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by: Aliyah Boyer

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# SPATIAL & MVG OBJ DB CIS 4930

Aliyah Boyer
UF
GPA 3.58

Jorg Peters

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COURSE
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Jorg Peters
TYPE
Class Notes
PAGES
2
WORDS
KARMA
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## Popular in Comm Sciences and Disorders

This 2 page Class Notes was uploaded by Aliyah Boyer on Friday September 18, 2015. The Class Notes belongs to CIS 4930 at University of Florida taught by Jorg Peters in Fall. Since its upload, it has received 15 views. For similar materials see /class/207030/cis-4930-university-of-florida in Comm Sciences and Disorders at University of Florida.

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Date Created: 09/18/15
Jorg s Graphics Lecture Notes 7 Discrete Quadratic Curvature Energies 1 Discrete Quadratic Curvature Energies Motivation thin shell surface bending and stretching as in cloth dynamics also think skin paper etc also 77fairing77 Willmore energy Dynamics For a surface at position xu vt E R3 moving in a velocity eld Vu v t E R3 a mass matrix 6 R3 the Hessian with respect to x denoted by Hess and the energies7 E5 Ep 6 R Newton7s formula for moving the surface S with position x subject to the forces FE and FD is x z I o v t v8 l0 erillFEltxltrgtgtlipltxltrgtgtl 1 FD I 7a1M agHessEb agHessEpV 2 FE 7VE1 7 V13 3 Here the suscript D stands for damping E for energy and b for bending and p for stretching The decisive ingredient in the motion of S is the de nition of the bending energy Ebl The equations may be solved by semiimplicit Euler stepsl Energies The stretching energy Ep penalizes stretching of the surface This is captured by looking at the rst fundamental form1 lli Since E1 is assumed in the model to be at least two orders of magnitude larger than Eb we may assume that the deformation minimizes E1 and is therefore almost isometric ll does not change i We choose to model bending as an integral of mean crvature H rather than Gauss curvature K whose integral is determined by the GaussBonnet formula or total curvature H H3 where H1 and H2 are the principle curvatures l Eb SH2dA Hz H1H2 4 1AXAXRSdA 5 2 5 Here Ax is the LaplaceBeltrami operator ie A V V where the differenti ation gradient is on the surface intrinsic rather than in the ambient space 3i If ll is constant E5 is quadratic in x Discretization The solution of the Dirichlet Problem AI 07Iasg where 85 is the boundary of S and I could be one coordinate of x is a minimizer of the Dirchlet energy Dz I SVzVzdA 6 D is D1 nonegative D2 zero iff zconst D3 scaleinvariant ifz is bivariatei R The area A scales by A2 and V by A 1 if z is scaled by A E 1 webesearch notions you do not know Jorg s Graphics Lecture Notes 7 Discrete Quadratic Curvature Energies 2 We discretize D and z With k samples in each parameter by the quadratic form L 6 R16 for Laplace as Dz m duu utLul 7 Where u E Rk is the vector of samples of 1 To match D17D27D3 L should be Ll symmetric positive semide nite7 L2 vanish exactly When u is constant and be L3 invariant under uniform scalingl note that du e7u e du7 u de7 e 2due and hence for dee 0 L27 due 0 must hold since otherwise due7 ue du7 u adu7 e lt 0 for suitably chosen a E R contradicting nonegativity of d We add L4 Lx 0 if x is part of a planet The solution of the Poisson Problem AI f7 9535 0 is minimized by the continuous analogue of duv utLv ftMV 8 With the mass matrix M providing a scaling of the inner product of f and vi Then M should be Ml symmetric positive definite7 M2 scale by A2 M3 M1 Al Then MilL satis es L17L2 and scales like bivariate Laplace Beltramil Finite Elements For a basis gym Lmn mendA MW someondA 9 S S Linear basis elements can for example be hot functions vertexbased Lagrange functions or CrouzeixRaviart functions midedge connected7 nonconforming Exercise check that MS holds for hat functions7 ilel crytically7 explained in class f100010dA Em Mm Al 6 check for the triangle A0mn With opening angle a and neighbor triangle acros m7 n With angle that l Lmn 7 c0ta cot 10 Abbreviate k I 7 0 t n cosa 10 ill a 11 T T sina

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