Discrete Differential GeometrySurfaces

2D/3D Shape Manipulation,3D Printing

CS 6501

Slides from Olga Sorkine, Eitan Grinspun

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Surfaces, Parametric Form

● Continuous surface

● Tangent plane at point p(u,v) is spanned by

n

p(u,v)pu

u

v

pv

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Isoparametric Lines

● Lines on the surface when keeping one parameter fixed

u

v

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Surface Normals

● Surface normal:

Assuming regular parameterization, i.e.,

n

p(u,v)pu

u

v

pv

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Normal Curvature

Direction t in the tangent plane (if pu and pv are orthogonal):

t

n

p

pupv

t

Tangent plane

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Normal Curvature

n

p

pupv

t

The curve is the intersection of the surface with the plane through n and t.

Normal curvature: n() = ((p))

t

Tangent plane

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Surface Curvatures

● Principal curvatures Maximal curvature Minimal curvature

● Mean curvature

● Gaussian curvature

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Principal Directions

● Principal directions:tangent vectorscorresponding to max and min

min curvature max curvature

tangent plane

t1 max

t2

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Euler’s Theorem: Planes of principal curvature are orthogonaland independent of parameterization.

Principal Directions

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Principal Directions

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Mean Curvature

● Intuition for mean curvature

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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

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Local Surface Shape By Curvatures

Isotropic:all directions are principal directions

spherical (umbilical) planar

K > 0, 1= 2

Anisotropic:2 distinct principal directions

elliptic parabolic hyperbolic

2 > 0, 1 > 02 = 0

1 > 0

2 < 0

1 > 0

K > 0 K = 0 K < 0

K = 0

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Gauss-Bonnet Theorem

● For a closed surface M:

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Gauss-Bonnet Theorem

● For a closed surface M:

● Compare with planar curves:

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Fundamental Forms

• First fundamental form

• Second fundamental form

• Together, they define a surface (given some compatibility conditions)

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Fundamental Forms

• I and II allow to measure– length, angles, area, curvature– arc element

– area element

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Intrinsic Geometry

• Properties of the surface that only depend on the first fundamental form– length– angles– Gaussian curvature (Theorema

Egregium)

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Laplace Operator

Laplaceoperator

gradientoperator

2nd partialderivatives

Cartesiancoordinatesdivergence

operatorfunction inEuclidean space

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Laplace-Beltrami Operator

● Extension of Laplace to functions on manifolds

Laplace-Beltrami

gradientoperator

divergenceoperator

function onsurface M

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Laplace-Beltrami Operator

mean curvature

unitsurfacenormal

Laplace-Beltrami

gradientoperator

divergenceoperator

function onsurface M

● For coordinate functions:

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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

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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

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Discrete Laplace-Beltrami

● Uniform discretization: L(v) or ∆v

● Depends only on connectivity= simple and efficient

● Bad approximation for irregular triangulations

vi

vj

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Discrete Laplace-Beltrami

● Intuition for uniform discretization

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Discrete Laplace-Beltrami

● Intuition for uniform discretizationvi vi+1

vi-1

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Discrete Laplace-Beltrami

vi

vj1 vj2

vj3

vj4vj5

vj6

● Intuition for uniform discretization

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Discrete Laplace-Beltrami

● Cotangent formula

Aivi

vi

vj vj

ij

ij

vi

vj

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Voronoi Vertex Area

● Unfold the triangle flap onto the plane (without distortion)

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θvi

vj

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Voronoi Vertex Area

θvi

cjvj

cj+1

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Flattened flapvi

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Discrete Laplace-Beltrami

● Cotangent formula

Accounts for mesh geometry

Potentially negative/infinite weights

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Discrete Laplace-Beltrami

● Cotangent formula

Can be derived using linear Finite Elements

Nice property: gives zero for planar 1-rings!

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Discrete Laplace-Beltrami

● Uniform Laplacian Lu(vi)● Cotangent Laplacian Lc(vi)

● Mean curvature normal

vi

vj

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Discrete Laplace-Beltrami

vi

vj

● Uniform Laplacian Lu(vi)● Cotangent Laplacian Lc(vi)

● Mean curvature normal

● For nearly equal edge lengthsUniform ≈ Cotangent

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Discrete Laplace-Beltrami

vi

vj

● Uniform Laplacian Lu(vi)● Cotangent Laplacian Lc(vi)

● Mean curvature normal

● For nearly equal edge lengthsUniform ≈ Cotangent

Cotan Laplacian allows computing discrete normal

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Discrete Curvatures

● Mean curvature (sign defined according to normal)

● Gaussian curvature

● Principal curvatures

Ai

j

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Discrete Gauss-Bonnet Theorem

● Total Gaussian curvature is fixed for a given topology

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Example: Discrete Mean Curvature

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Links and Literature

● M. Meyer, M. Desbrun, P. Schroeder, A. BarrDiscrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath, 2002

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● P. Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Code

● http://www-sop.inria.fr/geometrica/team/Pierre.Alliez/demos/curvature/

principal directions

Links and Literature

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Measuring Surface Smoothness

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Links and Literature

● Grinspun et al.:Computing discrete shape operators on general meshes, Eurographics 2006

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Reflection Lines as an Inspection Tool

● Shape optimization using reflection lines

E. Tosun, Y. I. Gingold, J. Reisman, D. ZorinSymposium on Geometry Processing 2007

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● Shape optimization using reflection lines

E. Tosun, Y. I. Gingold, J. Reisman, D. ZorinSymposium on Geometry Processing 2007

Reflection Lines as an Inspection Tool

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Thank You