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index.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>entry</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: "Lucida Console",Consolas,"Courier",monospace; --title-bar-height: 20px; }
.mac-os-11 { --title-bar-height: 28px; }
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<div id='write' class=''><h1><a name="appearance-mimicking-surfaces" class="md-header-anchor"></a><span>Appearance-Mimicking Surfaces</span></h1><p><span>Inspired by bas-reliefs, appearance-mimicking surfaces are thin surfaces, or 2.5D images whose normals approximate the normals of a 3D shape. Given a viewpoint and per-vertex depth bounds, the algorithm proposed </span><sup class='md-footnote'><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup><span> finds a globally optimal surface that preserves the appearance of the target shape when observed from the designated viewpoint, while satisfying the depth constraints. </span></p><h2><a name="problem-formulation" class="md-header-anchor"></a><span>Problem Formulation</span></h2><p><span>Let </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.549ex" height="2.066ex" viewBox="0 -783.2 1097.6 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2954-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E2954-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2954-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2954-MJMATHI-6F" x="925" y="513"></use></g></svg></span><script type="math/tex">S^{o}</script><span> be the original surface, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.498ex" height="2.066ex" viewBox="0 -783.2 645 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2960-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2960-MJMATHI-53" x="0" y="0"></use></g></svg></span><script type="math/tex">S</script><span> be the deformed surface when observed from viewpoint </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.335ex" height="1.461ex" viewBox="0 -522.8 575 629" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2966-MJMAINB-6F" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2966-MJMAINB-6F" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{o}</script><span>. Each point </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.168ex" height="2.55ex" viewBox="0 -835.3 933.5 1097.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2945-MJMAINB-70" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path><path stroke-width="0" id="E2945-MJMAIN-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2945-MJMAINB-70" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2945-MJMAIN-2032" x="903" y="513"></use></g></svg></span><script type="math/tex">\bold{p}'</script><span> of the deformed surface is constrained to stay on the ray emanating from the viewpoint in the direction of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.484ex" height="1.824ex" viewBox="0 -522.8 639 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2955-MJMAINB-70" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2955-MJMAINB-70" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{p}</script><span> (line </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.82ex" height="1.824ex" viewBox="0 -522.8 1214 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2947-MJMAINB-6F" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path stroke-width="0" id="E2947-MJMAINB-70" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2947-MJMAINB-6F" x="0" y="0"></use><use xlink:href="#E2947-MJMAINB-70" x="575" y="0"></use></g></svg></span><script type="math/tex">\bold{op}</script><span>).</span></p><p><img src="report/normals.png" alt="image-20201201115724580" style="zoom:20%;" /></p><p><span>The perceived difference </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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y="0"></use><use xlink:href="#E2927-MJMAINB-70" x="14599" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-405">d(S, S^{o}, \bold{o}) = \int_{S} \Vert \bold{n}^{S}_{\phi(\bold{p}, \bold{o})} - \bold{n}^{S^{o}}_\bold{p} \Vert^{2} d\bold{p}</script></div></div><p><span>Here, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.309ex" height="2.671ex" viewBox="0 -835.3 5299.7 1149.8" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E2951-MJMATHI-3D5" d="M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 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xlink:href="#E2951-MJMAIN-3D" x="3310" y="0"></use><g transform="translate(4366,0)"><use xlink:href="#E2951-MJMAINB-70" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2951-MJMAIN-2032" x="903" y="513"></use></g></g></svg></span><script type="math/tex">\phi(\bold{p}, \bold{o}) = \bold{p}'</script><span> on surface </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.498ex" height="2.066ex" viewBox="0 -783.2 645 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2960-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2960-MJMATHI-53" x="0" y="0"></use></g></svg></span><script type="math/tex">S</script><span>. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.519ex" height="3.154ex" viewBox="0 -991.6 1515.1 1358.2" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E2953-MJMAINB-6E" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 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x="0" y="0"></use><g transform="translate(639,359)"><use transform="scale(0.707)" xlink:href="#E2953-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E2953-MJMATHI-6F" x="925" y="610"></use></g><use transform="scale(0.707)" xlink:href="#E2953-MJMAINB-70" x="903" y="-218"></use></g></svg></span><script type="math/tex">\bold{n}^{S^{o}}_\bold{p}</script><span> is the normal of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.549ex" height="2.066ex" viewBox="0 -783.2 1097.6 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2954-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E2954-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2954-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2954-MJMATHI-6F" x="925" y="513"></use></g></svg></span><script type="math/tex">S^{o}</script><span> at point </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.484ex" height="1.824ex" viewBox="0 -522.8 639 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2955-MJMAINB-70" d="M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2955-MJMAINB-70" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{p}</script><span>. Our goal is to minimize </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.47ex" height="2.671ex" viewBox="0 -835.3 4507.9 1149.8" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E2956-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E2956-MJMAIN-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 stroke-width="0" id="E2956-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E2956-MJMAIN-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 stroke-width="0" id="E2956-MJMATHI-6F" 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 stroke-width="0" id="E2956-MJMAINB-6F" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path stroke-width="0" id="E2956-MJMAIN-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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2956-MJMATHI-64" x="0" y="0"></use><use xlink:href="#E2956-MJMAIN-28" x="523" y="0"></use><use xlink:href="#E2956-MJMATHI-53" x="912" y="0"></use><use xlink:href="#E2956-MJMAIN-2C" x="1557" y="0"></use><g transform="translate(2001,0)"><use xlink:href="#E2956-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2956-MJMATHI-6F" x="925" y="513"></use></g><use xlink:href="#E2956-MJMAIN-2C" x="3099" y="0"></use><use xlink:href="#E2956-MJMAINB-6F" x="3543" y="0"></use><use xlink:href="#E2956-MJMAIN-29" x="4118" y="0"></use></g></svg></span><script type="math/tex">d(S, S^{o}, \bold{o})</script><span>.</span></p><h2><a name="discretization" class="md-header-anchor"></a><span>Discretization</span></h2><h4><a name="preserving-surface-similarity" class="md-header-anchor"></a><span>Preserving Surface Similarity</span></h4><p><span>When surface </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.498ex" height="2.066ex" viewBox="0 -783.2 645 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2960-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2960-MJMATHI-53" x="0" y="0"></use></g></svg></span><script type="math/tex">S</script><span> is represented by a triangle mesh </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.441ex" height="1.945ex" viewBox="0 -783.2 1051 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2985-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2985-MJMATHI-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">M</script><span>, points </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.498ex" height="2.066ex" viewBox="0 -783.2 645 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2960-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2960-MJMATHI-53" x="0" y="0"></use></g></svg></span><script type="math/tex">S</script><span> are approximated by vertices </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span>. Each vertex </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span> can be written as:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n136" cid="n136" mdtype="math_block">
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transform="scale(0.707)" xlink:href="#E2928-MJMATHI-69" x="858" y="-213"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-406">\bold{v}_i = \bold{o} + \Vert \bold{v}_i - \bold{o} \Vert \frac{\bold{v}_i - \bold{o}}{\Vert \bold{v}_i - \bold{o} \Vert} = \bold{o} + \lambda_i \bold{\hat{v}}_i</script></div></div><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="2.429ex" viewBox="0 -783.2 951 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2970-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E2970-MJMAINB-5E" d="M207 632L287 694Q289 693 368 632T448 570T431 545T413 520Q410 520 350 559L287 597L224 559Q164 520 161 520Q160 520 143 544T126 570T207 632Z"></path><path stroke-width="0" id="E2970-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2970-MJMAINB-76" x="0" y="0"></use><use xlink:href="#E2970-MJMAINB-5E" x="16" y="2"></use><use transform="scale(0.707)" xlink:href="#E2970-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{\hat{v}}_i</script><span> is the unit vector pointing in the direction of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.501ex" height="1.703ex" viewBox="0 -522.8 1507.6 733.2" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E2964-MJMAINB-6F" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path><path stroke-width="0" id="E2964-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E2964-MJMAINB-69" d="M72 610Q72 649 98 672T159 695Q193 693 217 670T241 610Q241 572 217 549T157 525Q120 525 96 548T72 610ZM46 442L136 446L226 450H232V62H294V0H286Q271 3 171 3Q67 3 49 0H40V62H109V209Q109 358 108 362Q103 380 55 380H43V442H46Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2964-MJMAINB-6F" x="0" y="0"></use><g transform="translate(575,0)"><use xlink:href="#E2964-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2964-MJMAINB-69" x="858" y="-213"></use></g></g></svg></span><script type="math/tex">\bold{ov_i}</script><span>. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> measures the distance between </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.335ex" height="1.461ex" viewBox="0 -522.8 575 629" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2966-MJMAINB-6F" d="M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2966-MJMAINB-6F" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{o}</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span>. This representation is convenient because </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.441ex" height="1.945ex" viewBox="0 -783.2 1051 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2985-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2985-MJMATHI-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">M</script><span> (deformed mesh) and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.526ex" height="1.945ex" viewBox="0 -783.2 1518.3 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2987-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path stroke-width="0" id="E2987-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2987-MJMATHI-4D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2987-MJMATHI-6F" x="1520" y="513"></use></g></svg></span><script type="math/tex">M^o</script><span> (original mesh) share the same set of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="2.429ex" viewBox="0 -783.2 951 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2970-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E2970-MJMAINB-5E" d="M207 632L287 694Q289 693 368 632T448 570T431 545T413 520Q410 520 350 559L287 597L224 559Q164 520 161 520Q160 520 143 544T126 570T207 632Z"></path><path stroke-width="0" id="E2970-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2970-MJMAINB-76" x="0" y="0"></use><use xlink:href="#E2970-MJMAINB-5E" x="16" y="2"></use><use transform="scale(0.707)" xlink:href="#E2970-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{\hat{v}}_i</script><span>. Their differences are entirely expressed by </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.383ex" height="2.671ex" viewBox="0 -783.2 1025.9 1149.8" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E2972-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E2972-MJMATHI-6F" 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 stroke-width="0" id="E2972-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2972-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2972-MJMATHI-6F" x="824" y="498"></use><use transform="scale(0.707)" xlink:href="#E2972-MJMATHI-69" x="824" y="-429"></use></g></svg></span><script type="math/tex">\lambda_i^o</script><span>. Depth constraints for each vertex of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.441ex" height="1.945ex" viewBox="0 -783.2 1051 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2985-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2985-MJMATHI-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">M</script><span> can be specified as a upper bound and a lower bound on </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n138" cid="n138" mdtype="math_block">
<div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-407-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="116.371ex" height="3.033ex" viewBox="0 -887.4 50104 1306.1" role="img" focusable="false" style="vertical-align: -0.732ex; margin-bottom: -0.24ex; max-width: 100%;"><defs><path stroke-width="0" id="E2929-MJMAIN-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 stroke-width="0" id="E2929-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E2929-MJMAIN-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 stroke-width="0" id="E2929-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 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280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E2929-MJMATHI-69" 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 stroke-width="0" id="E2929-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E2929-MJMAIN-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 stroke-width="0" id="E2929-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 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transform="translate(48825,0)"><g id="mjx-eqn-3" transform="translate(0,-5)"><use xlink:href="#E2929-MJMAIN-28"></use><use xlink:href="#E2929-MJMAIN-33" x="389" y="0"></use><use xlink:href="#E2929-MJMAIN-29" x="889" y="0"></use></g></g><g transform="translate(21401,0)"><g transform="translate(-15,0)"><g transform="translate(0,-5)"><use xlink:href="#E2929-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-69" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-6E" x="1223" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-69" x="824" y="-429"></use><use xlink:href="#E2929-MJMAIN-2264" x="2249" y="0"></use><g transform="translate(3305,0)"><use xlink:href="#E2929-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-69" x="824" y="-213"></use></g><use xlink:href="#E2929-MJMAIN-2264" x="4510" y="0"></use><g transform="translate(5566,0)"><use xlink:href="#E2929-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-78" x="1406" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E2929-MJMATHI-69" x="824" y="-429"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-407">\lambda^{min}_i \leq \lambda_i \leq \lambda^{max}_i</script></div></div><p><span>Using this representation, Eq. (1) can be discretized and linearized as:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n140" cid="n140" mdtype="math_block">
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y="-429"></use></g><g transform="translate(4113,0)"><use xlink:href="#E2930-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-6F" x="1060" y="498"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-69" x="1060" y="-429"></use></g><use xlink:href="#E2930-MJMAIN-2225" x="5306" y="0"></use><use xlink:href="#E2930-MJMAIN-28" x="5806" y="0"></use><g transform="translate(6195,0)"><use xlink:href="#E2930-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-6F" x="978" y="584"></use></g><g transform="translate(7330,0)"><use xlink:href="#E2930-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-3BB" x="1247" y="-213"></use></g><g transform="translate(8724,0)"><use xlink:href="#E2930-MJMAINB-56" x="0" y="0"></use><use xlink:href="#E2930-MJMAINB-5E" x="147" y="244"></use></g><g transform="translate(9593,0)"><use xlink:href="#E2930-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-69" x="550" y="-213"></use></g><use xlink:href="#E2930-MJMAIN-2212" x="10549" y="0"></use><use xlink:href="#E2930-MJMAIN-28" x="11549" y="0"></use><g transform="translate(11938,0)"><use xlink:href="#E2930-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-6F" x="978" y="584"></use></g><g transform="translate(13073,0)"><use xlink:href="#E2930-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-174)"><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E2930-MJMATHI-6F" x="824" y="595"></use></g></g><g transform="translate(14780,0)"><use xlink:href="#E2930-MJMAINB-56" x="0" y="0"></use><use xlink:href="#E2930-MJMAINB-5E" x="147" y="244"></use></g><g transform="translate(15649,0)"><use xlink:href="#E2930-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-69" x="550" y="-213"></use></g><g transform="translate(16382,0)"><g transform="translate(120,0)"><rect stroke="none" width="1145" height="60" x="0" y="220"></rect><g transform="translate(109,676)"><use xlink:href="#E2930-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-69" x="824" y="-213"></use></g><g transform="translate(60,-686)"><use xlink:href="#E2930-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-6F" x="824" y="498"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMATHI-69" x="824" y="-429"></use></g></g></g><g transform="translate(17768,0)"><use xlink:href="#E2930-MJMAIN-2225" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2930-MJMAIN-32" x="707" y="583"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-408">\begin{align*}
d(M, M^o, \bold{o}) = \sum_{i \in \bold{V}} w_i^2 A_i \Vert\bold{n}_i - \bold{n}_i^o \Vert^2 \\
&= \sum_{i \in \bold{V}} w_i^2 A_i \Vert\frac{(\bold{L} \bold{V})_i}{H_i} - \frac{(\bold{L}^o \bold{V}^o)_i}{H_i^o} \Vert^2 \\
&= \sum_{i \in \bold{V}} w_i^2 A_i \Vert\frac{(\bold{L} \bold{D}_\lambda \bold{\hat{V}})_i}{H_i} - \frac{(\bold{L}^o \bold{D}_{\lambda^o} \bold{\hat{V}})_i}{H_i^o} \Vert^2 \\
&= \sum_{i \in \bold{V}} w_i^2 A_i^o \Vert\frac{(\bold{L}^o \bold{D}_\lambda \bold{\hat{V}})_i}{H_i^o} - \frac{(\bold{L}^o \bold{D}_{\lambda^o} \bold{\hat{V}})_i}{H_i^o} \frac{\lambda_i}{\lambda_i^o}\Vert^2 \\
&= \sum_{i \in \bold{V}} w_i^2 A_i^o \Vert(\bold{L}^o \bold{D}_\lambda \bold{\hat{V}})_i - (\bold{L}^o \bold{D}_{\lambda^o} \bold{\hat{V}})_i \frac{\lambda_i}{\lambda_i^o}\Vert^2 \\
\end{align*}</script></div></div><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.541ex" height="2.429ex" viewBox="0 -783.2 1094 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2975-MJMATHI-41" 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 stroke-width="0" id="E2975-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2975-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2975-MJMATHI-69" x="1060" y="-213"></use></g></svg></span><script type="math/tex">A_i</script><span> is the Voronoi area ssociated with </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span> and can be obtained from the mass matrix coefficients. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.462ex" height="1.824ex" viewBox="0 -522.8 1060 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2979-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path stroke-width="0" id="E2979-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2979-MJMATHI-77" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2979-MJMATHI-69" x="1012" y="-213"></use></g></svg></span><script type="math/tex">w_i</script><span> are weights denoting the relative importance of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span>. Visible vertices from the viewpoint are given more weight than occluded vertices. By default </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.462ex" height="1.824ex" viewBox="0 -522.8 1060 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2979-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path stroke-width="0" id="E2979-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2979-MJMATHI-77" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2979-MJMATHI-69" x="1012" y="-213"></use></g></svg></span><script type="math/tex">w_i</script><span> are 1. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.607ex" height="1.945ex" viewBox="0 -783.2 692 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2980-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2980-MJMAINB-4C" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{L}</script><span> is the discrete laplace operator of mesh </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.441ex" height="1.945ex" viewBox="0 -783.2 1051 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2985-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2985-MJMATHI-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">M</script><span>. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.238ex" height="2.429ex" viewBox="0 -783.2 1394.2 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2994-MJMAINB-44" 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><path stroke-width="0" id="E2994-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2994-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2994-MJMATHI-3BB" x="1247" y="-213"></use></g></svg></span><script type="math/tex">\bold{D}_\lambda</script><span> is a diagonal matrix with entries </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> on the diagonal. </span></p><blockquote><p><span>We follow these conventions for notations:</span></p><ul><li><span>If </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.018ex" height="1.945ex" viewBox="0 -783.2 869 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2984-MJMAINB-58" d="M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2984-MJMAINB-58" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{X}</script><span> is a property/operator for mesh </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.441ex" height="1.945ex" viewBox="0 -783.2 1051 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2985-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2985-MJMATHI-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">M</script><span>, then </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.047ex" height="1.945ex" viewBox="0 -783.2 1311.9 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2986-MJMAINB-58" d="M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z"></path><path stroke-width="0" id="E2986-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2986-MJMAINB-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2986-MJMATHI-6F" x="1228" y="584"></use></g></svg></span><script type="math/tex">\bold{X}^o</script><span> is the corresponding property/operator for mesh </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.526ex" height="1.945ex" viewBox="0 -783.2 1518.3 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E2987-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path stroke-width="0" id="E2987-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2987-MJMATHI-4D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2987-MJMATHI-6F" x="1520" y="513"></use></g></svg></span><script type="math/tex">M^o</script><span>.</span></li><li><span>If </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.41ex" height="1.34ex" viewBox="0 -522.8 607 576.9" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3035-MJMAINB-78" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3035-MJMAINB-78" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{x}</script><span> is a vector, then </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.278ex" height="2.308ex" viewBox="0 -783.2 1411.2 993.6" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E2989-MJMAINB-44" 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><path stroke-width="0" id="E2989-MJMAINB-78" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2989-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2989-MJMAINB-78" x="1247" y="-213"></use></g></svg></span><script type="math/tex">\bold{D_x}</script><span> is a diagonal matrix with </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.41ex" height="1.34ex" viewBox="0 -522.8 607 576.9" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3035-MJMAINB-78" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3035-MJMAINB-78" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{x}</script><span> on the diagonal. </span></li></ul></blockquote><p><span>Now our goal is to find </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="2.066ex" viewBox="0 -783.2 583 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3062-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3062-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\lambda</script><span> such that it minimizes </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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To do this, we extract the unknown variable </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script><span> from </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.238ex" height="2.429ex" viewBox="0 -783.2 1394.2 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E2994-MJMAINB-44" 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><path stroke-width="0" id="E2994-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2994-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2994-MJMATHI-3BB" x="1247" y="-213"></use></g></svg></span><script type="math/tex">\bold{D}_\lambda</script><span>. The above equation can be further vectorized as:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n150" cid="n150" mdtype="math_block">
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y="595"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-410">\bold{L}_\theta = \bold{D_{(S \lambda_o)}}^{-1} \bold{\tilde{L}}^o \bold{D_\hat{V}} \bold{S} \bold{\lambda}^o</script></div></div><p><span>We then construct all the components of matrix </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="30.769ex" height="3.517ex" viewBox="0 -1043.7 13247.8 1514.4" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E2995-MJMAINB-51" d="M64 339Q64 431 96 502T182 614T295 675T420 696Q469 696 481 695Q620 680 709 589T798 339Q798 255 768 184Q720 77 611 26L600 21Q635 -26 682 -26H696Q769 -26 769 0Q769 7 774 12T787 18Q805 18 805 -7V-13Q803 -64 785 -106T737 -171Q720 -183 697 -191Q687 -193 668 -193Q636 -193 613 -182T575 -144T552 -94T532 -27Q531 -23 530 -16T528 -6T526 -3L512 -5Q499 -7 477 -8T431 -10Q393 -10 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228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2995-MJMAINB-51" x="0" y="0"></use><use xlink:href="#E2995-MJMAIN-3D" x="1141" y="0"></use><g transform="translate(2197,0)"><use xlink:href="#E2995-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-322)"><use transform="scale(0.707)" xlink:href="#E2995-MJMAIN-221A" x="0" y="82"></use><rect stroke="none" width="843" height="42" x="589" y="581"></rect><g transform="translate(589,0)"><use transform="scale(0.707)" xlink:href="#E2995-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E2995-MJMATHI-6F" x="1060" y="626"></use></g></g></g><g transform="translate(4612,0)"><use xlink:href="#E2995-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2995-MJMATHI-77" x="1247" y="-213"></use></g><use xlink:href="#E2995-MJMAIN-28" x="6100" y="0"></use><g transform="translate(6489,0)"><use xlink:href="#E2995-MJMAINB-4C" x="0" y="0"></use><use xlink:href="#E2995-MJMAINB-7E" x="58" y="562"></use><use transform="scale(0.707)" xlink:href="#E2995-MJMATHI-6F" x="978" y="895"></use></g><g transform="translate(7624,0)"><use xlink:href="#E2995-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-342)"><use transform="scale(0.707)" xlink:href="#E2995-MJMAINB-56" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2995-MJMAINB-5E" x="147" y="276"></use></g></g><use xlink:href="#E2995-MJMAIN-2212" x="9443" y="0"></use><g transform="translate(10443,0)"><use xlink:href="#E2995-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-150)"><use transform="scale(0.707)" xlink:href="#E2995-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E2995-MJMATHI-3B8" x="978" y="-218"></use></g></g><use xlink:href="#E2995-MJMAIN-29" x="12219" y="0"></use><use xlink:href="#E2995-MJMAINB-53" x="12608" y="0"></use></g></svg></span><script type="math/tex">\bold{Q} = \bold{D}_\sqrt{A^o} \bold{D}_w (\bold{\tilde{L}}^o \bold{D_\hat{V}} - \bold{D}_{\bold{L}_\theta}) \bold{S}</script><span>.</span></p><p><span>Let </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.829ex" height="2.671ex" viewBox="0 -835.3 3801.5 1149.8" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E2996-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E2996-MJMAIN-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 stroke-width="0" id="E2996-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E2996-MJMAINB-56" d="M592 686H604Q615 685 631 685T666 684T700 684T724 683Q829 683 835 686H843V624H744L611 315Q584 254 546 165Q492 40 482 19T461 -6L460 -7H409Q398 -4 391 9Q385 20 257 315L124 624H25V686H36Q57 683 190 683Q340 683 364 686H377V624H289L384 403L480 185L492 212Q504 240 529 298T575 405L670 624H582V686H592Z"></path><path stroke-width="0" id="E2996-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2996-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E2996-MJMAIN-3D" x="877" y="0"></use><use xlink:href="#E2996-MJMAIN-7C" x="1933" y="0"></use><g transform="translate(2211,0)"><use xlink:href="#E2996-MJMAINB-56" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2996-MJMATHI-6F" x="1228" y="584"></use></g><use xlink:href="#E2996-MJMAIN-7C" x="3523" y="0"></use></g></svg></span><script type="math/tex">n = |\bold{V}^o|</script><span>, </span></p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.608ex" height="2.913ex" viewBox="0 -783.2 2414.6 1254" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E3045-MJMAINB-44" 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><path stroke-width="0" id="E3045-MJMATHI-41" 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 stroke-width="0" id="E3045-MJMATHI-6F" 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 stroke-width="0" id="E3045-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3045-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-322)"><use transform="scale(0.707)" xlink:href="#E3045-MJMAIN-221A" x="0" y="82"></use><rect stroke="none" width="843" height="42" x="589" y="581"></rect><g transform="translate(589,0)"><use transform="scale(0.707)" xlink:href="#E3045-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3045-MJMATHI-6F" x="1060" y="626"></use></g></g></g></svg></span><script type="math/tex">\bold{D}_\sqrt{A^o}</script><span> is a 3n x 3n matrix with the square root of mass matrix coefficients repeated 3 times (1 for each dimension) on the diagonal. </span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.457ex" height="2.313ex" viewBox="0 -768.4 1488.3 995.9" role="img" focusable="false" style="vertical-align: -0.528ex;"><defs><path stroke-width="0" id="E3044-MJMAINB-44" 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><path stroke-width="0" id="E3044-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3044-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3044-MJMATHI-77" x="1247" y="-213"></use></g></svg></span><script type="math/tex">\bold{D}_w</script><span> is a 3n x 3n matrix with the weight vector </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.93ex" height="1.461ex" viewBox="0 -522.8 831 629" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E2999-MJMAINB-77" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E2999-MJMAINB-77" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{w}</script><span> repeated 3 times on the diagonal.</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.636ex" height="2.55ex" viewBox="0 -1043.7 1134.9 1097.7" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3049-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path><path stroke-width="0" id="E3049-MJMAINB-7E" d="M343 202Q320 202 278 225T215 249Q181 249 146 214L134 202L115 219Q111 222 106 226T98 234L96 236Q158 306 165 313Q199 344 230 344Q239 344 244 343Q262 339 300 318T359 297Q393 297 428 332L440 344L459 327Q463 324 468 320T476 312L478 310Q416 240 409 233Q375 202 343 202Z"></path><path stroke-width="0" id="E3049-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3049-MJMAINB-4C" x="0" y="0"></use><use xlink:href="#E3049-MJMAINB-7E" x="58" y="562"></use><use transform="scale(0.707)" xlink:href="#E3049-MJMATHI-6F" x="978" y="895"></use></g></svg></span><script type="math/tex">\bold{\tilde{L}}^o</script><span>, also 3n x 3n, is the Kronecker product between the cotangent matrix and 3 x 3 identity matrix: </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.275ex" height="3.033ex" viewBox="0 -1043.7 5715.4 1306.1" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3001-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path><path stroke-width="0" id="E3001-MJMAINB-7E" d="M343 202Q320 202 278 225T215 249Q181 249 146 214L134 202L115 219Q111 222 106 226T98 234L96 236Q158 306 165 313Q199 344 230 344Q239 344 244 343Q262 339 300 318T359 297Q393 297 428 332L440 344L459 327Q463 324 468 320T476 312L478 310Q416 240 409 233Q375 202 343 202Z"></path><path stroke-width="0" id="E3001-MJMATHI-6F" 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 stroke-width="0" id="E3001-MJMAIN-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 stroke-width="0" id="E3001-MJMAIN-2297" d="M56 250Q56 394 156 488T384 583Q530 583 626 485T722 250Q722 110 625 14T390 -83Q249 -83 153 14T56 250ZM582 471Q531 510 496 523Q446 542 381 542Q324 542 272 519T196 471L389 278L485 375L582 471ZM167 442Q95 362 95 250Q95 137 167 58L359 250L167 442ZM610 58Q682 138 682 250Q682 363 610 442L418 250L610 58ZM196 29Q209 16 230 2T295 -27T388 -42Q409 -42 429 -40T465 -33T496 -23T522 -11T544 1T561 13T574 22T582 29L388 222L196 29Z"></path><path stroke-width="0" id="E3001-MJMAINB-49" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path stroke-width="0" id="E3001-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3001-MJMAINB-4C" x="0" y="0"></use><use xlink:href="#E3001-MJMAINB-7E" x="58" y="562"></use><use transform="scale(0.707)" xlink:href="#E3001-MJMATHI-6F" x="978" y="895"></use><use xlink:href="#E3001-MJMAIN-3D" x="1412" y="0"></use><g transform="translate(2468,0)"><use xlink:href="#E3001-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3001-MJMATHI-6F" x="978" y="584"></use></g><use xlink:href="#E3001-MJMAIN-2297" x="3825" y="0"></use><g transform="translate(4825,0)"><use xlink:href="#E3001-MJMAINB-49" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3001-MJMAIN-33" x="616" y="-213"></use></g></g></svg></span><script type="math/tex">\bold{\tilde{L}}^o = \bold{L}^o \otimes \bold{I}_3</script></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.484ex" height="2.066ex" viewBox="0 -783.2 639 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3051-MJMAINB-53" d="M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3051-MJMAINB-53" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{S}</script><span> is a 3n x n selector matrix that associates each </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> with the x, y, z coordinates of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span>: </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.299ex" height="2.913ex" viewBox="0 -939.5 5725.8 1254" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3054-MJMAINB-49" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path stroke-width="0" id="E3054-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3054-MJMAIN-2297" d="M56 250Q56 394 156 488T384 583Q530 583 626 485T722 250Q722 110 625 14T390 -83Q249 -83 153 14T56 250ZM582 471Q531 510 496 523Q446 542 381 542Q324 542 272 519T196 471L389 278L485 375L582 471ZM167 442Q95 362 95 250Q95 137 167 58L359 250L167 442ZM610 58Q682 138 682 250Q682 363 610 442L418 250L610 58ZM196 29Q209 16 230 2T295 -27T388 -42Q409 -42 429 -40T465 -33T496 -23T522 -11T544 1T561 13T574 22T582 29L388 222L196 29Z"></path><path stroke-width="0" id="E3054-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="0" id="E3054-MJMAIN-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 stroke-width="0" id="E3054-MJMAIN-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 stroke-width="0" id="E3054-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="0" id="E3054-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3054-MJMAINB-49" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3054-MJMATHI-6E" x="616" y="-213"></use><use xlink:href="#E3054-MJMAIN-2297" x="1182" y="0"></use><use xlink:href="#E3054-MJMAIN-5B" x="2182" y="0"></use><use xlink:href="#E3054-MJMAIN-31" x="2460" y="0"></use><use xlink:href="#E3054-MJMAIN-2C" x="2960" y="0"></use><use xlink:href="#E3054-MJMAIN-31" x="3405" y="0"></use><use xlink:href="#E3054-MJMAIN-2C" x="3905" y="0"></use><use xlink:href="#E3054-MJMAIN-31" x="4350" y="0"></use><g transform="translate(4850,0)"><use xlink:href="#E3054-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3054-MJMATHI-54" x="393" y="513"></use></g></g></svg></span><script type="math/tex">\bold{I}_n \otimes [1,1,1]^T</script></li></ul><h4><a name="allowing-disconnected-pieces" class="md-header-anchor"></a><span>Allowing Disconnected Pieces</span></h4><p><span>Some meshes are composed of disconnected pieces. There might be use cases where each deformed piece needs to occupy a different location. In this case, the </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script><span> bounds need to be discontinuous as well. For example, suppose mesh A is a "connected" mesh with 4 vertices. Its </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script><span> bounds could be:</span></p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.58ex" height="2.429ex" viewBox="0 -783.2 1972.1 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3011-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3011-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3011-MJMATHI-69" 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 stroke-width="0" id="E3011-MJMATHI-6E" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3011-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-69" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-6E" x="1223" y="0"></use></g></g></svg></span><script type="math/tex">\bold{\lambda}_{min}</script><span>: 10, 10, 10, 10</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.836ex" height="2.429ex" viewBox="0 -783.2 2082.4 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3012-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3012-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3012-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E3012-MJMATHI-78" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3012-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-78" x="1406" y="0"></use></g></g></svg></span><script type="math/tex">\bold{\lambda}_{max}</script><span>: 11, 11, 11, 11</span></li></ul><p><span>Suppose mesh B is a "disconnected" mesh with 4 vertices and 2 vertex groups. Its </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script><span> bounds could be:</span></p><ul><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.58ex" height="2.429ex" viewBox="0 -783.2 1972.1 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3011-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3011-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3011-MJMATHI-69" 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 stroke-width="0" id="E3011-MJMATHI-6E" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3011-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-69" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3011-MJMATHI-6E" x="1223" y="0"></use></g></g></svg></span><script type="math/tex">\bold{\lambda}_{min}</script><span>: 10, 10, 11, 11</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.836ex" height="2.429ex" viewBox="0 -783.2 2082.4 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3012-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3012-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3012-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E3012-MJMATHI-78" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3012-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3012-MJMATHI-78" x="1406" y="0"></use></g></g></svg></span><script type="math/tex">\bold{\lambda}_{max}</script><span>: 11, 11, 12, 12</span></li></ul><p><span>This complicates user input because users need to calculate different bounds for different vertex groups. To simplify user input for this use case, the paper introduces a vector </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.401ex" height="1.945ex" viewBox="0 -522.8 603 837.3" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3034-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3034-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\mu}</script><span>, where each element </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.421ex" height="2.066ex" viewBox="0 -522.8 1042.4 889.4" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3037-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3037-MJMATHI-67" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3037-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3037-MJMATHI-67" x="852" y="-213"></use></g></svg></span><script type="math/tex">\mu_g</script><span> is a scaling factor for an independent group of vertices such that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.606ex" height="2.913ex" viewBox="0 -887.4 9733.3 1254" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3058-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3058-MJMATHI-67" 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 stroke-width="0" id="E3058-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 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117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E3058-MJMATHI-78" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3058-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-67" x="852" y="-213"></use><g transform="translate(1042,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6E" x="1223" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-429"></use></g><use xlink:href="#E3058-MJMAIN-2264" x="3292" y="0"></use><g transform="translate(4348,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-213"></use></g><use xlink:href="#E3058-MJMAIN-2264" x="5552" y="0"></use><g transform="translate(6608,0)"><use xlink:href="#E3058-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-67" x="852" y="-213"></use></g><g transform="translate(7650,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-78" x="1406" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-429"></use></g></g></svg></span><script type="math/tex">\mu_g \lambda^{min}_i \leq \lambda_i \leq \mu_g \lambda^{max}_i</script><span> holds for all vertices </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.945ex" viewBox="0 -731.2 345 837.3" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3016-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3016-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> in group </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.115ex" height="1.824ex" viewBox="0 -522.8 480 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3020-MJMATHI-67" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3020-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex">g</script><span>. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.144ex" height="2.671ex" viewBox="0 -835.3 2214.8 1149.8" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3018-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E3018-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3018-MJMAIN-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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3018-MJMAIN-7C" x="0" y="0"></use><use xlink:href="#E3018-MJMATHI-3BC" x="278" y="0"></use><use xlink:href="#E3018-MJMAIN-7C" x="881" y="0"></use><use xlink:href="#E3018-MJMAIN-3D" x="1436" y="0"></use></g></svg></span><script type="math/tex">|\bold{\mu}| =</script><span> number of groups. </span></p><p><span>To obtain a unique solution for </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.421ex" height="2.066ex" viewBox="0 -522.8 1042.4 889.4" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3037-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3037-MJMATHI-67" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3037-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3037-MJMATHI-67" x="852" y="-213"></use></g></svg></span><script type="math/tex">\mu_g</script><span>, for each group </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.115ex" height="1.824ex" viewBox="0 -522.8 480 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3020-MJMATHI-67" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3020-MJMATHI-67" x="0" y="0"></use></g></svg></span><script type="math/tex">g</script><span>, we can fix the value of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.23ex" height="2.671ex" viewBox="0 -783.2 1390.8 1149.8" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3032-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3032-MJMATHI-67" 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 stroke-width="0" id="E3032-MJMATHI-6B" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3032-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-6B" x="480" y="0"></use></g></g></svg></span><script type="math/tex">\lambda_{gk}</script><span> for a (randomly selected) vertex </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.286ex" height="2.066ex" viewBox="0 -522.8 1414.8 889.4" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3022-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3022-MJMATHI-67" 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 stroke-width="0" id="E3022-MJMATHI-6B" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3022-MJMAINB-76" x="0" y="0"></use><g transform="translate(607,-150)"><use transform="scale(0.707)" xlink:href="#E3022-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3022-MJMATHI-6B" x="480" y="0"></use></g></g></svg></span><script type="math/tex">\bold{v}_{gk}</script><span>. The value of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.23ex" height="2.671ex" viewBox="0 -783.2 1390.8 1149.8" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3032-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3032-MJMATHI-67" 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 stroke-width="0" id="E3032-MJMATHI-6B" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3032-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-6B" x="480" y="0"></use></g></g></svg></span><script type="math/tex">\lambda_{gk}</script><span> then affects the value of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="2.066ex" viewBox="0 -783.2 583 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3062-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3062-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\lambda</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.401ex" height="1.945ex" viewBox="0 -522.8 603 837.3" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3026-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3026-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex">\mu</script><span>:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n337" cid="n337" mdtype="math_block">
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xlink:href="#E3080-MJMATHI-45" x="1175" y="-213"></use><use xlink:href="#E3080-MJMAIN-5B" x="1471" y="0"></use><use xlink:href="#E3080-MJMATHI-3BB" x="1749" y="0"></use><use xlink:href="#E3080-MJMATHI-3BC" x="2887" y="0"></use><g transform="translate(3490,0)"><use xlink:href="#E3080-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3080-MJMATHI-54" x="393" y="583"></use></g><use xlink:href="#E3080-MJMAIN-3D" x="4644" y="0"></use><use xlink:href="#E3080-MJMAINB-62" x="5700" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2461">\begin{align*}
& \min_{\bold{\lambda}, \bold{\mu}} \Vert \bold{Q} \bold{\lambda} \Vert^2 + \alpha \Vert \bold{\mu} \Vert^2 \\
\\
& \text{subject to } \\
& \bold{C}_I [\lambda \;\; \mu]^T \le \bold{d} \\
& \bold{C}_E [\lambda \;\; \mu]^T = \bold{b} \\
\end{align*}</script>
</div></div><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.461ex" viewBox="0 -522.8 640 629" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3027-MJMATHI-3B1" 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 stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3027-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> Is set to </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.654ex" height="2.429ex" viewBox="0 -939.5 2003.7 1045.7" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3028-MJMAIN-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 stroke-width="0" id="E3028-MJMAIN-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 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y="0"></use></g></g></svg></span><script type="math/tex">10^{-7}</script><span> in the paper. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.99ex" height="2.308ex" viewBox="0 -783.2 1287.4 993.6" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E3057-MJMAINB-43" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path stroke-width="0" id="E3057-MJMATHI-49" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 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y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-78" x="1406" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-429"></use></g></g></svg></span><script type="math/tex">\mu_g \lambda^{min}_i \leq \lambda_i \leq \mu_g \lambda^{max}_i</script><span>. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.417ex" height="2.308ex" viewBox="0 -783.2 1471.2 993.6" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E3031-MJMAINB-43" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path stroke-width="0" id="E3031-MJMATHI-45" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 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 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3031-MJMAINB-43" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3031-MJMATHI-45" x="1175" y="-213"></use></g></svg></span><script type="math/tex">\bold{C}_E</script><span> are the equality constraints derived from fixing </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.23ex" height="2.671ex" viewBox="0 -783.2 1390.8 1149.8" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3032-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3032-MJMATHI-67" 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 stroke-width="0" id="E3032-MJMATHI-6B" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3032-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,-150)"><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-67" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3032-MJMATHI-6B" x="480" y="0"></use></g></g></svg></span><script type="math/tex">\lambda_{gk}</script><span>.</span></p><p><span>Combining </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script><span> and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.401ex" height="1.945ex" viewBox="0 -522.8 603 837.3" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3034-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3034-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\mu}</script><span> into one unknown vector </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.41ex" height="1.34ex" viewBox="0 -522.8 607 576.9" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3035-MJMAINB-78" d="M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3035-MJMAINB-78" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{x}</script><span>, we now have quadratic programming problem that can be solved using the libigl active set solver:</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="c++"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">igl::active_set_params</span> <span class="cm-variable">as</span>;</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Eigen::VectorXd</span> <span class="cm-variable">x</span>; <span class="cm-comment">// (n + |mu|) x 1</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// B - linear coefficients, set to 0</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// b - index of lambdas to be fixed</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// Y - value of the fixed lambdas</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// Aeq, Beq - empty matrices</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// Aieq - inequality constraint matrix C_I</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// Bieq - inequality constraint d, set to a 0 vector with dimension (n + |mu|) x 1</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// lx, ux - empty vectors</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">// x - [lambda; mu]</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">igl::active_set</span>(<span class="cm-variable">F</span>.<span class="cm-variable">transpose</span>() <span class="cm-operator">*</span> <span class="cm-variable">F</span>, <span class="cm-variable">B</span>, <span class="cm-variable">b</span>, <span class="cm-variable">Y</span>, <span class="cm-variable">Aeq</span>, <span class="cm-variable">Beq</span>, <span class="cm-variable">Aieq</span>, <span class="cm-variable">Bieq</span>, <span class="cm-variable">lx</span>, <span class="cm-variable">ux</span>, <span class="cm-variable">as</span>, <span class="cm-variable">x</span>);</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 312px;"></div><div class="CodeMirror-gutters" style="display: none; height: 312px;"></div></div></div></pre><p><span>where F is:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n342" cid="n342" mdtype="math_block">
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\begin{bmatrix}
\bold{Q} & \bold{0} \\
\bold{0} & \sqrt{\alpha} \cdot \bold{I}
\end{bmatrix}</script></div></div><p><span>The deformed vertices are retrieved by multiplying each </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> with </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.421ex" height="2.066ex" viewBox="0 -522.8 1042.4 889.4" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3037-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3037-MJMATHI-67" 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></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3037-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3037-MJMATHI-67" x="852" y="-213"></use></g></svg></span><script type="math/tex">\mu_g</script><span>. </span></p><h2><a name="demo" class="md-header-anchor"></a><span>Demo</span></h2><figure><table><thead><tr><th><span>Original Mesh</span></th><th><span>Deformed Mesh</span></th><th><span>Original Mesh</span></th><th><span>Deformed Mesh</span></th><th><span>Original Mesh</span></th><th><span>Deformed Mesh</span></th></tr></thead><tbody><tr><td><img src="report/orig_bunny.png" alt="original bunny" style="zoom:20%;" /></td><td><img src="report/deform_bunny_front.png" alt="deformed bunny" style="zoom:12%;" /><img src="report/deform_bunny.png" alt="deformed bunny" style="zoom:20%;" /></td><td><img src="report/orig_knight.png" alt="original knight" style="zoom:20%;" /></td><td><img src="report/deform_knight.png" alt="deformed knight" style="zoom:20%;" /></td><td><img src="report/orig_dragon.png" alt="original dragon" style="zoom:20%;" /></td><td><img src="report/d_dragon.png" alt="deformed dragon" style="zoom:15%;" /><img src="report/d_dragon_front.png" alt="deformed dragon" style="zoom:15%;" /></td></tr></tbody></table></figure><p><span> </span><span>The example </span><code>main.cpp</code><span> deforms a mesh along the z-axis (front view). </span></p><h2><a name="implementation-details" class="md-header-anchor"></a><span>Implementation Details</span></h2><h4><a name="getting-bolddsqrtao" class="md-header-anchor"></a><span>Getting </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.608ex" height="2.913ex" viewBox="0 -783.2 2414.6 1254" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E3045-MJMAINB-44" 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><path stroke-width="0" id="E3045-MJMATHI-41" 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 stroke-width="0" id="E3045-MJMATHI-6F" 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 stroke-width="0" id="E3045-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3045-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-322)"><use transform="scale(0.707)" xlink:href="#E3045-MJMAIN-221A" x="0" y="82"></use><rect stroke="none" width="843" height="42" x="589" y="581"></rect><g transform="translate(589,0)"><use transform="scale(0.707)" xlink:href="#E3045-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3045-MJMATHI-6F" x="1060" y="626"></use></g></g></g></svg></span><script type="math/tex">\bold{D}_\sqrt{A^o}</script><span> </span></h4><ul><li><span>Dimension: 3n x 3n</span></li></ul><p><span>First, we construct a mass matrix </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.536ex" height="1.945ex" viewBox="0 -783.2 1092 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3040-MJMAINB-4D" d="M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3040-MJMAINB-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{M}</script><span>:</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="c++"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Eigen::SparseMatrix</span><span class="cm-operator"><</span><span class="cm-variable-3">double</span><span class="cm-operator">></span> <span class="cm-variable">M</span>;</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">igl::massmatrix</span>(<span class="cm-variable">V</span>, <span class="cm-variable">F</span>, <span class="cm-variable">igl::MASSMATRIX_TYPE_VORONOI</span>, <span class="cm-variable">M</span>);</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 52px;"></div><div class="CodeMirror-gutters" style="display: none; height: 52px;"></div></div></div></pre><p><span>Here, </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.536ex" height="1.945ex" viewBox="0 -783.2 1092 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3040-MJMAINB-4D" d="M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3040-MJMAINB-4D" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{M}</script><span> is a diagonal matrix, in which the diagonal entry </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.618ex" height="2.429ex" viewBox="0 -783.2 1557.9 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3041-MJMATHI-4D" d="M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"></path><path stroke-width="0" id="E3041-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3041-MJMATHI-4D" x="0" y="0"></use><g transform="translate(970,-150)"><use transform="scale(0.707)" xlink:href="#E3041-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3041-MJMATHI-69" x="345" y="0"></use></g></g></svg></span><script type="math/tex">M_{ii}</script><span> is the Voronoi area around </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span> in the mesh </span><sup class='md-footnote'><a href='#dfref-footnote-2' name='ref-footnote-2'>2</a></sup><span>. We then take the diagonal entry for each vertex, take the square root, and repeat it 3 times (1 for each dimension). The resulting matrix </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.608ex" height="2.913ex" viewBox="0 -783.2 2414.6 1254" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E3045-MJMAINB-44" 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><path stroke-width="0" id="E3045-MJMATHI-41" 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 stroke-width="0" id="E3045-MJMATHI-6F" 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 stroke-width="0" id="E3045-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3045-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-322)"><use transform="scale(0.707)" xlink:href="#E3045-MJMAIN-221A" x="0" y="82"></use><rect stroke="none" width="843" height="42" x="589" y="581"></rect><g transform="translate(589,0)"><use transform="scale(0.707)" xlink:href="#E3045-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3045-MJMATHI-6F" x="1060" y="626"></use></g></g></g></svg></span><script type="math/tex">\bold{D}_\sqrt{A^o}</script><span> should look like this:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n193" cid="n193" mdtype="math_block">
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y="3789"></use><g transform="translate(0,-5669)"><use xlink:href="#E2935-MJSZ1-221A" x="0" y="-32"></use><rect stroke="none" width="3922" height="60" x="1000" y="758"></rect><g transform="translate(1000,0)"><use xlink:href="#E2935-MJMATHI-4D" x="0" y="0"></use><g transform="translate(970,-150)"><use transform="scale(0.707)" xlink:href="#E2935-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMAIN-2212" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMAIN-31" x="1378" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMAIN-2C" x="1878" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMATHI-6E" x="2156" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMAIN-2212" x="2756" y="0"></use><use transform="scale(0.707)" xlink:href="#E2935-MJMAIN-31" x="3533" y="0"></use></g></g></g></g></g><g transform="translate(29851,6549)"><use xlink:href="#E2935-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-10850.84053346402) scale(1,15.119336434325614)"><use xlink:href="#E2935-MJSZ4-23A5"></use></g><use xlink:href="#E2935-MJSZ4-23A6" x="0" y="-11956"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-412">\bold{D}_\sqrt{A^o} =
\begin{bmatrix}
\sqrt{M_{00}} & 0 & \dots & & & & & 0\\
0 & \sqrt{M_{00}} & & & & & &\vdots \\
& & \sqrt{M_{00}} \\
& & & \sqrt{M_{11}} \\
& & & & \sqrt{M_{11}} \\
& & & & & \sqrt{M_{11}} \\
\vdots & & & & & & \ddots \\
0 & 0 & \dots & & & 0 & \dots & \sqrt{M_{n-1,n-1}}
\end{bmatrix}</script></div></div><h4><a name="getting-bolddw" class="md-header-anchor"></a><span>Getting </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.457ex" height="2.313ex" viewBox="0 -768.4 1488.3 995.9" role="img" focusable="false" style="vertical-align: -0.528ex;"><defs><path stroke-width="0" id="E3044-MJMAINB-44" 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><path stroke-width="0" id="E3044-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3044-MJMAINB-44" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3044-MJMATHI-77" x="1247" y="-213"></use></g></svg></span><script type="math/tex">\bold{D}_w</script></h4><ul><li><span>Dimension: 3n x 3n</span></li></ul><p><span>Similar to </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.608ex" height="2.913ex" viewBox="0 -783.2 2414.6 1254" role="img" focusable="false" style="vertical-align: -1.093ex;"><defs><path stroke-width="0" id="E3045-MJMAINB-44" 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><path stroke-width="0" id="E3045-MJMATHI-41" 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 stroke-width="0" id="E3045-MJMATHI-6F" 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 stroke-width="0" id="E3045-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3045-MJMAINB-44" x="0" y="0"></use><g transform="translate(882,-322)"><use transform="scale(0.707)" xlink:href="#E3045-MJMAIN-221A" x="0" y="82"></use><rect stroke="none" width="843" height="42" x="589" y="581"></rect><g transform="translate(589,0)"><use transform="scale(0.707)" xlink:href="#E3045-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E3045-MJMATHI-6F" x="1060" y="626"></use></g></g></g></svg></span><script type="math/tex">\bold{D}_\sqrt{A^o}</script><span>, we take the weight vector </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.663ex" height="1.461ex" viewBox="0 -522.8 716 629" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3046-MJMATHI-77" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3046-MJMATHI-77" x="0" y="0"></use></g></svg></span><script type="math/tex">w</script><span> of size n x 1 and repeat it along the diagonal:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n200" cid="n200" mdtype="math_block">
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transform="translate(13005,0)"><use xlink:href="#E2936-MJMAIN-22F1" x="0" y="-4157"></use><use xlink:href="#E2936-MJMAIN-2026" x="55" y="-5557"></use></g><g transform="translate(15287,0)"><use xlink:href="#E2936-MJMAIN-30" x="821" y="5463"></use><use xlink:href="#E2936-MJMAIN-22EE" x="932" y="3563"></use><g transform="translate(0,-5557)"><use xlink:href="#E2936-MJMATHI-77" x="0" y="0"></use><g transform="translate(716,-150)"><use transform="scale(0.707)" xlink:href="#E2936-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2936-MJMAIN-2212" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E2936-MJMAIN-31" x="1378" y="0"></use></g></g></g></g><g transform="translate(18432,6263)"><use xlink:href="#E2936-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-10278.982756057296) scale(1,14.169406571523748)"><use xlink:href="#E2936-MJSZ4-23A5"></use></g><use xlink:href="#E2936-MJSZ4-23A6" x="0" y="-11384"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-413">\bold{D}_w =
\begin{bmatrix}
w_0 & 0 & \dots & & & & & 0\\
0 & w_0 & & & & & &\vdots \\
& & w_0 \\
& & & w_1 \\
& & & & w_1 \\
& & & & & w_1 \\
\vdots & & & & & & \ddots \\
0 & 0 & \dots & & & 0 & \dots & w_{n-1}
\end{bmatrix}</script></div></div><h4><a name="getting-boldtildelo" class="md-header-anchor"></a><span>Getting </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.636ex" height="2.55ex" viewBox="0 -1043.7 1134.9 1097.7" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3049-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path><path stroke-width="0" id="E3049-MJMAINB-7E" d="M343 202Q320 202 278 225T215 249Q181 249 146 214L134 202L115 219Q111 222 106 226T98 234L96 236Q158 306 165 313Q199 344 230 344Q239 344 244 343Q262 339 300 318T359 297Q393 297 428 332L440 344L459 327Q463 324 468 320T476 312L478 310Q416 240 409 233Q375 202 343 202Z"></path><path stroke-width="0" id="E3049-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3049-MJMAINB-4C" x="0" y="0"></use><use xlink:href="#E3049-MJMAINB-7E" x="58" y="562"></use><use transform="scale(0.707)" xlink:href="#E3049-MJMATHI-6F" x="978" y="895"></use></g></svg></span><script type="math/tex">\bold{\tilde{L}}^o</script></h4><ul><li><span>Dimension: 3n x 3n</span></li></ul><p><span>First, we compute the cotangent Laplace-Beltrami operator </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.636ex" height="1.945ex" viewBox="0 -783.2 1134.9 837.3" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3048-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path><path stroke-width="0" id="E3048-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3048-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3048-MJMATHI-6F" x="978" y="584"></use></g></svg></span><script type="math/tex">\bold{L}^o</script><span>:</span></p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="c++"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="c++"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 8px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Eigen::SparseMatrix</span><span class="cm-operator"><</span><span class="cm-variable-3">double</span><span class="cm-operator">></span> <span class="cm-variable">cot</span>, <span class="cm-variable">M_inv</span>, <span class="cm-variable">L</span>;</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">igl::cotmatrix</span>(<span class="cm-variable">V</span>, <span class="cm-variable">F</span>, <span class="cm-variable">cot</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">igl::invert_diag</span>(<span class="cm-variable">M</span>, <span class="cm-variable">M_inv</span>);</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">L</span> <span class="cm-operator">=</span> <span class="cm-variable">M_inv</span> <span class="cm-operator">*</span> <span class="cm-variable">cot</span>;</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 104px;"></div><div class="CodeMirror-gutters" style="display: none; height: 104px;"></div></div></div></pre><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.636ex" height="2.55ex" viewBox="0 -1043.7 1134.9 1097.7" role="img" focusable="false" style="vertical-align: -0.126ex;"><defs><path stroke-width="0" id="E3049-MJMAINB-4C" d="M643 285Q641 280 629 148T612 4V0H39V62H147V624H39V686H51Q75 683 228 683Q415 685 425 686H439V624H304V62H352H378Q492 62 539 138Q551 156 558 178T569 214T576 255T581 289H643V285Z"></path><path stroke-width="0" id="E3049-MJMAINB-7E" d="M343 202Q320 202 278 225T215 249Q181 249 146 214L134 202L115 219Q111 222 106 226T98 234L96 236Q158 306 165 313Q199 344 230 344Q239 344 244 343Q262 339 300 318T359 297Q393 297 428 332L440 344L459 327Q463 324 468 320T476 312L478 310Q416 240 409 233Q375 202 343 202Z"></path><path stroke-width="0" id="E3049-MJMATHI-6F" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3049-MJMAINB-4C" x="0" y="0"></use><use xlink:href="#E3049-MJMAINB-7E" x="58" y="562"></use><use transform="scale(0.707)" xlink:href="#E3049-MJMATHI-6F" x="978" y="895"></use></g></svg></span><script type="math/tex">\bold{\tilde{L}}^o</script><span> is the Kronecker product between the cotangent matrix and 3 x 3 identity matrix:</span></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">
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transform="scale(0.707)" xlink:href="#E2938-MJMAIN-31" x="3533" y="0"></use></g></g></g><g transform="translate(9274,0)"><g transform="translate(0,-1543)"><use xlink:href="#E2938-MJMAINB-4C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMATHI-6F" x="978" y="584"></use><g transform="translate(692,-248)"><use transform="scale(0.707)" xlink:href="#E2938-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMAIN-2212" x="600" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMAIN-31" x="1378" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMAIN-2C" x="1878" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMATHI-6E" x="2156" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMAIN-2212" x="2756" y="0"></use><use transform="scale(0.707)" xlink:href="#E2938-MJMAIN-31" x="3533" y="0"></use></g></g></g></g><g transform="translate(13920,2427)"><use xlink:href="#E2938-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-2606.30846699413) scale(1,1.4241004435118438)"><use xlink:href="#E2938-MJSZ4-23A5"></use></g><use xlink:href="#E2938-MJSZ4-23A6" x="0" y="-3712"></use></g></g></g></g><g transform="translate(30519,5865)"><use xlink:href="#E2938-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-9481.61693398826) scale(1,12.844878627887475)"><use xlink:href="#E2938-MJSZ4-23A5"></use></g><use xlink:href="#E2938-MJSZ4-23A6" x="0" y="-10587"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-415">\bold{\tilde{L}}^o =
\begin{bmatrix}
\begin{bmatrix}
\textbf{L}^o_{00} && \\
&\textbf{L}^o_{00}& \\
&&\textbf{L}^o_{00} \\
\end{bmatrix} & \dots &
\begin{bmatrix}
\textbf{L}^o_{0,n-1} && \\
&\textbf{L}^o_{0,n-1}& \\
&&\textbf{L}^o_{0,n-1} \\
\end{bmatrix}\\
\vdots & \ddots &\vdots \\
\begin{bmatrix}
\textbf{L}^o_{n-1,0} && \\
&\textbf{L}^o_{n-1,0}& \\
&&\textbf{L}^o_{n-1,0} \\
\end{bmatrix} & &
\begin{bmatrix}
\textbf{L}^o_{n-1,n-1} && \\
&\textbf{L}^o_{n-1,n-1}& \\
&&\textbf{L}^o_{n-1,n-1} \\
\end{bmatrix} \\
\end{bmatrix}</script></div></div><h4><a name="getting-bolds" class="md-header-anchor"></a><span>Getting </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.484ex" height="2.066ex" viewBox="0 -783.2 639 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3051-MJMAINB-53" d="M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3051-MJMAINB-53" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{S}</script></h4><ul><li><span>Dimension: 3n x n</span></li><li><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.484ex" height="2.066ex" viewBox="0 -783.2 639 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3051-MJMAINB-53" d="M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3051-MJMAINB-53" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{S}</script><span> is a selector matrix that associates each </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> with the x, y, z coordinates of </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.209ex" height="1.824ex" viewBox="0 -522.8 951 785.2" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3053-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></path><path stroke-width="0" id="E3053-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3053-MJMAINB-76" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3053-MJMATHI-69" x="858" y="-213"></use></g></svg></span><script type="math/tex">\bold{v}_i</script><span>: </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; 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1\\
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</div></div><h4><a name="getting-boldci" class="md-header-anchor"></a><span>Getting </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.99ex" height="2.308ex" viewBox="0 -783.2 1287.4 993.6" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E3057-MJMAINB-43" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path stroke-width="0" id="E3057-MJMATHI-49" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3057-MJMAINB-43" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3057-MJMATHI-49" x="1175" y="-213"></use></g></svg></span><script type="math/tex">\bold{C}_I</script></h4><ul><li><span>Dimension: 2n x (n + </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.692ex" height="2.671ex" viewBox="0 -835.3 1159 1149.8" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3056-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E3056-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3056-MJMAIN-7C" x="0" y="0"></use><use xlink:href="#E3056-MJMATHI-3BC" x="278" y="0"></use><use xlink:href="#E3056-MJMAIN-7C" x="881" y="0"></use></g></svg></span><script type="math/tex">|\mu|</script><span>)</span></li></ul><p><span>We construct </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.99ex" height="2.308ex" viewBox="0 -783.2 1287.4 993.6" role="img" focusable="false" style="vertical-align: -0.489ex;"><defs><path stroke-width="0" id="E3057-MJMAINB-43" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path stroke-width="0" id="E3057-MJMATHI-49" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3057-MJMAINB-43" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3057-MJMATHI-49" x="1175" y="-213"></use></g></svg></span><script type="math/tex">\bold{C}_I</script><span> such that </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.606ex" height="2.913ex" viewBox="0 -887.4 9733.3 1254" role="img" focusable="false" style="vertical-align: -0.851ex;"><defs><path stroke-width="0" id="E3058-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3058-MJMATHI-67" 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 stroke-width="0" id="E3058-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3058-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3058-MJMATHI-69" 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 stroke-width="0" id="E3058-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E3058-MJMAIN-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 stroke-width="0" id="E3058-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E3058-MJMATHI-78" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3058-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-67" x="852" y="-213"></use><g transform="translate(1042,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6E" x="1223" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-429"></use></g><use xlink:href="#E3058-MJMAIN-2264" x="3292" y="0"></use><g transform="translate(4348,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-213"></use></g><use xlink:href="#E3058-MJMAIN-2264" x="5552" y="0"></use><g transform="translate(6608,0)"><use xlink:href="#E3058-MJMATHI-3BC" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-67" x="852" y="-213"></use></g><g transform="translate(7650,0)"><use xlink:href="#E3058-MJMATHI-3BB" x="0" y="0"></use><g transform="translate(583,352)"><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-61" x="878" y="0"></use><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-78" x="1406" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E3058-MJMATHI-69" x="824" y="-429"></use></g></g></svg></span><script type="math/tex">\mu_g \lambda^{min}_i \leq \lambda_i \leq \mu_g \lambda^{max}_i</script><span> can be transformed into </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.148ex" height="2.913ex" viewBox="0 -939.5 6091.3 1254" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3059-MJMAINB-43" d="M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z"></path><path stroke-width="0" id="E3059-MJMATHI-49" d="M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path><path stroke-width="0" id="E3059-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="0" id="E3059-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3059-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path><path stroke-width="0" id="E3059-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path><path stroke-width="0" id="E3059-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E3059-MJMAIN-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 stroke-width="0" id="E3059-MJMAINB-30" d="M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3059-MJMAINB-43" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3059-MJMATHI-49" x="1175" y="-213"></use><use xlink:href="#E3059-MJMAIN-5B" x="1287" y="0"></use><use xlink:href="#E3059-MJMATHI-3BB" x="1565" y="0"></use><use xlink:href="#E3059-MJMATHI-3BC" x="2703" y="0"></use><g transform="translate(3306,0)"><use xlink:href="#E3059-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3059-MJMATHI-54" x="393" y="513"></use></g><use xlink:href="#E3059-MJMAIN-2264" x="4460" y="0"></use><use xlink:href="#E3059-MJMAINB-30" x="5516" y="0"></use></g></svg></span><script type="math/tex">\bold{C}_I [\lambda \;\; \mu]^T \le \bold{0}</script><span>. Below is an example of an inequality constraint for a mesh with 4 vertices and 2 groups. </span></p><div mdtype="math_block" cid="n391" id="mathjax-n391" class="mathjax-block md-end-block md-math-block md-rawblock" spellcheck="false" contenteditable="false">
<div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
<div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-1959-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="116.371ex" height="27.106ex" viewBox="0 -6095.7 50104 11670.6" role="img" focusable="false" style="vertical-align: -12.778ex; margin-bottom: -0.17ex; max-width: 100%;"><defs><path stroke-width="0" id="E2940-MJMAIN-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 stroke-width="0" id="E2940-MJMAIN-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 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250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E2940-MJMAIN-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 stroke-width="0" id="E2940-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E2940-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 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0 & 0 & 0 & -1 & 0 & \lambda_{min,11} \\
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0 & 0 & 0 & 1 & 0 & -\lambda_{max,11} \\
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\begin{bmatrix}
\lambda_0 \\
\lambda_1 \\
\lambda_2 \\
\lambda_3 \\
\mu_0 \\
\mu_1 \\
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\le
\bold{0}</script>
</div></div><p> </p><h4><a name="getting-bounds-on-boldlambda" class="md-header-anchor"></a><span>Getting Bounds on </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="1.999ex" viewBox="0 -768.4 583 860.6" role="img" focusable="false" style="vertical-align: -0.214ex;"><defs><path stroke-width="0" id="E3060-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3060-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\lambda}</script></h4><p><img src="report/bounding_box.png" alt="bounding box" style="zoom:20%;" /></p><p><span>Bounds for </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.153ex" height="2.429ex" viewBox="0 -783.2 927 1045.7" role="img" focusable="false" style="vertical-align: -0.609ex;"><defs><path stroke-width="0" id="E3061-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path><path stroke-width="0" id="E3061-MJMATHI-69" 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="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3061-MJMATHI-3BB" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E3061-MJMATHI-69" x="824" y="-213"></use></g></svg></span><script type="math/tex">\lambda_i</script><span> varies for each vertex and each mesh. We first compute a bounding box </span><sup class='md-footnote'><a href='#dfref-footnote-3' name='ref-footnote-3'>3</a></sup><span> for the mesh to help with customizing </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.354ex" height="2.066ex" viewBox="0 -783.2 583 889.4" role="img" focusable="false" style="vertical-align: -0.247ex;"><defs><path stroke-width="0" id="E3062-MJMATHI-3BB" d="M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3062-MJMATHI-3BB" x="0" y="0"></use></g></svg></span><script type="math/tex">\lambda</script><span> bounds. </span></p><p><img src="report/viewpoint.png" alt="bounding box" style="zoom:20%;" /></p><p><span>The above figure shows how </span><code>main.cpp</code><span> deforms a mesh from a left viewpoint. We assume the mesh faces the positive z-axis and the bounding box is roughly square. </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n224" cid="n224" mdtype="math_block">
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transform="translate(11527,0)"><use xlink:href="#E2941-MJMAINB-6E" x="0" y="0"></use><use xlink:href="#E2941-MJMAINB-5E" x="32" y="8"></use><use transform="scale(0.707)" xlink:href="#E2941-MJMATHI-69" x="903" y="-213"></use></g><use xlink:href="#E2941-MJMAIN-29" x="12510" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-417">\begin{align*}
& \lambda_{max} = |OG|\\
& \lambda_{min} = |OE|\\
& (\lambda_i \bold{\hat{v}}_i) \; \cdot \; \bold{\hat{n}}_i \le \lambda_{max}
\rightarrow
\lambda_i \le \lambda_{max} / (\bold{\hat{v}}_i \; \cdot \; \bold{\hat{n}}_i)\\
& (\lambda_i \bold{\hat{v}}_i) \; \cdot \; \bold{\hat{n}}_i \ge \lambda_{min}
\rightarrow
\lambda_i \ge \lambda_{min} / (\bold{\hat{v}}_i \; \cdot \; \bold{\hat{n}}_i)\\
& lowerbound = \lambda_{max} / (\bold{\hat{v}}_i \; \cdot \; \bold{\hat{n}}_i) \\
& upperbound = \lambda_{min} / (\bold{\hat{v}}_i \; \cdot \; \bold{\hat{n}}_i)
\end{align*}</script></div></div><h2><a name="future-directions--challenges" class="md-header-anchor"></a><span>Future Directions & Challenges</span></h2><p><span>Although </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.401ex" height="1.945ex" viewBox="0 -522.8 603 837.3" role="img" focusable="false" style="vertical-align: -0.73ex;"><defs><path stroke-width="0" id="E3034-MJMATHI-3BC" d="M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E3034-MJMATHI-3BC" x="0" y="0"></use></g></svg></span><script type="math/tex">\bold{\mu}</script><span> frees the user from specifying individual depth constraints for each disconnected piece, I am still having trouble producing disconnected appearance mimicking surfaces. If the input mesh is all connected (like the bunny), I am not sure how to break up the faces after breaking the vertices into mulitple groups. If the input mesh has disconnected pieces, I am not sure how to determine which vertices belong to the same group, other than manually inspecting the mesh file. </span></p><h2><a name="references" class="md-header-anchor"></a><span>References</span></h2><div class='footnotes-area' ><hr/>
<div class='footnote-line'><span class='md-fn-count'>1</span> <span>Christian Schuller, Daniele Panozzo, Olga Sorkine-Hornung, </span><a href='https://cims.nyu.edu/gcl/papers/mimicking-2014.pdf'><em><span>Appearance-Mimicking Surfaces</span></em></a><span>, 2014</span> <a name='dfref-footnote-1' href='#ref-footnote-1' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>2</span> <span>Mark Meyer, Mathieu Desbrun, Peter Schröder and Alan H. Barr, </span><a href='https://www.google.com/search?q=Discrete+Differential-Geometry+Operators+for+Triangulated+2-Manifolds'><span>Discrete Differential-Geometry Operators for Triangulated 2-Manifolds</span></a><span>, 2003.</span> <a name='dfref-footnote-2' href='#ref-footnote-2' title='back to document' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>3</span> <span>Libigl tutorial on bounding boxes: </span><a href='https://github.com/libigl/libigl/blob/master/tutorial/105_Overlays/main.cpp' target='_blank' class='url'>https://github.com/libigl/libigl/blob/master/tutorial/105_Overlays/main.cpp</a> <a name='dfref-footnote-3' href='#ref-footnote-3' title='back to document' class='reversefootnote' >↩</a></div></div></div>
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