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Lecture04.html
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</style><title>Lecture04</title>
</head>
<body class='typora-export os-windows typora-export-show-outline typora-export-collapse-outline'><div class='typora-export-content'>
<div class="typora-export-sidebar"><div class="outline-content"><li class="outline-item-wrapper outline-h1"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#games101-lecture-04---transformation-cont">GAMES101 Lecture 04 - Transformation Cont.</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h2 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#i-rodrigues-rotation-formula">I. Rodrigues' Rotation Formula</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-mvp-transformation">II. MVP Transformation</a></div><ul class="outline-children"><li class="outline-item-wrapper outline-h3"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#mvp-transformation">MVP Transformation</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="#viewcamera-transformation">View/Camera Transformation</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h4"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#projection-transformation">Projection Transformation</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="#orthographic-projection">Orthographic projection:</a></div><ul class="outline-children"></ul></li><li class="outline-item-wrapper outline-h5 outline-item-single"><div class="outline-item"><span class="outline-expander"></span><a class="outline-label" href="#perspective-projection">Perspective projection:</a></div><ul class="outline-children"></ul></li></ul></li></ul></li></ul></li></ul></li></div></div><div id='write' class=''><h1 id='games101-lecture-04---transformation-cont'><span>GAMES101 Lecture 04 - Transformation Cont.</span></h1><p><a href='https://sites.cs.ucsb.edu/~lingqi/teaching/resources/GAMES101_Lecture_04.pdf'><span>GAMES101_Lecture_04.pdf</span></a></p><h2 id='i-rodrigues-rotation-formula'><span>I. 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data-mjx-texclass="OPEN">(</mo><mi>α</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><munder><mrow data-mjx-texclass="OP"><munder><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>n</mi><mi>z</mi></msub></mtd><mtd><msub><mi>n</mi><mi>y</mi></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>n</mi><mi>z</mi></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>n</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>n</mi><mi>y</mi></msub></mtd><mtd><msub><mi>n</mi><mi>x</mi></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow data-mjx-texclass="ORD"><mtext mathvariant="bold">N</mtext></mrow></munder></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h2 id='ii-mvp-transformation'><span>II. MVP Transformation</span></h2><h3 id='mvp-transformation'><span>MVP Transformation</span></h3><p><span>Model-Viewing-Projection Transformation.</span></p><ul><li><p><strong><span>View/Camera Transformation</span></strong><span>: Converts points in </span><em><span>canonical coordinates</span></em><span> (or </span><em><span>world coordinates</span></em><span>) to </span><em><span>camera coordinates</span></em><span> (or places them in </span><em><span>camera space</span></em><span>).</span></p><ul><li><p><em><span>Camera Space - Eye Space</span></em></p></li></ul></li><li><p><strong><span>Projection Transformation</span></strong><span>: Moves points from camera space to the </span><em><span>canonical view volume</span></em><span>.</span></p><ul><li><p><span>Sometimes called Viewing Transformation</span></p></li><li><p><em><span>Canonical View Volume - Clip Space or Normalized Device Coordinates</span></em></p></li></ul></li><li><p><strong><span>Viewport Transformation</span></strong><span>: Maps the canonical view volume to </span><em><span>screen space</span></em><span>.</span></p><ul><li><p><em><span>Screen Space - Pixel Coordinates</span></em></p></li></ul></li></ul><p> </p><h4 id='viewcamera-transformation'><span>View/Camera Transformation</span></h4><p><span>Let the following attributes be that of the camera:</span></p><ul><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.054ex" height="1.937ex" role="img" focusable="false" viewBox="0 -845 466 856" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-25-TEX-I-1D452" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path id="MJX-25-TEX-N-20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></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="1D452" xlink:href="#MJX-25-TEX-I-1D452"></use></g><g data-mml-node="mo" transform="translate(288.6,31) translate(-250 0)"><use data-c="20D7" xlink:href="#MJX-25-TEX-N-20D7"></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>e</mi><mo stretchy="false">→</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\vec{e}</script><span>: the position of the camera</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.079ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 477 1015" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-26-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-26-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></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="1D454" xlink:href="#MJX-26-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(266.3,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-26-TEX-N-5E"></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>g</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{g}</script><span>: the gaze direction</span></p></li><li><p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="2.274ex" role="img" focusable="false" viewBox="0 -994 361 1005" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-33-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-33-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></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="1D461" xlink:href="#MJX-33-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(263.8,200) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-33-TEX-N-5E"></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>t</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{t}</script><span>: up direction, assuming perpendicular to the gaze direction</span></p></li></ul><p><span>The target of the view transformation is to transform the coordinates such that:</span></p><ul><li><p><span>the camera is at the origin, and</span></p></li><li><p><span>its up direction is the new </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-32-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-32-TEX-I-1D466"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">y</script><span> axis and it looks at the direction of </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.812ex" height="1.505ex" role="img" focusable="false" viewBox="0 -583 1243 665" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.186ex;"><defs><path id="MJX-29-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-29-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-29-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D467" xlink:href="#MJX-29-TEX-I-1D467"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi>z</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">-z</script><span>.</span></p></li></ul><p><span>When deducting the matrix for view transformation, consider the inverse rotation which rotates the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-30-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-30-TEX-I-1D465"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">x</script><span> axis to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="4.662ex" height="2.713ex" role="img" focusable="false" viewBox="0 -994 2060.4 1199" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-31-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-31-TEX-N-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path><path id="MJX-31-TEX-N-D7" d="M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z"></path><path id="MJX-31-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D454" xlink:href="#MJX-31-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(266.3,16) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-31-TEX-N-5E"></use></g></g></g><g data-mml-node="mo" transform="translate(699.2,0)"><use data-c="D7" xlink:href="#MJX-31-TEX-N-D7"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1699.4,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-31-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(263.8,200) translate(-250 0)"><use data-c="5E" xlink:href="#MJX-31-TEX-N-5E"></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>g</mi><mo stretchy="false">^</mo></mover></mrow><mo>×</mo><mrow data-mjx-texclass="ORD"><mover><mi>t</mi><mo stretchy="false">^</mo></mover></mrow></math></mjx-assistive-mml></mjx-container><script type="math/tex">\hat{g} \times \hat{t}</script><span>, the </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-32-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-32-TEX-I-1D466"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">y</script><span> axis to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="2.274ex" role="img" focusable="false" viewBox="0 -994 361 1005" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.025ex;"><defs><path id="MJX-33-TEX-I-1D461" d="M26 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420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-34-TEX-I-1D467"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">z</script><span> axis to </span><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.839ex" height="2.296ex" role="img" focusable="false" viewBox="0 -810 1255 1015" xmlns:xlink="http://www.w3.org/1999/xlink" 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576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-42-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 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