D6FF"></use></g><g data-mml-node="mo" transform="translate(4495.8,0)"><use data-c="28" xlink:href="#MJX-589-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(4884.8,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-589-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-589-TEX-I-210E"></use></g></g><g data-mml-node="mo" transform="translate(6219.4,0)"><use data-c="2212" xlink:href="#MJX-589-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(7219.6,0)"><use data-c="28" xlink:href="#MJX-589-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(7608.6,0)"><use data-c="30" xlink:href="#MJX-589-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(8108.6,0)"><use data-c="2C" xlink:href="#MJX-589-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(8553.3,0)"><use data-c="30" xlink:href="#MJX-589-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(9053.3,0)"><use data-c="2C" xlink:href="#MJX-589-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(9497.9,0)"><use data-c="31" xlink:href="#MJX-589-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(9997.9,0)"><use data-c="29" xlink:href="#MJX-589-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(10386.9,0)"><use data-c="29" xlink:href="#MJX-589-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>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">D(\omega_h) = \delta (\omega_h - (0, 0, 1))</script></p></li></ul><p><span>Microfacet distribution functions must be</span></p><ul><li><p><strong><span>Normalized</span></strong><span>: Given a differential area of the microsurface, </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-602-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 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data-mml-node="msub" transform="translate(7171.2,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-471-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-471-TEX-I-210E"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="OP" transform="translate(8297.2,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="64" xlink:href="#MJX-471-TEX-N-64"></use></g></g><g data-mml-node="msub" transform="translate(556,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-471-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-471-TEX-I-210E"></use></g></g></g><g data-mml-node="mo" transform="translate(10243.3,0)"><use data-c="3D" xlink:href="#MJX-471-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(11299,0)"><use data-c="31" xlink:href="#MJX-471-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mtext mathvariant="bold">n</mtext><mo stretchy="false">)</mo></mrow></msub><mi>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>h</mi></msub></mrow><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></div></div><p><img src="../images/Lecture17-img-35.png" alt="image-20230719171037939" style="zoom:50%;" /></p></li></ul><h4 id='beckmann-distribution'><span>Beckmann Distribution</span></h4><p><span>A widely used microfacet distribution function based on a Gaussian distribution of microfacet slops is due to Beckmann and Spizzichino.</span></p><p><span>The traditional definition of the Beckmann-Spizzichino model is </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n405" cid="n405" 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="20.827ex" height="5.688ex" role="img" focusable="false" viewBox="0 -1628.7 9205.6 2514.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.003ex;"><defs><path id="MJX-472-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 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data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-472-TEX-I-210E"></use></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(3952.2,0)"><g data-mml-node="mo"><use data-c="2F" xlink:href="#MJX-472-TEX-N-2F"></use></g></g><g data-mml-node="msup" transform="translate(4452.2,0)"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-472-TEX-I-1D6FC"></use></g><g data-mml-node="mn" transform="translate(673,363) scale(0.707)"><use data-c="32" xlink:href="#MJX-472-TEX-N-32"></use></g></g></g></g><g data-mml-node="mrow" transform="translate(220,-727.7)"><g data-mml-node="mi"><use data-c="1D70B" xlink:href="#MJX-472-TEX-I-1D70B"></use></g><g data-mml-node="msup" transform="translate(570,0)"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-472-TEX-I-1D6FC"></use></g><g data-mml-node="mn" transform="translate(673,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-472-TEX-N-32"></use></g></g><g data-mml-node="msup" transform="translate(1813.2,0)"><g data-mml-node="mi"><use data-c="63" xlink:href="#MJX-472-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-472-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-472-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mn" transform="translate(1371,289) scale(0.707)"><use data-c="34" xlink:href="#MJX-472-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(3587.8,0)"><use data-c="2061" xlink:href="#MJX-472-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(3754.4,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-472-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-472-TEX-I-210E"></use></g></g></g><rect width="4913.7" 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>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>tan</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msup><mi>α</mi><mn>2</mn></msup></mrow></msup><mrow><mi>π</mi><msup><mi>α</mi><mn>2</mn></msup><msup><mi>cos</mi><mn>4</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>where if </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.292ex" height="1ex" role="img" focusable="false" viewBox="0 -431 571 442" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-592-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></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="1D70E" xlink:href="#MJX-592-TEX-I-1D70E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\sigma</script><span> is the RMS slope of the microfacets, then </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.818ex" height="2.398ex" role="img" focusable="false" 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display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>tan</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub><mo stretchy="false">(</mo><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>ϕ</mi><mi>h</mi></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msubsup><mi>α</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>ϕ</mi><mi>h</mi></msub><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msubsup><mi>α</mi><mi>y</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></msup><mrow><mi>π</mi><msub><mi>α</mi><mi>x</mi></msub><msub><mi>α</mi><mi>y</mi></msub><msup><mi>cos</mi><mn>4</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>h</mi></msub></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><span>Note that the original isotropic variant falls out when </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.988ex" height="1.986ex" role="img" focusable="false" viewBox="0 -583 3530.5 878" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.667ex;"><defs><path id="MJX-594-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 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transform="translate(673,-150) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-594-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(1405.2,0)"><use data-c="3D" xlink:href="#MJX-594-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(2461,0)"><g data-mml-node="mi"><use data-c="1D6FC" xlink:href="#MJX-594-TEX-I-1D6FC"></use></g><g data-mml-node="mi" transform="translate(673,-150) scale(0.707)"><use data-c="1D466" xlink:href="#MJX-594-TEX-I-1D466"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mi>x</mi></msub><mo>=</mo><msub><mi>α</mi><mi>y</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\alpha_x = \alpha_y</script></p></li></ul><p> </p><p><span>When programming, the algorithm directly translates the above equation, but pay special attention to the following issues:</span></p><ul><li><p><span>Infinity value of 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Zero should be explicitly returned.</span></p></li></ul><p> </p><h4 id='trowbridge-reitz-distribution'><span>Trowbridge-Reitz Distribution</span></h4><p><span>Anisotropic variant given by</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n423" cid="n423" 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="62.036ex" height="5.794ex" role="img" focusable="false" viewBox="0 -1342 27420 2560.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.758ex;"><defs><path id="MJX-474-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 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</span><strong><span>roughness</span></strong><span>, which is between </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.526ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2000.7 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-598-TEX-N-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path id="MJX-598-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-598-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-598-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-598-TEX-N-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="5B" xlink:href="#MJX-598-TEX-N-5B"></use></g><g data-mml-node="mn" transform="translate(278,0)"><use data-c="30" xlink:href="#MJX-598-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(778,0)"><use data-c="2C" xlink:href="#MJX-598-TEX-N-2C"></use></g><g data-mml-node="mn" transform="translate(1222.7,0)"><use data-c="31" xlink:href="#MJX-598-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1722.7,0)"><use data-c="5D" xlink:href="#MJX-598-TEX-N-5D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">[0, 1]</script><span>, and value close to zero correspond to near-perfect specular reflection.</span></p><p> </p><h3 id='masking-and-shadowing'><span>Masking and Shadowing</span></h3><p><strong><span>Smith's Masking-Shadowing Function</span></strong><span>: Some microfacets will be </span><em><span>invisible</span></em><span> from a given viewing or illumination direction because,</span></p><ol start='' ><li><p><span>They are back-facing</span></p></li><li><p><span>Some of the forward-facing microfacet area will be hidden due to being shadowed by back-facing microfacets.</span></p></li></ol><p><span>This is described by</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n442" cid="n442" 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="9.456ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4179.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-475-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-475-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-475-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-475-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><path id="MJX-475-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-475-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path id="MJX-475-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="msub"><g data-mml-node="mi"><use data-c="1D43A" xlink:href="#MJX-475-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-475-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1222.6,0)"><use data-c="28" xlink:href="#MJX-475-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1611.6,0)"><use data-c="1D714" xlink:href="#MJX-475-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(2233.6,0)"><use data-c="2C" xlink:href="#MJX-475-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2678.2,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-475-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-475-TEX-I-210E"></use></g></g><g data-mml-node="mo" transform="translate(3790.5,0)"><use data-c="29" xlink:href="#MJX-475-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"><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>which gives the fraction of microfacets with normal </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.517ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1112.3 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-638-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><path id="MJX-638-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D714" xlink:href="#MJX-638-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-638-TEX-I-210E"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_h</script><span> that are visible from direction </span><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-612-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-612-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><span>.</span></p><ul><li><p><span>Note that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="17.753ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7846.6 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-601-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-601-TEX-N-2264" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z"></path><path id="MJX-601-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-601-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-601-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-601-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><path id="MJX-601-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-601-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path id="MJX-601-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="mn"><use data-c="30" xlink:href="#MJX-601-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(777.8,0)"><use data-c="2264" xlink:href="#MJX-601-TEX-N-2264"></use></g><g data-mml-node="msub" transform="translate(1833.6,0)"><g data-mml-node="mi"><use data-c="1D43A" xlink:href="#MJX-601-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-601-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(3056.1,0)"><use data-c="28" xlink:href="#MJX-601-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(3445.1,0)"><use data-c="1D714" xlink:href="#MJX-601-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(4067.1,0)"><use data-c="2C" xlink:href="#MJX-601-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(4511.8,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-601-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-601-TEX-I-210E"></use></g></g><g data-mml-node="mo" transform="translate(5624.1,0)"><use data-c="29" xlink:href="#MJX-601-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6290.8,0)"><use data-c="2264" xlink:href="#MJX-601-TEX-N-2264"></use></g><g data-mml-node="mn" transform="translate(7346.6,0)"><use data-c="31" xlink:href="#MJX-601-TEX-N-31"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container><script type="math/tex">0 \leq G_1(\omega, \omega_h) \leq 1</script><span>.</span></p></li></ul><p> </p><p><strong><span>Normalization Constraint</span></strong><span>: A differential 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-602-TEX-N-64" d="M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z"></path><path id="MJX-602-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 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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-606-TEX-N-2061" d=""></path><path id="MJX-606-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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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-612-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><span> that makes an angle </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-605-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-605-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> with the surface normal. 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data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>H</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mtext mathvariant="bold">n</mtext><mo stretchy="false">)</mo></mrow></msub><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><mo>,</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mo data-mjx-texclass="OP" movablelimits="true">max</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>ω</mi><mo>⋅</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><msub><mi>ω</mi><mi>h</mi></msub></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><img src="../images/Lecture17-img-37.png" alt="img-37" style="zoom:50%;" /></p><p> </p><p><strong><span>Compute </span><mjx-container class="MathJax" jax="SVG" style="position: 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581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-608-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="msub"><g data-mml-node="mi"><use data-c="1D43A" xlink:href="#MJX-608-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-608-TEX-N-31"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">G_1</script><span> using </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z"></path><path id="MJX-625-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-625-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><path id="MJX-625-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="39B" xlink:href="#MJX-625-TEX-N-39B"></use></g><g data-mml-node="mo" transform="translate(694,0)"><use data-c="28" xlink:href="#MJX-625-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1083,0)"><use data-c="1D714" xlink:href="#MJX-625-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(1705,0)"><use data-c="29" xlink:href="#MJX-625-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 mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Lambda(\omega)</script></strong><span>: Because the microfacets form a heightfield, every back-facing microfacet shadows a forward-facing microfacet of equal projected area in the direction </span><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-612-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-612-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><span>. If </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.297ex" height="2.32ex" role="img" focusable="false" viewBox="0 -775.2 2783.1 1025.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-611-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><path id="MJX-611-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-611-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-611-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><path id="MJX-611-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="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-611-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(783,363) scale(0.707)"><use data-c="2B" xlink:href="#MJX-611-TEX-N-2B"></use></g></g><g data-mml-node="mo" transform="translate(1383.1,0)"><use data-c="28" xlink:href="#MJX-611-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1772.1,0)"><use data-c="1D714" xlink:href="#MJX-611-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(2394.1,0)"><use data-c="29" xlink:href="#MJX-611-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">A^+(\omega)</script><span> is the projected area of forward-facing microfacets as seen from the direction </span><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-612-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-612-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><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.297ex" height="2.32ex" role="img" focusable="false" viewBox="0 -775.2 2783.1 1025.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-613-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><path id="MJX-613-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-613-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-613-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><path id="MJX-613-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="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-613-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(783,363) scale(0.707)"><use data-c="2212" xlink:href="#MJX-613-TEX-N-2212"></use></g></g><g data-mml-node="mo" transform="translate(1383.1,0)"><use data-c="28" xlink:href="#MJX-613-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1772.1,0)"><use data-c="1D714" xlink:href="#MJX-613-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(2394.1,0)"><use data-c="29" xlink:href="#MJX-613-TEX-N-29"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">A^-(\omega)</script><span> is the projected area of backward-facing microfacets, then </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="22.841ex" height="2.32ex" role="img" focusable="false" viewBox="0 -775.2 10095.9 1025.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-614-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-614-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-614-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-614-TEX-N-2061" d=""></path><path id="MJX-614-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><path id="MJX-614-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-614-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><path id="MJX-614-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-614-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 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xlink:href="#MJX-614-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(5079.4,0)"><use data-c="1D714" xlink:href="#MJX-614-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(5701.4,0)"><use data-c="29" xlink:href="#MJX-614-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6312.6,0)"><use data-c="2212" xlink:href="#MJX-614-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(7312.8,0)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-614-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(783,363) scale(0.707)"><use data-c="2212" xlink:href="#MJX-614-TEX-N-2212"></use></g></g><g data-mml-node="mo" transform="translate(8695.9,0)"><use data-c="28" xlink:href="#MJX-614-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(9084.9,0)"><use data-c="1D714" xlink:href="#MJX-614-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(9706.9,0)"><use data-c="29" xlink:href="#MJX-614-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>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo>=</mo><msup><mi>A</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>−</mo><msup><mi>A</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\cos\theta = A^+(\omega) - A^-(\omega)</script><span>.</span></p><p><span>We can thus alternatively write the masking-shadowing function as the </span><strong><span>ratio</span></strong><span> of visible microfacets area to total forward-facing microfacet area:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n454" cid="n454" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div 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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><path id="MJX-477-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-477-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-477-TEX-N-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></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="1D43A" 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data-c="29" xlink:href="#MJX-477-TEX-N-29"></use></g></g><g data-mml-node="mrow" transform="translate(2222.8,-710)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-477-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(783,289) scale(0.707)"><use data-c="2B" xlink:href="#MJX-477-TEX-N-2B"></use></g></g><g data-mml-node="mo" transform="translate(1383.1,0)"><use data-c="28" xlink:href="#MJX-477-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1772.1,0)"><use data-c="1D714" xlink:href="#MJX-477-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(2394.1,0)"><use data-c="29" xlink:href="#MJX-477-TEX-N-29"></use></g></g><rect width="6988.7" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(11184.8,0)"><use data-c="2E" xlink:href="#MJX-477-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>A</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>−</mo><msup><mi>A</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>A</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>Shadowing-masking functions are traditionally expressed in terms of an auxiliary function </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 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xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Lambda(\omega)</script><span>, which measures invisible masked microfacet area </span><strong><span>per visible microfacet area</span></strong><span>.</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n456" cid="n456" 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="34.418ex" height="5.532ex" role="img" focusable="false" viewBox="0 -1485.2 15212.9 2445.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.172ex;"><defs><path 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xlink:href="#MJX-478-TEX-N-29"></use></g></g><rect width="6988.7" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(10934,0)"><use data-c="3D" xlink:href="#MJX-478-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(11989.8,0)"><g data-mml-node="mrow" transform="translate(220,710)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-478-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(783,363) scale(0.707)"><use data-c="2212" xlink:href="#MJX-478-TEX-N-2212"></use></g></g><g data-mml-node="mo" transform="translate(1383.1,0)"><use data-c="28" xlink:href="#MJX-478-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1772.1,0)"><use data-c="1D714" xlink:href="#MJX-478-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(2394.1,0)"><use data-c="29" xlink:href="#MJX-478-TEX-N-29"></use></g></g><g data-mml-node="mrow" transform="translate(624.7,-686)"><g 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stretchy="false">)</mo><mo>−</mo><msup><mi>A</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>A</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><mrow><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>After some algebra we have</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n458" cid="n458" 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="19.209ex" height="5.208ex" role="img" focusable="false" viewBox="0 -1342 8490.6 2302" 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transform="translate(8212.6,0)"><use data-c="2E" xlink:href="#MJX-479-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><strong><span>Specifying </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 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stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Lambda(\omega)</script></strong><span>: The microfacet distribution alone doesn't impose enough conditions to imply a specific </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z"></path><path id="MJX-625-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 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unique </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z"></path><path id="MJX-625-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-625-TEX-I-1D714" d="M495 384Q495 406 514 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data-c="28" xlink:href="#MJX-625-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1083,0)"><use data-c="1D714" xlink:href="#MJX-625-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(1705,0)"><use data-c="29" xlink:href="#MJX-625-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 mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Lambda(\omega)</script><span> given </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.15ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2718.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-619-TEX-I-1D437" d="M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 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stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">D(\omega_h)</script><span>.</span></p><ul><li><p><span>For many microfacet models, a closed-form expression can be found.</span></p></li><li><p><span>Although this isn't true in reality, the resulting </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.738ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2094 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-625-TEX-N-39B" d="M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z"></path><path id="MJX-625-TEX-N-28" d="M94 250Q94 319 104 381T127 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data-c="2061" xlink:href="#MJX-480-TEX-N-2061"></use></g><g data-mml-node="mrow" transform="translate(2000,0)"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-480-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D44E" xlink:href="#MJX-480-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(918,0)"><use data-c="29" xlink:href="#MJX-480-TEX-N-29"></use></g></g><g data-mml-node="mo" transform="translate(3529.2,0)"><use data-c="2212" xlink:href="#MJX-480-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(4529.4,0)"><use data-c="31" xlink:href="#MJX-480-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5251.7,0)"><use data-c="2B" xlink:href="#MJX-480-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(6251.9,0)"><g data-mml-node="msup" transform="translate(305.1,676)"><g data-mml-node="mi"><use data-c="1D452" xlink:href="#MJX-480-TEX-I-1D452"></use></g><g data-mml-node="TeXAtom" transform="translate(499,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-480-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(778,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-480-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,363) scale(0.707)"><use data-c="32" xlink:href="#MJX-480-TEX-N-32"></use></g></g></g></g><g data-mml-node="mrow" transform="translate(220,-797.5)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-480-TEX-I-1D44E"></use></g><g data-mml-node="msqrt" transform="translate(529,0)"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D70B" xlink:href="#MJX-480-TEX-I-1D70B"></use></g></g><g data-mml-node="mo" transform="translate(0,-22.5)"><use data-c="221A" xlink:href="#MJX-480-TEX-N-221A"></use></g><rect width="570" height="60" x="853" y="717.5"></rect></g></g><rect width="2152" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8643.9,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-480-TEX-S4-29"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>erf</mi><mo data-mjx-texclass="NONE"></mo><mrow><mo data-mjx-texclass="OPEN">(</mo><mi>a</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></msup><mrow><mi>a</mi><msqrt><mi>π</mi></msqrt></mrow></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow></math></mjx-assistive-mml></mjx-container></div></div><p><span>where </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg 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xlink:href="#MJX-621-TEX-N-74"></use><use data-c="61" xlink:href="#MJX-621-TEX-N-61" transform="translate(389,0)"></use><use data-c="6E" xlink:href="#MJX-621-TEX-N-6E" transform="translate(889,0)"></use></g><g data-mml-node="mo" transform="translate(5614.2,0)"><use data-c="2061" xlink:href="#MJX-621-TEX-N-2061"></use></g><g data-mml-node="mi" transform="translate(5780.9,0)"><use data-c="1D703" xlink:href="#MJX-621-TEX-I-1D703"></use></g><g data-mml-node="mo" transform="translate(6249.9,0)"><use data-c="29" xlink:href="#MJX-621-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>α</mi><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>α</mi><mi>tan</mi><mo data-mjx-texclass="NONE"></mo><mi>θ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\alpha = 1/(\alpha \tan\theta)</script><span> and 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display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>erf</mi><mo data-mjx-texclass="NONE"></mo><mrow><mo data-mjx-texclass="OPEN">(</mo><mi>x</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mn>2</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msqrt><mi>π</mi></msqrt><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msubsup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><msup><mi>u</mi><mn>2</mn></msup></mrow></msup><mrow data-mjx-texclass="OP"><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">d</mi></mrow><mi>u</mi></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\erf(x) = 2/\sqrt{\pi}\int_{0}^{x}e^{-u^2}\dd{u}</script><span>.</span></p><p><span>In </span><code>pbrt</code><span>, a rational polynomial approximation is used, to avoid calling </span><code>std::erf()</code><span> and </span><code>std::exp()</code><span> which are fairly expensive to evaluate.</span></p></li></ul></li></ul><p> </p><p><strong><span>Computing the interpolated </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 640 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-628-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g 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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="39B" xlink:href="#MJX-625-TEX-N-39B"></use></g><g data-mml-node="mo" transform="translate(694,0)"><use data-c="28" xlink:href="#MJX-625-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1083,0)"><use data-c="1D714" xlink:href="#MJX-625-TEX-I-1D714"></use></g><g data-mml-node="mo" transform="translate(1705,0)"><use data-c="29" xlink:href="#MJX-625-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 mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">\Lambda(\omega)</script><span> by taking their corresponding isotropic function and stretching the underlying surface according to the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.551ex" height="1.357ex" role="img" focusable="false" viewBox="0 -442 1127.5 599.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-626-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 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x="1020" y="1192.8"></rect></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>α</mi><mo>=</mo><msqrt><msup><mi>cos</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>ϕ</mi><mi>h</mi></msub><mo>∗</mo><msubsup><mi>α</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo data-mjx-texclass="NONE"></mo><msub><mi>ϕ</mi><mi>h</mi></msub><mo>∗</mo><msubsup><mi>α</mi><mi>y</mi><mn>2</mn></msubsup></msqrt></math></mjx-assistive-mml></mjx-container></div></div><p><img src="../images/Lecture17-img-38.png" alt="img-38" style="zoom:50%;" /></p><ul><li><p><span>The function is close to one over much of the domain, but </span><em><span>falls to zero at </span><strong><span>grazing angles</span></strong></em><span>.</span></p></li><li><p><span>Increasing surface roughness causes the function to fall off more quickly.</span></p></li></ul><p> </p><p><strong><span>Computing 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380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path id="MJX-629-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="1D43A" xlink:href="#MJX-629-TEX-I-1D43A"></use></g><g data-mml-node="mo" transform="translate(786,0)"><use data-c="28" xlink:href="#MJX-629-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(1175,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-629-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-629-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(2222.9,0)"><use data-c="2C" xlink:href="#MJX-629-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2667.6,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-629-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-629-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3616.6,0)"><use data-c="29" xlink:href="#MJX-629-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>G</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">G(\omega_o, \omega_i)</script></strong><span>: This function gives the fraction of microfacets in a differential area that are visible from both directions </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1047.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-634-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><path id="MJX-634-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D714" xlink:href="#MJX-634-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-634-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_o</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 949 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-635-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><path id="MJX-635-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="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-635-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-635-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_i</script><span>. Defining </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-633-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></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="1D43A" xlink:href="#MJX-633-TEX-I-1D43A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">G</script><span> requires some additional assumptions. For starters, </span></p><ul><li><p><span>If we </span><strong><span>assume</span></strong><span> that the probability of a microfacet being visible from both directions is the probability that it is visible from each direction independently, then we have</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n487" cid="n487" 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="26.279ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 11615.1 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-482-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></path><path id="MJX-482-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-482-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><path id="MJX-482-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path id="MJX-482-TEX-N-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path id="MJX-482-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><path id="MJX-482-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><path 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data-c="1D714" xlink:href="#MJX-482-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-482-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(2222.9,0)"><use data-c="2C" xlink:href="#MJX-482-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(2667.6,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-482-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-482-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(3616.6,0)"><use data-c="29" xlink:href="#MJX-482-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4283.3,0)"><use data-c="3D" xlink:href="#MJX-482-TEX-N-3D"></use></g><g data-mml-node="msub" transform="translate(5339.1,0)"><g data-mml-node="mi"><use data-c="1D43A" xlink:href="#MJX-482-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-482-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(6561.7,0)"><use data-c="28" xlink:href="#MJX-482-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(6950.7,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-482-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-482-TEX-I-1D45C"></use></g></g><g data-mml-node="mo" transform="translate(7998.6,0)"><use data-c="29" xlink:href="#MJX-482-TEX-N-29"></use></g><g data-mml-node="msub" transform="translate(8387.6,0)"><g data-mml-node="mi"><use data-c="1D43A" xlink:href="#MJX-482-TEX-I-1D43A"></use></g><g data-mml-node="mn" transform="translate(819,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-482-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(9610.2,0)"><use data-c="28" xlink:href="#MJX-482-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(9999.2,0)"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-482-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-482-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(10948.1,0)"><use data-c="29" xlink:href="#MJX-482-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(11337.1,0)"><use data-c="2E" xlink:href="#MJX-482-TEX-N-2E"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>G</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo><msub><mi>G</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>.</mo></math></mjx-assistive-mml></mjx-container></div></div><p><span>In practice, </span></p><ul><li><p><span>This often </span><strong><span>underestimates</span></strong><span> </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.05ex;"><defs><path id="MJX-633-TEX-I-1D43A" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z"></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="1D43A" xlink:href="#MJX-633-TEX-I-1D43A"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">G</script><span>.</span></p></li><li><p><span>The closer together the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.371ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1047.9 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-634-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><path id="MJX-634-TEX-I-1D45C" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></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="1D714" xlink:href="#MJX-634-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-634-TEX-I-1D45C"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>o</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_o</script><span> and </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.147ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 949 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-635-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><path id="MJX-635-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="msub"><g data-mml-node="mi"><use data-c="1D714" xlink:href="#MJX-635-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-635-TEX-I-1D456"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_i</script><span> are, the more correlation there is between </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.897ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3048.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-636-TEX-I-1D43A" 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stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><script type="math/tex">G_1(\omega_i)</script><span>.</span></p></li></ul></li><li><p><span>A </span><strong><span>more accurate</span></strong><span> model can be derived, </span><strong><span>assuming</span></strong><span> that microfacet visibility is more likely the higher up a given point on a microfacet is. 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xlink:href="#MJX-484-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-484-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-484-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(2004.7,0)"><use data-c="2061" xlink:href="#MJX-484-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(2171.3,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-484-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D45C" xlink:href="#MJX-484-TEX-I-1D45C"></use></g></g><g data-mml-node="mi" transform="translate(3232.9,0)"><use data-c="63" xlink:href="#MJX-484-TEX-N-63"></use><use data-c="6F" xlink:href="#MJX-484-TEX-N-6F" transform="translate(444,0)"></use><use data-c="73" xlink:href="#MJX-484-TEX-N-73" transform="translate(944,0)"></use></g><g data-mml-node="mo" transform="translate(4570.9,0)"><use data-c="2061" xlink:href="#MJX-484-TEX-N-2061"></use></g><g data-mml-node="msub" transform="translate(4737.6,0)"><g data-mml-node="mi"><use data-c="1D703" xlink:href="#MJX-484-TEX-I-1D703"></use></g><g data-mml-node="mi" transform="translate(502,-150) scale(0.707)"><use data-c="1D456" xlink:href="#MJX-484-TEX-I-1D456"></use></g></g></g><rect width="9794.7" 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"><msub><mi>f</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>h</mi></msub><mo stretchy="false">)</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>F</mi><mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi><mi>o</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>o</mi></msub><mi>cos</mi><mo data-mjx-texclass="NONE"></mo><msub><mi>θ</mi><mi>i</mi></msub></mrow></mfrac></math></mjx-assistive-mml></mjx-container></div></div><p><span>For very detailed derivation, refer to the PBR book. Basically, we assume that individual microfacets are perfectly specular, and therefore only those that have the direction of their normals matched with the orientation of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.517ex" height="1.359ex" role="img" focusable="false" viewBox="0 -443 1112.3 600.8" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.357ex;"><defs><path id="MJX-638-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><path id="MJX-638-TEX-I-210E" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></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="1D714" xlink:href="#MJX-638-TEX-I-1D714"></use></g><g data-mml-node="mi" transform="translate(655,-150) scale(0.707)"><use data-c="210E" xlink:href="#MJX-638-TEX-I-210E"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ω</mi><mi>h</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">\omega_h</script><span> will reflect light.</span></p><p> </p></div></div>
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