CSE 274 Discrete Differential Geometryalchern/teaching/cse274_2020/...Course Information Discrete...

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CSE 274 Discrete Differential Geometry (Selected Topics in Computer Graphics) Instructor: Albert Chern TA: Mohammad Nezhad

Transcript of CSE 274 Discrete Differential Geometryalchern/teaching/cse274_2020/...Course Information Discrete...

  • CSE 274 Discrete Differential Geometry(Selected Topics in Computer Graphics)

    Instructor: Albert Chern
TA: Mohammad Nezhad

  • Course InformationDiscrete Differential Geometry

    • Goal: Differential geometric notions and their discrete theories for geometry processing and modeling.

    • Prerequisite: Linear algebra, Multivariable calculus, (computer graphics).• Grade: 4 homework assignments (theory+implementation) (90%) and

    participation (10%).

    • Collaboration: Team of one (individual) or two people.

  • Course Information• UCSD Canvas (and Gradescope) (Zoom links, homework submission)

  • Course Information• Piazza (Q&A discussion)‣ https://piazza.com/signup/eng.ucsd

    https://piazza.com/signup/eng.ucsd

  • Course Information• http://cseweb.ucsd.edu/~alchern/teaching/cse274_2020/ (static page for schedule, homework, lecture notes)

    http://cseweb.ucsd.edu/~alchern/teaching/cse274_2020/

  • Course Information

    • Office hour: we will make a poll (until 10/4 Sun) on Piazza for the time slot.

  • Course Information

    • Lecture note: http://cseweb.ucsd.edu/~alchern/teaching/DDG.pdf

  • Course Information

    • Implementation part of the homework: Houdini Software

  • Course Information

    https://youtu.be/T_0ICxROM0QNCSA Advanced Visualization Lab

  • Course Information

    • Implementation part of the homework: Houdini Software

    ‣ Tutorial next Monday‣ HW0 due next-next

    Monday: upload a screenshot/render of your surface.

  • Course Information

    • Install Houdini & SciPy (numerical linear algebra for sparse matrices)

  • Course Information

    • Install Houdini & SciPy (numerical linear algebra for sparse matrices)

  • Questions?

  • Discrete Differential Geometry

  • What is Geometry?

    • Relationships between shapes, order of spaces• Symmetries and invariants

  • What is Geometry?

    • Relationships between shapes, order of spaces

  • What is Geometry?

    • Relationships between shapes, order of spaces (18th, 19th century)

    Schwarz 1890 photo courtesy Capanna

    • Engineering design• General relativity• Computer graphics

    Differential geometry

    Gauss, Riemann, Poincaré, …

  • What is Geometry?

    • Relationships between shapes, order of spaces (18th, 19th century)Non-Euclidean projective geometry

    Lobachevsky, Möbius, Grassmann,
Beltrami, Lie, Plücker, Cayley, Klein,…

    ✓Two points connect into a line✓Center and distance gives a circle✓90° is well-defined

    Two lines meet unless parallel

  • What is Geometry?

    • Relationships between shapes, order of spaces (18th, 19th century)

    ✓Two points connect into a line✓Center and distance gives a circle✓90° is well-defined

    Two lines meet unless parallel

    • Linear algebra• Quantum mechanics• Algebraic geometry

    Non-Euclidean projective geometry

    Lobachevsky, Möbius, Grassmann,
Beltrami, Lie, Plücker, Cayley, Klein,…

  • What is Geometry?

    • Relationships between shapes, order of spaces• Symmetries and invariants

    Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations• Invariant notions

    Felix Klein 1872 Emmy Noether 1915 Albert Eistein 1916

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions8<:p=

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    rotation

    translation

    ‣ straight lines, planes‣ distance‣ circles, spheres‣ angles‣ area‣ parallelism

    • Non-invariant notions‣ additions p1 + p2AAAU/3iczZhbj9vGFceVtGlSNWmcZt/6QnexaJGsFqISe23ACWzvylBc7cUrr2N4qRgjciTR4m1nRrK0Az70U/S1ecpbkNd8lAD9MD0z5BySkhigQB8qQOTM73/OmTmcCy+jJPC5aLd/eefd3/z2vd+9/8Hvm3/48KM/fnzrkz+94PGcufTSjYOYvRwRTgM/opfCFwF9mTBKwlFAvx3NjpT+7YIy7sfRc7FK6DAkk8gf+y4RgIbOKHltW59b6tx5fWu3fdDWP2uzYOeF3Ub+O3/9yaf/drzYnYc0Em5AOL+y24kYSsKE7wY0bTpzThPizsiEXkExIiHlQ6l7nVp7QDxrHDP4R8LStOwhScj5KhyBZUjElK9rCm7TruZifG8o/SiZCxq5WUPjeWCJ2FKXwPJ8Rl0RrKBAXOZDXy13ShhxBVyoSiQ+5i4JqPeVYHO6D8Wv3BCO+8ocjIdSNe5R7k+iarKZ21BOabCgArRjCheI0QHkEwdPIF85goKXysvUlMJURtssHwXJlIyokI5qLTfOTk0nom/dOAxJ5EknjmhaMpJ2WumUXI4hx/3Id6kqVDXiujCIPG029/asQa/76MJ6fPayO9AtcLKgo3gpnXEcw3mt2QWfUsIUv+oMrw7uDJuWJffgYFkO5LDUJUQa83I0udtJS9qUz/zEeevlciEk3lh1pAqogHyZv5S2tWtbMEvLoYKAJFJlWW6ubAARGOUiZiaqFtfym/6f5NdWGf5v8oMx7p4eV8YZmPpZx2eXj/td6/HFo6O/d5/D+IdkRgnsKwImPM5ONS+fkNAPVvLs5GUqT6JsvnZTuXWyn0R9wiY54emGUz73B1T8ilM289ddR5VloxwHU5LQ7U1IfR0etO5+bVmo3VHwwd3WYRne1fCwda8MDzW817pfhvc0vN+y218X8D7AB3a7ZXdK0G5r2mmBe4l2muWrVp/BqDaD1mO4NFvT0MrWXLSyNSGtbM1KK1tT08r2/DIpTxLmkZq20SSgzsN55NEx3Li8jDO2yfOLcgKb2jHc5EIfZqE0EfLdLk5oVJr1ld/GJPqrfffLdDutay3vV96aG8Sc/hfNHdrbmgNqllYsprCy8uWnluXZk41V2Gw6cEmcV6/M/i5fqc1dsYsLZBeGnZ4iOzVsMEA2MOz8HNm5Yb0esp5hR0fIjgx7+hTZU7Qr4h1hwIsCXhStFLCH8PmJfO7cPjG1U1U7NbW+qvVN7ZmqPctrI5IHG0tiYkFgw0bIxmOEY4QTZBNkPjIf2Rtkb5DNkM2QzZHNkS2QLZC9RfYW2QAZR7ZEtkS2QrZCdoPsBhlDxpCJAjrCDzyqNCNGqEWFAzKBLEGWILtGdm2YD3b7jh8LkgO6zKOxUEIZ7SI3mKuHVDB/kzPPV8O17+VVN86AOn7neGQyocx4exjT9yBk3p8AHoM9mBrqvsH10oOdQzPscjivymExZPBQF645a4QGINxQFmPSbdXyntYWhMH+IcAdSnD7zva1287t3LfQ9ckXps+8m4XjLpNd0xLvFRCXJD8p4AnCQQEHeCEiM6ijsBhVIQoo0BQeDGf5elDFYkWo2rQkTMuCKAlFrB4JcVigjPvQKkyQq0qlCeOTNVJ4qTpHT63yzLdZ2bbPEspgS2WfSQd23NCHG2Z+znvlcelxM8OE9JxwRlnUsg/uhHNLGGG5JiyNMFLKqKgyVWVYXanqCqtCy6LQr1X9GquJqiamulprFOPcrAk3RrheE0xkV2UJKyZ6U+wjHoX5VjF3KC5LDxZCVYOVkEvJ1F/TgOAVnG6G1Qy9+YY3R+8JgWfsNVkzY0DUu8+agWbGAFbSZv8yaEzybaBqku8D+VpQOeR7ob4omN2W5IrchMrNgfdQPkVvlVzhvpm8KLJ3y+56rCrersprTZ6WgrubwV0dfOt6gKcYeIC5oGmN6sPj8Tdhncrh/VbySVSnE0HUQoNjpzZEQiJ4m0nlAAr1vSCTVPq1/QhoNIGXftnPznW9YZSk8pE61lgs4iCVL+BQo8Nreyp7tFZ3k1S6SZ3q+Rzy1McaC9gRpNoVsqHszgPKcEvUNTPK3wyeoABlg/t9pP2+gcc+891pQIsb7LHeiyufCBaUjQhLr+yhzKaOlU0dJahvSLI8oXbtci0tO1TekT/LJmgWNW9C7tobrfveUtvkN6mxtlk3Cmb4WI+JAEudKNZP5HzdYcKIt+miadmp4sO3O/Ff93LnLNh00rTiw2jZy/MXm04K1rbTP15zUPfzfr39lHK+2URP0frLliQwSR2hPgVBRX2yIyq5vAS3wvT1rV17/evfZuFF58D+4qDz7Mvdh6f5l8EPGn9u/KXxt4bdOGw8bPQa543Lhtu4bvyz8a/G9zv/2Plh58ednzLTd9/JfT5tVH47P/8H482Kng==‣ dot product p1 · 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Geometry in Euclidean space

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions8<:p=

    24

    xyz

    35 2 R39=;

    AAAVOXiczZhJj9vIFcc1M1kmyuaZ6VsudBqNBINWQ+wZu23AHtjulqFx1Islt8dwU2MUyZJYbm5dVZKlLvAT5SPkU+SY5ZJbkGu+QF4VyUdSywABcogAkVW//3uv6rEWLm4aMiG73b989PEnP/rxT3766c/aP//FL3/16zufff5aJDPu0UsvCRP+xiWChiyml5LJkL5JOSWRG9Lv3OtjrX83p1ywJH4llykdR2QaswnziAT07k7PCelEOspxU+ux1XZcOmWxciMiOVtk7YXjWEv437YdGvvIHRY7w+H3XzmcTQPpZO/u7HYPuuZnrRfsorDbKn4X7z774u+On3iziMbSC4kQV3Y3lWNFuGReSLO2MxM0Jd41mdIrKMYkomKsTL6ZtQfEtyYJh38sLUPrHopEQiwjFyyhw4FY1TTcpF3N5OTBWLE4nUkae3lDk1loycTSF8/yGaeeDJdQIB5n0FfLCwgnnoRL3IgkJsIjIfUfSz6j+1B87EVw3NfmYDxWunGfCjaNm8nmbmMV0HBOJWgnFC4QpyPIJwmfQ77KhYKfqcusLEWZijdZPg3TgLhUKke3Vhjnp7YT0w9eEkUEhtVJYprVjJSdNTqlFhPIcT9mHtWFpkY8DwZRZO323p416veeDq1n5296I9OCIHPqJgvlTJIEzivNzkVACdf86nB8dXBv3LYstQcHy3Igh4UpITJY1KOp3cOspgXimqXOB7+QKyH1J7ojTUBlPpmVbe3aFszSeqgwJKnSWdabqxtABE6FTHgZ1Ygr+QX/J/l1dYb/m/xgjHtnJ41xBqZ/1sn55bNBz3o2fHr8h94rGP+IXFMCO5KECY+zU8/L5yRi4VKdn77J1Gmcz9depjZO9tN4QPi0ICJbcyrm/ojKH3DKZ/6qq9tYNtpxFJCUbm5CmevwqHP/G8tC7Z6Gj+53jurwvoFHnQd1eGTgg87DOnxg4MOO3f2mgg8BPrK7HfuwBu2uoYcdcK/Rw3b9qm3PwN2aQecZXJqNaRhlYy5G2ZiQUTZmZZSNqRllc365VCQJ80hP23gaUufJLPbpBG55fs45X+fFRTmFTe0Ebo8Rg1moygjFbpekNK7N+sZvbRL9zr7/dbaZbmut6FfRmhcmgv4XzR3Zm5oDWi6tRAawsorlp5fl+fO1VdhuO3BJnLdvy/1dvdWbu2bDIbJhyc7OkJ2VbDRCNirZxQWyi5L1+8j6JTs+RnZcshcvkL1AuyreMQYcVnBYtVLBPsJXp+qVc/e0rJ3p2llZG+jaoKy91LWXRc0lRbCJImUsCFwyF9lkgnCCcIpsiowhY8jeI3uP7BrZNbIZshmyObI5sg/IPiAbIRPIFsgWyJbIlshukd0i48g4MllBR7LQp1orxRi1uHJAJpGlyFJkN8huSsbAbt9hiSQFoIsiGo8UlNEu9sKZfrwF8/cF85kern2/qHpJDvTxe8cn0ynlpbePMZkPIYv+hPAA7cPU0PcNYZYe7ByGYZejWVOOqiGDh7poxdkgNADhlvIEk+7qlveMNicc9g8J7lCC23e+r9117ha+lW5OTJZ9Fr08nPC46pUtiX4FcUmK0wqeIhxVcIQXIi4H1Y2qUZWyghJN4cHwulgPulitCF0LakJQF2RNqGL1SYTDAmXch5ZRilxXGk2UPnkjlZeuC/Q0qsh9241t+zylHLZU/qVyYMeNGNwwi3PRK18oX5QzTCrfia4pjzv2wb1oZslSWKwIi1JwteJWVa6rHKtLXV1iVRpZVvqNrt9gNdXVtKwuVxrFOLcrwm0p3KwIZWRPZwkrJn5f7SM+hfnWMHcoLksfFkJTg5VQSGnAVjQgeAWD9bCGobdY8xboPSXwjL0iG1YaEP3us2JgWGkAK2m9fzksTYptoGlS7APFWtA5FHuhuSiY3Ybkqtykzs2B91ARoLdOrnJfT15W2Xt1dzNWDW9P57UiB7Xg3npwzwTfuB7gKQYeYIY026IyeDz+NtqmCni/VWIab9OJJHqhwfFwa4iUxPA2k6kRFLb3gkwzxbb2I6TxFF761SA/b+sNpyRTT/Vxi8U8CTP1Gg5bdHhtz1SfbtW9NFNeuk31mYA8zXGLBewISu8K+VD2ZiHluCWaWjnK346eowDlEg8GSAeDEp4wzrwgpNUN9sTsxY1PBHPKXcKzK3us8qlj5VNHC/rrk6pPqF27XsvqDo135C/zCZpHLZpQu/Za68xfGJviJjUxNqtG4TU+1mMiwDInTswTuVh1mHLir7sYWndq+IjNTuKHvbwZD9edDG34cFr38tl83UnDre0MTlYc9P18sN0+oEKsN9HXdPtlS1OYpI7Un4Kgoj/ZEZ1cUYJbof4eaK9+/VsvvD48sL86OHz59e6Ts+LL4Ket37R+2/p9y24dtZ60+q2L1mXLa/2x9efWX1t/2/nTzj92/rnzr9z0448Kny9ajd/Ov/8Du4iiwg==

    rotation

    ‣ straight lines, planes‣ distance‣ circles, spheres‣ angles‣ area‣ parallelism

    • Non-invariant notions

    ‣ additions p1 + 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‣ dot product p1 · 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

    Geometry in Euclidean vector space

    p̃= 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

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions8<:p=

    24

    xyz

    35 2 R39=;

    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

    invertible matrix

    ‣ straight lines, planes

    ‣ distance‣ circles, spheres‣ angles‣ area

    ‣ parallelism

    • Non-invariant notions

    ‣ additions p1 + 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

    ‣ dot product p1 · 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Geometry in vector space

    p̃= 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

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions8<:p=

    24

    xyz

    35 2 R39=;

    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

    invertible matrix

    ‣ straight lines, planes

    ‣ distance‣ circles, spheres‣ angles‣ area

    ‣ parallelism

    • Non-invariant notions

    ‣ additions p1 + 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Geometry in affine space

    p̃= Ap+ 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

    translationp1+p2

    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

    ‣ midpoint

    ‣ dot product p1 · 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

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions‣ straight lines

    ‣ intersection‣ point

    ‣ oriented contact

    • Non-invariant notionsLaguerre geometry of oriented circles

    oriented circles in R2AAAU93iczZhbj9vGFceV9JaqNyfZt77QXSxaBKuFqMReG3AC27syFFd7seR1DC8VY0SOJFq87cxIlnbAz9DX5ilvRV/7cQr0w/TMkHNISmKAAHmIAJEzv/85Z+ZwLryMk8Dnot3+7wcf/uKXv/r1bz76bfN3v//DH/905+NPXvF4wVx65cZBzF6PCaeBH9Er4YuAvk4YJeE4oN+M5ydK/2ZJGffj6KVYJ3QUkmnkT3yXCEBXzmDwbeftnf32UVv/rO2CnRf2G/nv8u3Hn/7P8WJ3EdJIuAHh/NpuJ2IkCRO+G9C06Sw4TYg7J1N6DcWIhJSPpO5tah0A8axJzOAfCUvTsockIefrcAyWIREzvqkpuEu7XojJg5H0o2QhaORmDU0WgSViS6VueT6jrgjWUCAu86GvljsjjLgCLlAlEp9wlwTU+1KwBT2E4pduCMdDZQ7GI6ka9yj3p1E12cxtJGc0WFIB2imFC8ToEPKJg2eQrxxDwUvlVWpKYSqjXZZPgmRGxlRIR7WWG2enphPR924chiTypBNHNC0ZSTutdEquJpDjYeS7VBWqGnFdGESeNpsHB9aw130ysJ5evO4OdQucLOk4XklnEsdw3mh2yWeUMMWvO6Pro3ujpmXJAzhYlgM5rHQJkca8HE3ud9KSNuNzP3Hee7lcCIk3UR2pAiogX+avpG3t2xbM0nKoICCJVFmWmysbQARGuYiZiarFjfxmP5P82irDnyY/GOPu+WllnIGpn3V6cfW037WeDp6c/L37EsY/JHNKYD8RMOFxdqp5+YyEfrCWF2evU3kWZfO1m8qdk/0s6hM2zQlPt5zyuT+k4gecspm/6TquLBvlOJyRhO5uQurr8Kh1/yvLQu2ego/ut47L8L6Gx60HZXis4YPWwzJ8oOHDlt3+qoAPAT6y2y27U4J2W9NOC9xLtNMsX7X6DMa1GbSewqXZmYZWduailZ0JaWVnVlrZmZpWdueXSXmSMI/UtI2mAXUeLyKPTuCG5WWcsW2eX5Qz2NRO4eYW+jALpYmQ73ZxQqPSrK/8tibRX+37X6S7aV1reb/y1twg5vRHNHds72oOqFlasZjBysqXn1qWF8+2VmGz6cAlcd68Mfu7fKM2d8UGA2QDw87PkZ0bNhwiGxp2eYns0rBeD1nPsJMTZCeGPX+O7DnaFfFOMOCggIOilQL2EL48ky+du2emdq5q56bWV7W+qb1QtRd5bUzyYBNJTCwIbNgY2WSCcIJwimyKzEfmI3uH7B2yObI5sgWyBbIlsiWy98jeIxsi48hWyFbI1sjWyG6R3SJjyBgyUUBH+IFHlWbECLWocEAmkCXIEmQ3yG4M88Hu0PFjQXJAV3k0Fkooo13kBgv1cArm73Lm+Wq4Dr286sYZUMdvHY9Mp5QZbw9j+h6EzPsTwOOvB1ND3Te4Xnqwc2iGXQ4XVTkshgwe6sINZ43QAIRbymJMuq1aPtDakjDYPwS4Qwlu39m+dte5m/sWuj75wvSZd7Nw3GWya1rivQLikuRnBTxDOCzgEC9EZAZ1HBajKkQBBZrCg+E8Xw+qWKwIVZuVhFlZECWhiNUjIQ4LlHEfWocJclWpNGF8skYKL1Xn6KlVnvk2K9v2RUIZbKnsM+nAjhv6cMPMz3mvPC49bmaYkJ4TzimLWvbRvXBhCSOsNoSVEcZKGRdVpqoMq2tVXWNVaFkU+o2q32A1UdXEVNcbjWKc2w3h1gg3G4KJ7KosYcVE74p9xKMw3yrmDsVl6cFCqGqwEnIpmfkbGhC8grPtsJqhN9/y5ug9JfCMvSFrZgyIevfZMNDMGMBK2u5fBo1Jvg1UTfJ9IF8LKod8L9QXBbPbkVyRm1C5OfAeymforZIr3LeTF0X2btldj1XF21V5bcizUnB3O7irg+9cD/AUAw8wA5rWqD48Hn8d1qkc3m8ln0Z1OhFELTQ4dmpDJCSCt5lUDqFQ3wsyTaVf24+ARlN46Zf97FzXG0ZJKp+oY43FMg5S+QoONTq8tqeyR2t1N0mlm9Spns8hT32ssYAdQapdIRvK7iKgDLdEXTOj/PXwGQpQNrjfR9rvG3jqM9+dBbS4wZ7qvbjyiWBJ2Ziw9NoeyWzqWNnUUYL6diTLE2rfLtfSskPlHfmzbIJmUfMm5L691brvrbRNfpOaaJtNo2COj/WYCLDUiWL9RM43HaaMeNsumpadKj58txP/YS93wYJtJ00rPoyWvTx/ue2kYG07/dMNB3U/79fbzyjn2030FK2/bEkCk9QR6lMQVNQnO6KSy0twK0zf3tm3N7/+bRdedY7sz486L77Yf3yafxn8qPHnxl8af2vYjePG40avcdm4argNv/GPxj8b3+2t977f+9fevzPTDz/IfT5tVH57//k/2hGH7A== ‣ rotate, translate‣ normal offset 


    (propagate wave) ‣ lengths of 
common tangents

  • What is Geometry?Klein’s Erlangen program 1872:

    • Collection of objectsIngredients of geometry:

    • Group of transformations • Invariant notions‣ tangents, normals

    ‣ particle speed‣ particle acceleration

    ‣ curvatures

    • Non-invariant notions

    Differential geometry of curves

    Particle moving 
along a trajectory

    Reparametrizations

  • Differential GeometryWhy do we care?

    Jiang, Wang, Rist, Wallner & Pottmann 2020

    Sageman-Furnas, Chern, Ben-Chen, Vaxman 2019

    Konaković, Crane, Deng, Bouaziz,
Piker & Pauly 2016

    • geometry of surfaces

  • Differential GeometryWhy do we care?

    • geometry of surfaces• mothertongue of physical theories

    Elasticity of lipid bilayer membrane (Helfrich 1973)

    E =Z

    S

    ↵ dA+ �(H � H0)2dA+ �K 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

    mean curvature Gaussian curvature

  • Differential GeometryWhy do we care?

    • geometry of surfaces• mothertongue of physical theories• computation: simulation, processing

    Chern, Knöppel, Pinkall & Schröder 2017 Bergou, Wardetzky, Robinson, 
Audoly & Grinspun 2008

    Stein, Grinspun, Wardetzky & Jacobson 2018

    Solomon, de Goes, Peyré, Cuturi, Butscher, 
Nguyen, Du & Guibas 2015

  • Differential GeometryHow do we apply differential geometry ideas?

    • Surfaces are collections of samples and connectivity• Apply continuous ideas


    BUT: setting is discrete

    • What is the right way?‣ Discrete v.s. Discretized

  • Discretized v.s. Discrete

    modeling
& analysis

    @ u@

    t+r u

    u=�r

    p

    r ·u=

    0

    Invariants,Symmetries

    discretization

    discretedifferential
geometry

    Approximation
theory

  • DiscretizedBuild smooth manifold structure

    • Collection of charts
with compatibility on their overlaps

    • Realize everything as smooth functions

    • Numerical analysis paradigm‣ Polynomial, rational functions‣ Spectral basis

    ‣ Truncation error‣ Stability

  • DiscretizedExample: pendulum equation.

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    m✓̈ = �mg sin✓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

    ✓̇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

    ✓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

    ✓Energy conservation

    ✓Integrable system

    12

    m✓̇ 2 �mg cos✓ = 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

  • DiscretizedExample: pendulum equation.

    m✓̈ = �mg sin✓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

    ✓̇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

    ✓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

    • High order differential equation solver
(4th order Runge–Kutta method)

    Given

    ✓ ⇤i+1/2 = ✓i +�t2

    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

    (✓i , vi = ✓̇i)AAAVEniczZjdjtvGFceVNG1TpW2cZu8KBHQWi6bBaiFuY68N2IHtXRmKq/3wyusYXqqLETmSaPFrZ0aytAPe5RHyFLlNr3oX5LYvUKAP0zNDziEpiQEC5KICRM38/uecmTNfJDVMAp+Ldvs/77z7q/d+/Zvfvv+75ge//8MfP7z10Z9e8njGXHrhxkHMXg0Jp4Ef0Qvhi4C+Shgl4TCgXw+nh0r/ek4Z9+PohVgmdBCSceSPfJcIQFe3PvnMERMqyJW/a82vfOuh5XixMOyvV7e223tt/bHWC3Ze2G7kn7Orjz7+LwRwZyGNhBsQzi/tdiIGkjDhuwFNm86M04S4UzKml1CMSEj5QOpEUmsHiGeNYgbfSFialj0kCTlfhkOwDImY8FVNwU3a5UyM7g2kHyUzQSM3a2g0CywRW2pULM9n1BXBEgrEZT701XInhBFXwNhVIvERd0lAvYeCzeguFB+6IVx3lTkYD6Rq3KPcH0fVZDO3gZzQYE4FaEcUBojRPuQTB08hXzmEgpfKi9SUwlRGmywfB8mEDKmQjmotN85+mk5E37pxGJLIk04c0bRkJO200im5GEGOu5HvUlWoasR1YRJ52mzu7Fj9bufxufXk9FWnr1vgZE6H8UI6oziG35Vm53xCCVP8cn9wuXdn0LQsuQMXy3Igh4UuIdKYl6PJ7f20pE341E+ct14uF0LijVRHqoAKyJf5C2lb27YFq7QcKghIIlWW5ebKBhCBUS5iZqJqcSW/yf9Jfm2V4S+TH8xx5+SoMs/A1Mc6Or140utYT84fH/698wLmPyRTSuCoEbDgcXWqdfmUhH6wlKfHr1J5HGXrtZPKjYv9OOoRNs4JT9ec8rXfp+InnLKVv+o6rGwb5difkIRubkLqcXjQuvulZaF2R8EHd1sHZXhXw4PWvTI80PBe634Z3tPwfstuf1nA+wAf2O2WvV+CdlvT/Ra4l+h+szxq9RkMazNoPYGh2ZiGVjbmopWNCWllY1Za2ZiaVjbnl0l5krCO1LKNxgF1Hs0ij47gXuZlnLF1ng/KMRxqR3DfC31YhdJEyE+7OKFRadVXPmuL6C/23S/SzbSutbxfeWtuEHP6M5o7sDc1B9RsrRjuv8xsP7UtT5+u7cJm04EhcV6/Nue7fK0Od8XOz5GdG3ZyguzEsH4fWd+wszNkZ4Z1u8i6hh0eIjs07NkzZM/Qroh3iAHPC3hetFLALsIXx/KFc/vY1E5U7cTUeqrWM7XnqvY8rw1JHmwkiYkFgQ0bIhuNEI4QjpGNkfnIfGRvkL1BNkU2RTZDNkM2RzZH9hbZW2R9ZBzZAtkC2RLZEtkNshtkDBlDJgroCD/wqNKMGKEWFQ7IBLIEWYLsGtm1YT7Y7Tp+LEgO6CKPxkIJZbSL3GCmnlvB/E3OPF9N166XV904A+r6D8cj4zFlxtvDmL4HIfP+BPBk7MHSUPcNrrcenByaYZfDWVUOiymDh7pwxVkjNADhhrIYk26rlne0NicMzg8B7lCC23d2rt12bue+ha5/fGH6zDtZOO4y2TEt8W4BcUvy4wIeI+wXsI8DEZlJHYbFrApRQIGm8GA4zfeDKhY7QtUmJWFSFkRJKGJ1SYjTAmU8h5ZhglxVKk0Yn6yRwkvVOXpqlWe+zcqxfZpQBkcq+1w6cOKGPtww89+8Vx6XHjcrTEjPCaeURS177044s4QRFivCwghDpQyLKlNVhtWlqi6xKrQsCv1a1a+xmqhqYqrLlUYxzs2KcGOE6xXBRHZVlrBjojfFOeLBS17V3KG4LT3YCFUNdkIuJRN/RQOCIzhZD6sZevM1b47eYwLP2CuyZsaAqHefFQPNjAHspPX+ZdCY5MdA1SQ/B/K9oHLIz0I9KJjdhuSK3ITKzYH3UD5Bb5Vc4b6evCiyd8vueq4q3q7Ka0WelIK768FdHXzjfoCnGHiAOadpjerD4/FXYZ3K4f1W8nFUpxNB1EaD635tiIRE8DaTyj4U6ntBxqn0a/sR0GgML/2yl/3W9YZRksrH6lpjMY+DVL6ES40Or+2p7NJa3U1S6SZ1qudzyFNfayzgRJDqVMimsjMLKMMjUdfMLH/Vf4oClA3u9ZD2egYe+cx3JwEtbrBH+iyu/EUwp2xIWHppD2S2dKxs6ShB/a0kywtq2y7X0rJD5R3582yBZlHzJuS2vda67y20TX6TGmmbVaNgio/1mAiw1Ili/UTOVx3GjHjrLpqWnSo+fLMT/2kvd8aCdSdNKz6Mlr08f77upGBtO72jFQd1P+/V208o5+tNdBWtH7YkgUXqCPVXEFTUX3ZEJZeX4FaYXt3atlf//VsvvNzfs/+2t//8i+1HR/k/g+83/tz4tPFZw24cNB41uo2zxkXDbXzT+K7xfeOfW99u/Wvrh60fM9N338l9Pm5UPlv//h8Y0ZH2

    v⇤i+1/2 = vi ��t2

    sin✓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

    ‣ ✓ ⇤⇤i+1/2 = ✓i +

    �t2

    v⇤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