CS 6501 2D/3D Shape Manipulation,3D Printing Discrete Differential Geometry Surfaces Slides from Olga Sorkine, EitanGrinspun
Surfaces, Parametric Form • Continuous surface • Tangent plane at point p(u,v) is spanned by n pv pu p(u,v) v u
Isoparametric Lines • Lines on the surface when keeping one parameter fixed v u
Surface Normals • Surface normal: • Assuming regular parameterization, i.e., n pv pu 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 The curve is the intersection of the surface with the plane through n and t. Normal curvature: n() = ((p)) n pv pu p t t Tangent plane
Surface Curvatures • Principal curvatures • Maximal curvature • Minimal curvature • Mean curvature • Gaussian curvature
Principal Directions • Principal directions:tangent vectorscorresponding to max and min t2 t1 max min curvature max curvature tangent plane
Principal Directions Euler’s Theorem: Planes of principal curvature are orthogonal and independent of parameterization.
Principal Directions
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 iffK = 0
Isotropic:all directions are principal directions planar spherical (umbilical) K > 0 K = 0 K < 0 2< 0 2= 0 Anisotropic:2 distinct principal directions 2> 0, 1> 0 1> 0 1> 0 elliptic parabolic hyperbolic Local Surface Shape By Curvatures K > 0, 1= 2 K = 0
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)
Fundamental Forms • I and II allow to measure • length, angles, area, curvature • arc element • area element
Intrinsic Geometry • Properties of the surface that only depend on the first fundamental form • length • angles • Gaussian curvature (TheoremaEgregium)
Laplace Operator gradientoperator 2nd partialderivatives Laplaceoperator Cartesiancoordinates function inEuclidean space divergenceoperator
Laplace-Beltrami Operator • Extension of Laplace to functions on manifolds gradientoperator Laplace-Beltrami function onsurface M divergenceoperator
Laplace-Beltrami Operator • For coordinate functions: gradientoperator mean curvature Laplace-Beltrami unitsurfacenormal function onsurface M divergenceoperator
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 vi ij vi Ai vi ij vj vj vj
Voronoi Vertex Area • Unfold the triangle flap onto the plane (without distortion) vi θ vj
Voronoi Vertex Area Flattened flap vi vi cj+1 θ cj vj
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 curvaturenormal • FornearlyequaledgelengthsUniform≈Cotangent b a vj
Discrete Laplace-Beltrami vi • Uniform Laplacian Lu(vi) • Cotangent Laplacian Lc(vi) • Mean curvaturenormal • FornearlyequaledgelengthsUniform≈Cotangent b a vj CotanLaplacian 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. BarrDiscrete 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.:Computingdiscrete shape operators on general meshes, Eurographics 2006
Reflection Lines as an Inspection Tool • Shape optimization using reflection linesE. Tosun, Y. I. Gingold, J. Reisman, D. ZorinSymposium on Geometry Processing 2007
Reflection Lines as an Inspection Tool • Shape optimization using reflection linesE. Tosun, Y. I. Gingold, J. Reisman, D. ZorinSymposium on Geometry Processing 2007
Thank You
