05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo

Transcription

1 05 - Surfaces Acknowledgements: Olga Sorkine-Hornung

2 Reminder Curves

3 Turning Number Theorem Continuous world Discrete world k:

4 Curvature is scale dependent is scale-independent

5 Discrete Curvature Integrated Quantity! Cannot view α i as pointwise curvature α 1 α 2 It is integrated curvature over a local area associated with vertex i

6 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1

7 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1 A 2

8 Discrete Curvature Integrated Quantity! Integrated over a local area associated with vertex i α 1 α 2 A 1 A 2 The vertex areas A i form a covering of the curve. They are pairwise disjoint (except endpoints).

9 Structure Preservation Arbitrary discrete curve discrete analogue of continuous theorem total signed curvature obeys discrete turning number theorem even coarse mesh (curve) which continuous theorems to preserve? that depends on the application

10 Recap Structure- preservation Convergence For an arbitrary (even coarse) discrete curve, the discrete measure of curvature obeys the discrete turning number theorem. In the limit of a refinement sequence, discrete measures of length and curvature agree with continuous measures.

11 Surfaces

12 Surfaces, Parametric Form Continuous surface n p u p v p(u,v) Tangent plane at point p(u,v) is spanned by v u These vectors don t have to be orthogonal

13 Isoparametric Lines Lines on the surface when keeping one parameter fixed v u

14 Surface Normals Surface normal: n p u p v p(u,v) Assuming regular parameterization, i.e., v u

15 Normal Curvature n p u p v t p Unit-length direction t in the tangent plane (if p u and p v are orthogonal): ϕ t Tangent plane

16 Normal Curvature n The curve γ is the intersection of the surface with the plane p u p v through n and t. γ t p Normal curvature: κ n (ϕ) = κ(γ(p)) ϕ t Tangent plane

17 Surface Curvatures Principal curvatures Minimal curvature Maximal curvature Mean curvature Gaussian curvature

18 Principal Directions Principal directions: tangent vectors corresponding to ϕ and ϕ max min t 2 t 1 ϕ min tangent plane min curvature max curvature Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Lévy, and Mathieu Desbrun Anisotropic polygonal remeshing. ACM Trans. Graph. 22, 3 (July 2003), DOI:

19 Principal Directions Euler s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. By Eric Gaba (Sting) - Based upon a drawing in a book, CC BY-SA 3.0,

20 Principal Directions

21 Mean Curvature Intuition for mean curvature

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

23 Local Surface Shape By Curvatures K > 0, κ 1 = κ 2 K = 0 Isotropic: all directions are principal directions spherical (umbilical) planar K > 0 K = 0 K < 0 Anisotropic: 2 distinct κ 1 = 0 κ 1 < 0 principal κ 1 > 0, κ 2 > 0 κ 2 > 0 κ 2 > 0 directions elliptic parabolic hyperbolic

24 Usage example: define fair surfaces FiberMesh s.t. M interpolates the curves Andrew Nealen, Takeo Igarashi, Olga Sorkine and Marc Alexa, "FiberMesh: Designing Freeform Surfaces with 3D Curves", ACM Transactions on Computer Graphics, ACM SIGGRAPH 2007, San Diego, USA, 2007

25 Gauss-Bonnet Theorem For a closed surface M:

26 Gauss-Bonnet Theorem For a closed surface M: Compare with planar curves:

27 Fundamental Forms First fundamental form Second fundamental form Together, they define a surface (if some compatibility conditions hold)

28 Fundamental Forms I allows to measure length, angles, area, curvature arc element area element

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

30 Interesting examples Hyperbolic geometry spaces with constant negative Gauss curvature Constant mean curvature surfaces Minimal surfaces (H = 0) Lobachevsky plane Soap surfaces (minimal surfaces) By Anders Sandberg - Own work, CC BY-SA 3.0,

31 Laplace Operator Laplace operator gradient operator 2nd partial derivatives Intuitive Explanation The Laplacian Δf(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the function in Cartesian radius of the sphere grows. Euclidean space divergence coordinates operator

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

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

34 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

35 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 (v) = k-ring neighborhood k

36 Discrete Laplace-Beltrami Uniform discretization: v i Depends only on connectivity = simple and efficient Bad approximation for irregular triangulations v j

37 Discrete Laplace-Beltrami Intuition for uniform discretization

38 Discrete Laplace-Beltrami Intuition for uniform discretization v i-1 v i v i+1 γ

39 Discrete Laplace-Beltrami v j1 v j2 v j6 v i v j3 v j5 v j4

40 Discrete Laplace-Beltrami Cotangent formula v i v i A i α ij v i v j v j v j β ij

41 Voronoi Vertex Area v i θ v j Unfold the triangle flap onto the plane (without distortion)

42 Voronoi Vertex Area v i v i Flattened flap θ c j+1 v j c j

43 Discrete Laplace-Beltrami Cotangent formula Accounts for mesh geometry Potentially negative/ infinite weights

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

45 Discrete Laplace-Beltrami v i Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Normal β α v j

46 Discrete Laplace-Beltrami v i Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Normal α β For nearly equal edge lengths Uniform Cotangent v j

47 Discrete Laplace-Beltrami v i Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Normal α β For nearly equal edge lengths Uniform Cotangent v j Cotan Laplacian allows computing discrete normal

48 Discrete Curvatures Mean curvature (sign defined according to normal) Gaussian curvature A i θ j Principal curvatures

49 Discrete Gauss-Bonnet Theorem Total Gaussian curvature is fixed for a given topology

50 Example: Discrete Mean Curvature

51 Links and Literature M. Meyer, M. Desbrun, P. Schroeder, A. Barr Discrete Differential-Geometry Operators for Triangulated 2- Manifolds, VisMath, 2002

52 Links and Literature Discrete Differential Geometry: An Applied Introduction Keenan Crane

53 Links and Literature libigl implements many discrete differential operators See the tutorial! Principal Directions

54 Thank you