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Lecture21.html
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</style><title>Lecture21</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-21---animation">GAMES101 Lecture 21 - Animation</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="#outline">Outline</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="#animation">Animation</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-history">I. History</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#ii-keyframe-animation">II. Keyframe Animation</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="#keyframe-interpolation">Keyframe Interpolation</a></div><ul class="outline-children"></ul></li></ul></li><li class="outline-item-wrapper outline-h2"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#iii-physical-simulation">III. Physical Simulation</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"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#mass-spring-system-example-of-modeling-a-dynamic-system">Mass Spring System: Example of Modeling a Dynamic System</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="#spring-with-internal-damping">Spring with Internal Damping</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="#structures-from-springs">Structures from Springs</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="#finite-element-method">Finite Element Method</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="#particle-systems">Particle Systems</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="#particle-system-forces">Particle System Forces</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="#gravitational-attraction">Gravitational Attraction</a></div><ul class="outline-children"></ul></li></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="#simulated-flocking-as-an-ode">Simulated Flocking as an ODE</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="#forward-kinematics">Forward Kinematics</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="#inverse-kinematics">Inverse Kinematics</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="#numerical-solution-to-general-n-link-ik-problem">Numerical Solution to general N-Link IK Problem</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="#rigging">Rigging</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="#blend-shapes">Blend Shapes</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="#motion-capture">Motion Capture</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="#facial-motion-capture">Facial Motion Capture</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="#challenge-of-facial-animation">Challenge of Facial Animation</a></div><ul class="outline-children"></ul></li></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="#the-production-pipeline">The Production Pipeline</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-21---animation'><span>GAMES101 Lecture 21 - Animation</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_21.pdf'><span>GAMES101_Lecture_21.pdf</span></a></p><h2 id='outline'><span>Outline</span></h2><ul><li><p><span>History</span></p></li><li><p><span>Keyframe Animation</span></p></li><li><p><span>Physical Animation</span></p></li><li><p><span>Kinematics</span></p></li><li><p><span>Rigging</span></p></li></ul><p> </p><h3 id='animation'><span>Animation</span></h3><p><em><span>Bring things to life.</span></em></p><ul><li><p><span>Communication tool</span></p></li><li><p><span>Aesthetic issues often dominate technical issues</span></p></li></ul><p><em><span>An extension of modeling</span></em></p><ul><li><p><span>Represent models as a function of time</span></p></li></ul><p><em><span>Output: A sequence of images that when viewed sequentially provides a sense of motion.</span></em></p><ul><li><p><span>Film: 24 frames per second (usually with motion blur)</span></p></li><li><p><span>Video (in general): 30 fps (usually with motion blur)</span></p></li><li><p><span>Virtual reality: 90 fps</span></p></li></ul><p> </p><h2 id='i-history'><span>I. History</span></h2><ul><li><p><span>First Animation (Shahr-e Sukhteh, Iran 3200 BCE) </span></p></li><li><p><strong><span>Phenakistoscope</span></strong><span> (1831): By showing part of a rotating disk:</span></p><p><img src="../images/Lecture21-img-1.png" alt="image-20230725111902802" style="zoom:33%;" /></p></li><li><p><strong><span>First Film</span></strong><span> (Edward Muybrdige, </span><em><span>"Sallie Gardner"</span></em><span>, 1878): Used as scientific tool rater than for entertainment</span></p></li><li><p><strong><span>First Hand-Drawn Feature-Length</span></strong><span> (>40 mins) Animation (Disney, "Snow White and the Seven Dwarfs", 1937)</span></p></li><li><p><strong><span>First Digital-Computer-Generated Animation</span></strong><span> (Ivan Sutherland, "Sketchpad", 1963): Light pen, vector display</span></p><p><img src="../images/Lecture21-img-2.png" alt="image-20230725112427046" style="zoom:33%;" /></p></li><li><p><strong><span>Early Computer Animation</span></strong><span> (Ed Catmull & Frederick Parke, "Computer Animated Faces", 1972)</span></p></li><li><p><strong><span>Digital Dinasours</span></strong><span> ("Jurassic Park", 1993)</span></p></li><li><p><strong><span>First CG Feature-Length Film</span></strong><span> (Pixar, "Toy Story", 1995)</span></p><p><img src="../images/Lecture21-img-3.png" alt="image-20230725112848276" style="zoom:33%;" /></p><ul><li><p><em><span>Milestone</span></em></p></li><li><p><em><span>Rasterization only</span></em></p></li></ul></li><li><p><span>Computer Animation - </span><strong><span>10 Years Ago</span></strong><span> (Sony Pictures Animation, "Cloudy With a Chance of Meatballs")</span></p><p><img src="../images/Lecture21-img-4.png" alt="image-20230725112935212" style="zoom:33%;" /></p></li><li><p><span>Computer Animation - </span><strong><span>2019</span></strong><span> (Walt Disney Animation Studios, "Frozen 2", 2019)</span></p><p><img src="../images/Lecture21-img-5.png" alt="image-20230725113029868" style="zoom:67%;" /></p><ul><li><p><span>Plants, after effects, ...</span></p></li></ul></li></ul><p> </p><h2 id='ii-keyframe-animation'><span>II. Keyframe Animation</span></h2><p><img src="../images/Lecture21-img-6.png" alt="image-20230725113153997" style="zoom:50%;" /></p><ul><li><p><strong><span>Animator</span></strong><span> (e.g. lead animator) creates keyframes</span></p></li><li><p><strong><span>Assistant (person or computer)</span></strong><span> creates in-between frames</span></p><ul><li><p><em><strong><span>tweening</span></strong></em></p></li></ul></li></ul><p> </p><h3 id='keyframe-interpolation'><span>Keyframe Interpolation</span></h3><p><img src="../images/Lecture21-img-7.png" alt="image-20230725113453202" style="zoom: 50%;" /></p><ul><li><p><span>Each frame is seen as a </span><strong><span>vector of parameter values</span></strong></p><ul><li><p><span>Linear interpolation is usually not good enough</span></p></li><li><p><strong><span>Splines</span></strong><span> for smooth/controllable interpolation</span></p></li></ul></li></ul><p> </p><h2 id='iii-physical-simulation'><span>III. Physical Simulation</span></h2><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n95" cid="n95" 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="7.895ex" height="1.724ex" role="img" focusable="false" viewBox="0 -680 3489.6 762" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-2065-TEX-I-1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 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y="220"></rect></g><g data-mml-node="mo" transform="translate(6749,0)"><use data-c="28" xlink:href="#MJX-2066-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(7138,0)"><use data-c="394" xlink:href="#MJX-2066-TEX-N-394"></use></g><g data-mml-node="mi" transform="translate(7971,0)"><use data-c="1D461" xlink:href="#MJX-2066-TEX-I-1D461"></use></g><g data-mml-node="msup" transform="translate(8332,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-2066-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-2066-TEX-N-32"></use></g></g><g data-mml-node="mi" transform="translate(9157.6,0)"><use data-c="1D482" xlink:href="#MJX-2066-TEX-BI-1D482"></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">Δ</mi><mi mathvariant="bold-italic">x</mi><mo>=</mo><mi mathvariant="bold-italic">v</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mi mathvariant="bold-italic">a</mi></math></mjx-assistive-mml></mjx-container></div></div><ul><li><p><span>Building the correct physical model leads to correct physical simulation.</span></p></li></ul><p><img src="../images/Lecture21-img-8.png" alt="image-20230725114233632" style="zoom:33%;" /></p><p><img src="../images/Lecture21-img-9.png" alt="image-20230725114252324" style="zoom:50%;" /></p><p align="center">Macklin and Müller, Position Based Fluids</p><p> </p><h3 id='mass-spring-system-example-of-modeling-a-dynamic-system'><span>Mass Spring System: Example of Modeling a Dynamic System</span></h3><ul><li><p><span>Mass Spring Rope,</span></p></li><li><p><span>Hair,</span></p></li><li><p><strong><span>Mass Spring Mesh</span></strong><span> (Cloth)</span></p><p><img src="../images/Lecture21-img-10.png" alt="image-20230725114656663" style="zoom: 50%;" /></p></li></ul><h4 id='spring-with-internal-damping'><span>Spring with Internal Damping</span></h4><p><em><span>One type of spring-mass system with damping:</span></em></p><p><img src="../images/Lecture21-img-11.png" alt="image-20230725131148290" style="zoom: 50%;" /></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n117" cid="n117" 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="31.213ex" height="5.475ex" role="img" focusable="false" viewBox="0 -1460 13796 2420" 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-2068-TEX-N-74" transform="translate(250,0)"></use><use data-c="6F" xlink:href="#MJX-2068-TEX-N-6F" transform="translate(639,0)"></use><use data-c="A0" xlink:href="#MJX-2068-TEX-N-A0" transform="translate(1139,0)"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2022,0)"><g data-mml-node="mi"><use data-c="1D483" xlink:href="#MJX-2068-TEX-BI-1D483"></use></g></g></g></g></g></g></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 rowspacing=".5em" columnspacing="1em" displaystyle="true"><mtr><mtd><munder><mrow data-mjx-texclass="OP"><munder><msub><mi mathvariant="bold-italic">f</mi><mi mathvariant="bold-italic">b</mi></msub><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtable data-mjx-smallmatrix="true" columnspacing="0em" rowspacing="0.1em"><mtr><mtd><mtext>Damping force</mtext></mtd></mtr><mtr><mtd><mrow><mtext>applied on </mtext><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">b</mi></mrow></mrow></mtd></mtr></mtable></mrow></munder><mo>=</mo><mo>−</mo><msub><mi>k</mi><mi>d</mi></msub><munder><mrow data-mjx-texclass="OP"><munder><mrow><mfrac><mrow><mi mathvariant="bold-italic">b</mi><mo>−</mo><mi mathvariant="bold-italic">a</mi></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN" symmetric="true">‖</mo><mi mathvariant="bold-italic">b</mi><mo>−</mo><mi mathvariant="bold-italic">a</mi><mo data-mjx-texclass="CLOSE" symmetric="true">‖</mo></mrow></mfrac><mo>⋅</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo>˙</mo></mover></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">a</mi><mo>˙</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtable data-mjx-smallmatrix="true" columnspacing="0em" rowspacing="0.1em"><mtr><mtd><mtext>Relative speed projected</mtext></mtd></mtr><mtr><mtd><mrow><mtext>to the direction from </mtext><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow><mtext> to </mtext><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">b</mi></mrow></mrow></mtd></mtr></mtable></mrow></munder><mo>⋅</mo><munder><mrow data-mjx-texclass="OP"><munder><mfrac><mrow><mi mathvariant="bold-italic">b</mi><mo>−</mo><mi mathvariant="bold-italic">a</mi></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN" symmetric="true">‖</mo><mi mathvariant="bold-italic">b</mi><mo>−</mo><mi mathvariant="bold-italic">a</mi><mo data-mjx-texclass="CLOSE" symmetric="true">‖</mo></mrow></mfrac><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtable data-mjx-smallmatrix="true" columnspacing="0em" rowspacing="0.1em"><mtr><mtd><mtext>Direction from</mtext></mtd></mtr><mtr><mtd><mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">a</mi></mrow><mtext> to </mtext><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">b</mi></mrow></mrow></mtd></mtr></mtable></mrow></munder></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>Here we assume that </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.432ex" height="1.041ex" role="img" focusable="false" viewBox="0 -452 633 460" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.018ex;"><defs><path id="MJX-2083-TEX-BI-1D482" d="M222 -8Q140 -8 89 34T38 158Q38 191 48 227Q72 329 151 390T327 452Q361 452 385 443T421 425T433 416H434L441 421Q448 426 460 430T486 435Q509 435 523 422T538 386Q538 380 522 315T488 179T467 93Q466 87 466 72Q466 42 483 42Q505 42 521 75Q531 94 541 134Q546 155 550 158T571 162H576H587Q607 162 607 148Q606 142 604 132T590 94T566 47T528 9T474 -8Q396 -8 358 40Q295 -8 222 -8ZM404 351Q383 401 324 401Q300 401 270 385T221 330Q206 296 186 220Q166 136 166 106Q166 72 184 58T228 43Q256 43 284 57T328 84T343 103Q343 106 374 228L404 351Z"></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="1D482" xlink:href="#MJX-2083-TEX-BI-1D482"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\boldsymbol{a}</script><span> is a static vector (by subtracting the (projected) velocity of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.588ex" role="img" focusable="false" viewBox="0 -694 521 702" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.018ex;"><defs><path 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</span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.432ex" height="1.041ex" role="img" focusable="false" viewBox="0 -452 633 460" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.018ex;"><defs><path id="MJX-2083-TEX-BI-1D482" d="M222 -8Q140 -8 89 34T38 158Q38 191 48 227Q72 329 151 390T327 452Q361 452 385 443T421 425T433 416H434L441 421Q448 426 460 430T486 435Q509 435 523 422T538 386Q538 380 522 315T488 179T467 93Q466 87 466 72Q466 42 483 42Q505 42 521 75Q531 94 541 134Q546 155 550 158T571 162H576H587Q607 162 607 148Q606 142 604 132T590 94T566 47T528 9T474 -8Q396 -8 358 40Q295 -8 222 -8ZM404 351Q383 401 324 401Q300 401 270 385T221 330Q206 296 186 220Q166 136 166 106Q166 72 184 58T228 43Q256 43 284 57T328 84T343 103Q343 106 374 228L404 351Z"></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="1D482" xlink:href="#MJX-2083-TEX-BI-1D482"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\boldsymbol{a}</script><span>).</span></p><p><span>In this type of damping, the resistance is </span><strong><span>proportional to the relative velocity on projected direction</span></strong><span>.</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.117ex" height="1.926ex" role="img" focusable="false" viewBox="0 -694 935.6 851.1" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.355ex;"><defs><path id="MJX-2084-TEX-I-1D458" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path id="MJX-2084-TEX-I-1D460" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></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="1D458" xlink:href="#MJX-2084-TEX-I-1D458"></use></g><g data-mml-node="mi" transform="translate(554,-150) scale(0.707)"><use data-c="1D460" xlink:href="#MJX-2084-TEX-I-1D460"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>s</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">k_s</script><span> - </span><strong><span>stiffness</span></strong></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.674ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 298 705" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-2085-TEX-I-1D459" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></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="1D459" xlink:href="#MJX-2085-TEX-I-1D459"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">l</script><span> - </span><strong><span>rest length</span></strong></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="2.324ex" role="img" focusable="false" viewBox="0 -1019 521 1027" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.018ex;"><defs><path id="MJX-2086-TEX-BI-1D483" d="M220 -8Q142 -8 94 35T45 155V167Q45 187 52 218T104 426L153 622H149Q148 622 144 622T134 623T122 624T111 624T101 624T96 625Q84 628 84 642Q84 647 88 661T94 679Q98 684 109 685T185 690Q258 694 272 694Q289 694 293 679Q293 676 263 553L232 429L244 434Q256 440 281 446T331 452Q417 452 465 407T513 285Q513 235 494 184T439 90T346 20T220 -8ZM385 337Q385 400 318 400Q269 400 226 360Q214 349 211 341T191 268Q162 149 162 113Q162 44 226 44Q269 44 299 76T339 135T362 215Q364 222 365 226Q385 303 385 337Z"></path><path id="MJX-2086-TEX-N-2D9" d="M190 609Q190 637 208 653T252 669Q275 667 292 652T309 609Q309 579 292 564T250 549Q225 549 208 564T190 609Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D483" xlink:href="#MJX-2086-TEX-BI-1D483"></use></g><g data-mml-node="mo" transform="translate(260.5,250) translate(-250 0)"><use data-c="2D9" xlink:href="#MJX-2086-TEX-N-2D9"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">b</mi><mo>˙</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\dot{\boldsymbol{b}}</script><span>, </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.432ex" height="1.776ex" role="img" focusable="false" viewBox="0 -777 633 785" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.018ex;"><defs><path id="MJX-2087-TEX-BI-1D482" d="M222 -8Q140 -8 89 34T38 158Q38 191 48 227Q72 329 151 390T327 452Q361 452 385 443T421 425T433 416H434L441 421Q448 426 460 430T486 435Q509 435 523 422T538 386Q538 380 522 315T488 179T467 93Q466 87 466 72Q466 42 483 42Q505 42 521 75Q531 94 541 134Q546 155 550 158T571 162H576H587Q607 162 607 148Q606 142 604 132T590 94T566 47T528 9T474 -8Q396 -8 358 40Q295 -8 222 -8ZM404 351Q383 401 324 401Q300 401 270 385T221 330Q206 296 186 220Q166 136 166 106Q166 72 184 58T228 43Q256 43 284 57T328 84T343 103Q343 106 374 228L404 351Z"></path><path id="MJX-2087-TEX-N-2D9" d="M190 609Q190 637 208 653T252 669Q275 667 292 652T309 609Q309 579 292 564T250 549Q225 549 208 564T190 609Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D482" xlink:href="#MJX-2087-TEX-BI-1D482"></use></g><g data-mml-node="mo" transform="translate(316.5,8) translate(-250 0)"><use data-c="2D9" xlink:href="#MJX-2087-TEX-N-2D9"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">a</mi><mo>˙</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\dot{\boldsymbol{a}}</script><span> - </span><strong><span>velocity</span></strong></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.198ex" height="1.926ex" role="img" focusable="false" viewBox="0 -694 971.7 851.1" 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xlink:href="#MJX-2088-TEX-I-1D451"></use></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>k</mi><mi>d</mi></msub></math></mjx-assistive-mml></mjx-container><script type="math/tex">k_d</script><span> - </span><strong><span>damping coefficient</span></strong></p></li></ul><p> </p><h4 id='structures-from-springs'><span>Structures from Springs</span></h4><p><img src="../images/Lecture21-img-12.png" alt="image-20230725151913916" style="zoom:33%;" /></p><ul><li><p><strong><span>Behavior</span></strong><span> is determined by </span><strong><span>structure linkages</span></strong></p></li><li><p><img src="../images/Lecture21-img-13.png" alt="image-20230725152532246" style="zoom:33%;" /></p><p><span>Resistance to </span><strong><span>shearing</span></strong><span> and </span><strong><span>out-of-plane bending</span></strong><span>.</span></p><ul><li><p><em><span>Red springs should be much weaker.</span></em></p></li></ul></li></ul><p> </p><h4 id='finite-element-method'><span>Finite Element Method</span></h4><p><img src="../images/Lecture21-img-14.png" alt="image-20230725153210763" style="zoom: 50%;" /></p><p> </p><h3 id='particle-systems'><span>Particle Systems</span></h3><p><span>Model dynamical systems as collections of large numbers of particles</span></p><ul><li><p><strong><span>Dynamical</span></strong><span> - involving modeling particles described by differential equations.</span></p></li><li><p><span>Each particle's motion is defined by a set of physical (or non-physical) forces</span></p></li><li><p><strong><span>Easy</span></strong><span> to understand and implement</span></p></li><li><p><span>Scalable: fewer particles for speed, more for higher complexity</span></p></li></ul><p><strong><span>Challenges</span></strong><span>:</span></p><ul><li><p><span>May need </span><em><span>many</span></em><span> particles (e.g., fluids)</span></p></li><li><p><span>May need acceleration structures (e.g. to find nearest particles for interactions)</span></p></li></ul><p> </p><p><span>For each frame in animation</span></p><ul><li><p><span>[If needed] Create new particles</span></p></li><li><p><span>Calculate forces on each particles</span></p></li><li><p><span>Update each particle's position and velocity</span></p></li><li><p><span>[If needed] Remove dead particles</span></p></li><li><p><span>Render particles</span></p></li></ul><p> </p><p><strong><span>Examples</span></strong></p><ul><li><p><span>Fluid</span></p></li><li><p><span>Flocking</span></p></li><li><p><span>Molecular Dynamics</span></p></li><li><p><span>...</span></p></li></ul><p> </p><h4 id='particle-system-forces'><span>Particle System Forces</span></h4><p><strong><span>Attraction and repulsion forces</span></strong></p><ul><li><p><span>Gravity, eletromagenetism, ...</span></p></li><li><p><span>Springs, propulsion, ...</span></p></li></ul><p><strong><span>Damping forces</span></strong></p><ul><li><p><span>Friction, air drag, viscosity, ...</span></p></li></ul><p><strong><span>Collisions</span></strong></p><ul><li><p><span>Walls, containers, fixed objects, ...</span></p></li><li><p><span>Dynamic objects, character body parts, ...</span></p></li></ul><p> </p><h5 id='gravitational-attraction'><span>Gravitational Attraction</span></h5><p><img src="../images/Lecture21-img-15.png" alt="image-20230725154721645" style="zoom:33%;" /></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n208" cid="n208" 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" width="full" style="min-width: 96.95ex; position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="100%" height="4.421ex" role="img" focusable="false" 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transform="translate(1063.8,0)"><use data-c="3D" xlink:href="#MJX-2089-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2119.6,0)"><use data-c="36" xlink:href="#MJX-2089-TEX-N-36"></use><use data-c="2E" xlink:href="#MJX-2089-TEX-N-2E" transform="translate(500,0)"></use><use data-c="36" xlink:href="#MJX-2089-TEX-N-36" transform="translate(778,0)"></use><use data-c="37" xlink:href="#MJX-2089-TEX-N-37" transform="translate(1278,0)"></use><use data-c="34" xlink:href="#MJX-2089-TEX-N-34" transform="translate(1778,0)"></use><use data-c="32" xlink:href="#MJX-2089-TEX-N-32" transform="translate(2278,0)"></use><use data-c="38" xlink:href="#MJX-2089-TEX-N-38" transform="translate(2778,0)"></use></g><g data-mml-node="mo" transform="translate(5619.8,0)"><use data-c="D7" xlink:href="#MJX-2089-TEX-N-D7"></use></g><g data-mml-node="msup" transform="translate(6620,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-2089-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-2089-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,393.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-2089-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-2089-TEX-N-31"></use><use data-c="31" xlink:href="#MJX-2089-TEX-N-31" transform="translate(500,0)"></use></g></g></g><g data-mml-node="msup" transform="translate(8960.2,0)"><g data-mml-node="mtext"><use data-c="4E" xlink:href="#MJX-2089-TEX-N-4E"></use><use data-c="6D" xlink:href="#MJX-2089-TEX-N-6D" transform="translate(750,0)"></use></g><g data-mml-node="mn" transform="translate(1616,410.1) scale(0.707)"><use data-c="32" xlink:href="#MJX-2089-TEX-N-32"></use></g></g><g data-mml-node="msup" transform="translate(10979.8,0)"><g data-mml-node="mtext"><use data-c="6B" xlink:href="#MJX-2089-TEX-N-6B"></use><use data-c="67" xlink:href="#MJX-2089-TEX-N-67" transform="translate(528,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1061,421.1) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-2089-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="32" xlink:href="#MJX-2089-TEX-N-32"></use></g></g></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mo>=</mo><mn>6.67428</mn><mo>×</mo><msup><mn>10</mn><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>11</mn></mrow></msup><msup><mtext>Nm</mtext><mn>2</mn></msup><msup><mtext>kg</mtext><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>2</mn></mrow></msup></math></mjx-assistive-mml></mjx-container><script type="math/tex">G = 6.67428 \times 10^{-11} \text{Nm}^2\text{kg}^{-2}</script></p></li></ul><p> </p><h4 id='simulated-flocking-as-an-ode'><span>Simulated Flocking as an ODE</span></h4><p><img src="../images/Lecture21-img-16.png" alt="image-20230725154923470" style="zoom:50%;" /></p><p> </p><h3 id='forward-kinematics'><span>Forward Kinematics</span></h3><p><img src="../images/Lecture21-img-17.png" alt="image-20230725155345443" style="zoom:33%;" /></p><p><img src="../images/Lecture21-img-18.png" alt="image-20230725155826122" style="zoom:50%;" /></p><p><strong><span>Forward Kinematics</span></strong><span>: Obtain position and velocity of </span><strong><span>end effector </span></strong><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-2091-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></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="1D45D" xlink:href="#MJX-2091-TEX-I-1D45D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">p</script><span>, given the known joint angles and angular velocities.</span></p><p><strong><span>Articulated Skeleton</span></strong></p><ul><li><p><span>Topology (what's connected to what)</span></p></li><li><p><span>Geometric relations from joints</span></p></li><li><p><span>Tree structure (in absence of loops)</span></p></li></ul><p><strong><span>Joint Types</span></strong></p><ul><li><p><span>Pin (1D rotation)</span></p></li><li><p><span>Ball (2D rotation)</span></p></li><li><p><strong><span>Prismatic joint</span></strong><span> (translation)</span></p></li></ul><p><span>#### </span></p><p><strong><span>Pros</span></strong></p><ul><li><p><span>Direct control is convenient</span></p></li><li><p><span>Implementation is straightforward</span></p></li></ul><p><strong><span>Cons</span></strong></p><ul><li><p><span>Animation may be inconsistent with physics</span></p></li><li><p><span>Time consuming for artists</span></p></li></ul><p> </p><h3 id='inverse-kinematics'><span>Inverse Kinematics</span></h3><p><strong><span>Inverse Kinematics</span></strong><span>: Given </span><strong><span>the position of the end effector</span></strong><span> </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-2091-TEX-I-1D45D" d="M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path></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="1D45D" xlink:href="#MJX-2091-TEX-I-1D45D"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">p</script><span>, calculate the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. (Wikipedia)</span></p><ul><li><p><strong><span>Multiple solutions</span></strong><span> in configuration space</span></p><p><img src="../images/Lecture21-img-19.png" alt="image-20230725160200036" style="zoom:50%;" /></p></li><li><p><span>Solutions </span><strong><span>may not</span></strong><span> always exist</span></p><p><img src="../images/Lecture21-img-20.png" alt="image-20230725160222786" style="zoom: 50%;" /></p></li></ul><h4 id='numerical-solution-to-general-n-link-ik-problem'><span>Numerical Solution to general N-Link IK Problem</span></h4><ul><li><p><span>Choose an </span><strong><span>initial configuration</span></strong></p></li><li><p><span>Define an </span><strong><span>error metric</span></strong><span> (e.g. square of distance between goal and current position)</span></p></li><li><p><span>Compute gradient of error as function of configuration</span></p></li><li><p><span>Apply </span><strong><span>gradient descent</span></strong><span> or other optimization procedure</span></p></li></ul><p> </p><h3 id='rigging'><span>Rigging</span></h3><p><img src="../images/Lecture21-img-21.png" alt="image-20230725160720523" style="zoom:50%;" /></p><p><strong><span>Rigging</span></strong><span>: A set of higher level controls on a character that allow mroe rapid & intuitive modification of pose, deformations, expressions, etc.</span></p><ul><li><p><span>Captures all meaningful character changes</span></p></li><li><p><span>Varies from character to character</span></p></li></ul><p><span>Expensive to create</span></p><ul><li><p><span>Manual effort</span></p></li><li><p><span>Requires bth artistic and technical training</span></p></li></ul><p> </p><h3 id='blend-shapes'><span>Blend Shapes</span></h3><p><img src="../images/Lecture21-img-22.png" alt="image-20230725160845374" style="zoom:50%;" /></p><p align="center">Courtesy: Félix Ferrand</p><p><span>Instead of skeleton, </span><strong><span>interpolate directly between surfaces</span></strong></p><ul><li><p><span>Interpolate vertex positions</span></p></li><li><p><span>Splines used to control choice of weights over time</span></p></li></ul><p> </p><h3 id='motion-capture'><span>Motion Capture</span></h3><p><img src="../images/Lecture21-img-23.png" alt="image-20230725161134900" style="zoom:50%;" /></p><p align="center">Motion capture room for ShaqFu</p><p><span>Record real-world performances, extract pose as a functin of time from data collected and apply them to a model.</span></p><ul><li><p><strong><span>Optics</span></strong></p></li><li><p><strong><span>Magnetic</span></strong><span>: Infer position/orientation by sensing magnetic fields</span></p></li><li><p><strong><span>Mechanical</span></strong><span>: Measure motion directly</span></p></li></ul><p><strong><span>Strengths</span></strong></p><ul><li><p><span>Can capture large amounts of real data quickly</span></p></li><li><p><span>Realism can be high</span></p></li></ul><p><strong><span>Weaknesses</span></strong></p><ul><li><p><span>Complex and costly set-ups</span></p></li><li><p><span>Captrued animation may not meet artistic needs, requiring alterations</span></p></li></ul><p><img src="../images/Lecture21-img-24.png" alt="image-20230725161417391" style="zoom:50%;" /></p><p> </p><h4 id='facial-motion-capture'><span>Facial Motion Capture</span></h4><p><img src="../images/Lecture21-img-25.png" alt="image-20230725161613484" style="zoom:50%;" /></p><p align="center">Discovery, "Avatar: Motion Capture Mirrors Emotions", <a>https://youtu.be/1wK1Ixr-UmM</a></p><h5 id='challenge-of-facial-animation'><span>Challenge of Facial Animation</span></h5><ul><li><p><strong><span>Uncanny Valley</span></strong></p></li></ul><p> </p><h3 id='the-production-pipeline'><span>The Production Pipeline</span></h3><p><img src="../images/Lecture21-img-26.png" alt="image-20230725161747349" style="zoom:67%;" /></p><p> </p></div></div>
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