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Discrete Differential Geometry Surfaces 2D/3D Shape Manipulation, 3D Printing CS 6501 Slides from Olga Sorkine, Eitan Grinspun

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# Surfaces, Parametric Form ● Continuous surface ● Tangent plane at point p(u,v) is spanned by n p(u,v) pupu u v pvpv 2

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# Isoparametric Lines ● Lines on the surface when keeping one parameter fixed u v 3

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# Surface Normals ● Surface normal: Assuming regular parameterization, i.e., n p(u,v) pupu u v pvpv 4

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# Normal Curvature Direction t in the tangent plane (if p u and p v are orthogonal): t n p pupu pvpv t Tangent plane 5

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# Normal Curvature n p pupu pvpv t The curve is the intersection of the surface with the plane through n and t. Normal curvature: n ( ) = ( (p)) t Tangent plane 6

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# Surface Curvatures ● Principal curvatures Maximal curvature Minimal curvature ● Mean curvature ● Gaussian curvature 7

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# Principal Directions ● Principal directions: tangent vectors corresponding to max and min min curvaturemax curvature tangent plane t1t1 max t2t2 8

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# Euler’s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. Principal Directions 9

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# 10

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# Mean Curvature ● Intuition for mean curvature 11

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# Classification ● A point p on the surface is called Elliptic, if K > 0 Parabolic, if K = 0 Hyperbolic, if K < 0 Umbilical, if ● Developable surface iff K = 0 12

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# Local Surface Shape By Curvatures Isotropic: all directions are principal directions spherical (umbilical) planar K > 0, 1 = 2 Anisotropic: 2 distinct principal directions ellipticparabolic hyperbolic 2 > 0, 1 > 0 2 = 0 1 > 0 2 < 0 1 > 0 K > 0 K = 0 K < 0 K = 0 13

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# Gauss-Bonnet Theorem ● For a closed surface M : 14

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# Gauss-Bonnet Theorem ● For a closed surface M : ● Compare with planar curves: 15

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# Fundamental Forms ● First fundamental form ● Second fundamental form ● Together, they define a surface (given some compatibility conditions) 16

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# Fundamental Forms ● I and II allow to measure length, angles, area, curvature arc element area element 17

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# Intrinsic Geometry ● Properties of the surface that only depend on the first fundamental form length angles Gaussian curvature (Theorema Egregium) 18

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# Laplace Operator Laplace operator gradient operator 2nd partial derivatives Cartesian coordinates divergence operator function in Euclidean space 19

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# Laplace-Beltrami Operator ● Extension of Laplace to functions on manifolds Laplace- Beltrami gradient operator divergence operator function on surface M 20

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# Laplace-Beltrami Operator mean curvature unit surface normal Laplace- Beltrami gradient operator divergence operator function on surface M ● For coordinate functions: 21

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# Differential Geometry on Meshes ● Assumption: meshes are piecewise linear approximations of smooth surfaces ● Can try fitting a smooth surface locally (say, a polynomial) and find differential quantities analytically ● But: it is often too slow for interactive setting and error prone 22

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# Discrete Differential Operators ● Approach: approximate differential properties at point v as spatial average over local mesh neighborhood N(v) where typically v = mesh vertex N k (v) = k -ring neighborhood 23

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# Discrete Laplace-Beltrami ● Uniform discretization: L(v) or ∆v ● Depends only on connectivity = simple and efficient ● Bad approximation for irregular triangulations vivi vjvj 24

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# Discrete Laplace-Beltrami ● Intuition for uniform discretization 25

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# Discrete Laplace-Beltrami ● Intuition for uniform discretization vivi v i+1 v i-1 26

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# Discrete Laplace-Beltrami vivi vj1vj1 vj2vj2 vj3vj3 vj4vj4 vj5vj5 vj6vj6 ● Intuition for uniform discretization 27

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# Discrete Laplace-Beltrami ● Cotangent formula AiAi vivi vivi vjvj vjvj ij ij vivi vjvj 28

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# Voronoi Vertex Area ● Unfold the triangle flap onto the plane (without distortion) 29 θ vivi vjvj

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# Voronoi Vertex Area θ vivi cjcj vjvj c j+1 30 Flattened flap vivi

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# Discrete Laplace-Beltrami ● Cotangent formula Accounts for mesh geometry Potentially negative/ infinite weights 31

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# Discrete Laplace-Beltrami ● Cotangent formula Can be derived using linear Finite Elements Nice property: gives zero for planar 1-rings! 32

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# Discrete Laplace-Beltrami ● Uniform Laplacian L u (v i ) ● Cotangent Laplacian L c (v i ) ● Mean curvature normal vivi vjvj 33

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# Discrete Laplace-Beltrami vivi vjvj ● Uniform Laplacian L u (v i ) ● Cotangent Laplacian L c (v i ) ● Mean curvature normal ● For nearly equal edge lengths Uniform ≈ Cotangent 34

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# Discrete Laplace-Beltrami vivi vjvj ● Uniform Laplacian L u (v i ) ● Cotangent Laplacian L c (v i ) ● Mean curvature normal ● For nearly equal edge lengths Uniform ≈ Cotangent Cotan Laplacian allows computing discrete normal 35

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# Discrete Curvatures ● Mean curvature (sign defined according to normal) ● Gaussian curvature ● Principal curvatures AiAi jj 36

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# Discrete Gauss-Bonnet Theorem ● Total Gaussian curvature is fixed for a given topology 37

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# Example: Discrete Mean Curvature 38

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# Links and Literature ● M. Meyer, M. Desbrun, P. Schroeder, A. Barr Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath,

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# ● P. Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Code ● sop.inria.fr/geometrica/team/Pierre.Alliez/demos/cu rvature/ sop.inria.fr/geometrica/team/Pierre.Alliez/demos/cu rvature/ principal directions Links and Literature 40

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# Measuring Surface Smoothness 41

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# Links and Literature ● Grinspun et al.:Computing discrete shape operators on general meshes, Eurographics 2006Computing discrete shape operators on general meshes 42

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# Reflection Lines as an Inspection Tool ● Shape optimization using reflection lines E. Tosun, Y. I. Gingold, J. Reisman, D. Zorin Symposium on Geometry Processing 2007 Shape optimization using reflection lines 43

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# ● Shape optimization using reflection lines E. Tosun, Y. I. Gingold, J. Reisman, D. Zorin Symposium on Geometry Processing 2007 Shape optimization using reflection lines Reflection Lines as an Inspection Tool 44

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