CS 523: Computer Graphics, Spring Shape Modeling. Differential Geometry of Surfaces

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1 CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, /22/2011

2 Differential Geometry of Surfaces Continuous and Discrete

3 Motivation Smoothness Mesh smoothing Adaptive tessellation Mesh decimation Shape preserving mesh editing

4 Surfaces Parametric form Continuous surface x( u,v) p( u,v) = y( u,v), ( u, v) R ( ) z u,v Tangent plane at point p(u,v) is spanned by 2 p u v n p v p(u,v) p u = p( u,v) p( u,v), pv = u v u

5 Surfaces Isoparametric lines Lines on the surface when keeping one parameter fixed γ γ u v 0 0 ( v) ( u) = = p( u 0 p( u, v, v) 0 ) v u

6 Surfaces Surface normal: n n( u,v) = p p u u p p v v p u p v p(u,v) Assuming regular parameterization, i.e., v p u p v 0 u

7 Normal curvature p u n p v n = p p u u p p v v t p Direction t in the tangent plane: t p u = cos φ + p u sinφ p p v v p v / p v ϕ t p u / p u

8 Normal curvature p u n p v The curve γis the intersection of the surface with the plane through nand t. γ t p Normal curvature: κ(γ(p)) p v / p v ϕ t p u / p u

9 Surface curvatures Principal curvatures Maximal curvature Minimal curvature Mean curvature Gaussian curvature

10 Mean curvature Intuition for mean curvature

11 Surface curvatures

12 Classification A point pon the surface is called Elliptic, if K > 0 Parabolic, if K = 0 Hyperbolic, if K < 0 Umbilical, if Developable surface K = 0

13 Laplace operator

14 Laplace-Beltrami operator Extension of Laplace to functions on manifolds

15 Laplace-Beltrami operator Extension of Laplace to functions on manifolds

16 Discrete differential operators Assumption: meshes are piecewise linear approximations of smooth surfaces Approach: approximate differential properties at point xas spatial average over local mesh neighborhood N(x)where typically x= mesh vertex N k (x) = k-ring neighborhood or local geodesic ball

17 Discrete Laplace-Beltrami Uniform discretization -L(v) or v Depends only on connectivity = simple and efficient v Bad approximation for irregular triangulations v i

18 Discrete Laplace-Beltrami Intuition for uniform discretization v i-1 v i v i+1 γ H 2π = 0 κ( θ ) dθ κ = && γ && γ v v v v = v + v v ( ) ( ) i 1 i i i+ 1 i 1 i+ 1 2 i

19 Discrete Laplace-Beltrami Intuition for uniform discretization v j1 v j2 v j6 v i v j3 v j5 v j4 H 2π = 0 κ( θ ) dθ v + v v + j1 j4 2 i j2 j5 2 i j3 j6 2 i 6 v + v v + v + v v = = vj 6 vi = L( vi ) k= 1 k Andrew Nealen, Rutgers, /22/2011

20 Discrete Laplace-Beltrami Cotangent formula

21 Discrete Laplace-Beltrami Cotangent formula Problems Potentially negative weights Depends on geometry

22 Discrete Laplace-Beltrami α v i β Laplacian operators Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Mean curvature normal v j A i Andrew Nealen, Rutgers, /22/2011

23 Discrete Laplace-Beltrami α A i v i β v j Laplacian operators Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Mean curvature normal Cotangent Laplacian= mean curvature normal x vertex area (A i ) For nearly equal edge lengths Uniform Cotangent Andrew Nealen, Rutgers, /22/2011

24 Discrete Laplace-Beltrami α v i A i β v j Laplacian operators Uniform Laplacian L u (v i ) Cotangent Laplacian L c (v i ) Mean curvature normal Cotangent Laplacian= mean curvature normal x vertex area (A i ) For nearly equal edge lengths Uniform Cotangent Andrew Nealen, Rutgers, /22/2011

25 Discrete curvatures Mean curvature Gaussian curvature Principal curvatures

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

27 Links and literature P. Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Code a/team/pierre.alliez/d emos/curvature/

28 Links and literature Grinspunet al.: Computing discrete shape operators on general meshes, Eurographics 2006