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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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Presentation on theme: "Discrete Differential Geometry Surfaces 2D/3D Shape Manipulation, 3D Printing CS 6501 Slides from Olga Sorkine, Eitan Grinspun."— Presentation transcript:

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

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

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

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

5 # 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

6 # 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

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

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

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

10 # 10

11 # Mean Curvature ● Intuition for mean curvature 11

12 # 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

13 # 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

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

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

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

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

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

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

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

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

22 # 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

23 # 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

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

25 # Discrete Laplace-Beltrami ● Intuition for uniform discretization 25

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

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

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

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

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

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

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

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

34 # 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

35 # 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

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

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

38 # Example: Discrete Mean Curvature 38

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

40 # ● 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

41 # Measuring Surface Smoothness 41

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

43 # 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

44 # ● 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

45 Thank You


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