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

Surfaces, Parametric Form Continuous surface Tangent plane at point p(u,v) is spanned by n pu pv p(u,v) v u

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

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

Normal Curvature n pv pu p t Direction t in the tangent plane (if pu and pv are orthogonal): t  Tangent plane

Normal Curvature n() = ((p)) n pv pu p t  The curve  is the intersection of the surface with the plane through n and t. Normal curvature: n() = ((p)) Make video of the normal plane? t  Tangent plane

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

Principal Directions Principal directions: tangent vectors corresponding to max and min t2 t1  max min curvature max curvature tangent plane

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

Principal Directions תמונה

Mean Curvature Intuition for mean curvature

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

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

Gauss-Bonnet Theorem For a closed surface M:

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

Fundamental Forms First fundamental form Second fundamental form Together, they define a surface (given some compatibility conditions) Intuitive explanation for I and II!

Fundamental Forms I and II allow to measure length, angles, area, curvature arc element area element here is just I, maybe add things that II measures…

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

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

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

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

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

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

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

Discrete Laplace-Beltrami Intuition for uniform discretization

Discrete Laplace-Beltrami Intuition for uniform discretization vi-1 vi vi+1 

Discrete Laplace-Beltrami Intuition for uniform discretization vj1 vj2 vi vj6 vj3 vj5 vj4

Discrete Laplace-Beltrami Cotangent formula Ai vi vj ij ij

Voronoi Vertex Area Unfold the triangle flap onto the plane (without distortion) vi θ vj

Voronoi Vertex Area vi vj needs a better picture and definition! θ Flattened flap vi vi θ cj+1 cj vj needs a better picture and definition!

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

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

Discrete Laplace-Beltrami vi Uniform Laplacian Lu(vi) Cotangent Laplacian Lc(vi) Mean curvature normal b a vj

Discrete Laplace-Beltrami vi Uniform Laplacian Lu(vi) Cotangent Laplacian Lc(vi) Mean curvature normal For nearly equal edge lengths Uniform ≈ Cotangent b a vj

Discrete Laplace-Beltrami vi Uniform Laplacian Lu(vi) Cotangent Laplacian Lc(vi) Mean curvature normal For nearly equal edge lengths Uniform ≈ Cotangent b a vj Cotan Laplacian allows computing discrete normal

Discrete Curvatures Mean curvature (sign defined according to normal) Gaussian curvature Principal curvatures Ai j

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

Example: Discrete Mean Curvature

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

Links and Literature P. Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Code http://www-sop.inria.fr/geometrica/team/Pierre.Alliez/demos/curvature/ principal directions

Measuring Surface Smoothness

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

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

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

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