Discrete Differential Geometry - PowerPoint Presentation

Slide1

Discrete Differential GeometrySurfaces

2D/3D Shape Manipulation,3D Printing

CS 6501

Slides from Olga

Sorkine

,

Eitan

GrinspunSlide2

Surfaces, Parametric Form

Continuous surface

Tangent plane at point p(

u,v

)

is spanned by

n

p

(

u,v

)

p

u

u

v

p

v

2Slide3

Isoparametric Lines

Lines on the surface when keeping one parameter fixed

u

v

3Slide4

Surface Normals

Surface normal:

Assuming regular parameterization, i.e.,

n

p

(

u,v

)

p

u

u

v

p

v

4Slide5

Normal Curvature

Direction

t in the tangent plane (if pu and

p

v

are orthogonal):

t



n

p

p

u

p

v

t

Tangent plane

5Slide6

Normal Curvature

n

p

p

u

p

v

t



The curve

 is the intersection

of the surface with the plane

through

n

and

t

.

Normal curvature:



n() = ((p))tTangent plane6Slide7

Surface Curvatures

Principal curvaturesMaximal curvatureMinimal curvature

Mean curvatureGaussian curvature

7Slide8

Principal Directions

Principal directions:tangent vectorscorresponding to

max and min

min curvature

max curvature

tangent plane

t

1



max

t2

8Slide9

Euler

’

s Theorem:

Planes of principal curvature are

orthogonal

and independent of parameterization.

Principal Directions

9Slide10

Principal Directions

10Slide11

Mean Curvature

Intuition for mean curvature

11Slide12

Classification

A point

p on the surface is calledElliptic, if K

> 0

Parabolic, if

K

= 0Hyperbolic, if K < 0Umbilical, ifDevelopable surface iff K = 0

12Slide13

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 > 02 = 01 > 02 < 01 > 0

K > 0

K = 0

K < 0K = 0

13Slide14

Gauss-Bonnet Theorem

For a closed surface M:

14Slide15

Gauss-Bonnet Theorem

For a closed surface M:

Compare with planar curves:

15Slide16

Fundamental Forms

First fundamental form

Second fundamental form

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

16Slide17

Fundamental Forms

I

and II allow to measurelength, angles, area, curvature

arc element

area element

17Slide18

Intrinsic Geometry

Properties of the surface that only depend on the first fundamental form

lengthanglesGaussian curvature (Theorema

Egregium

)

18Slide19

Laplace Operator

Laplaceoperator

gradientoperator

2nd partial

derivatives

Cartesian

coordinatesdivergenceoperatorfunction inEuclidean space

19Slide20

Laplace-Beltrami Operator

Extension of Laplace to functions on manifolds

Laplace-Beltrami

gradient

operator

divergence

operatorfunction onsurface M

20Slide21

Laplace-Beltrami Operator

mean curvature

unit

surface

normal

Laplace-

Beltramigradientoperator

divergenceoperatorfunction onsurface M

For coordinate functions:

21Slide22

Differential Geometry on Meshes

Assumption: meshes are piecewise linear approximations of smooth surfacesCan 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 prone22Slide23

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

23Slide24

Discrete Laplace-Beltrami

Uniform discretization: L

(v) or

∆

v

Depends only on connectivity

= simple and efficientBad approximation for irregular triangulations

v

i

v

j

24Slide25

Discrete Laplace-Beltrami

Intuition for uniform discretization

25Slide26

Discrete Laplace-Beltrami

Intuition for uniform discretization

v

i

v

i

+1

v

i

-1



26Slide27

Discrete Laplace-Beltrami

v

i

v

j

1

v

j

2

v

j

3

v

j

4

v

j

5

v

j6

Intuition for uniform discretization

27Slide28

Discrete Laplace-Beltrami

Cotangent formula

A

i

v

i

v

i

v

j

v

j



ij



ij

v

i

v

j28Slide29

Voronoi

Vertex Area

Unfold the triangle flap onto the plane (without distortion)

29

θ

v

i

v

jSlide30

Voronoi

Vertex Area

θ

v

i

c

j

v

j

c

j

+1

30

Flattened flap

v

iSlide31

Discrete Laplace-Beltrami

Cotangent formulaAccounts for mesh

geometryPotentially negative/infinite weights

31Slide32

Discrete Laplace-Beltrami

Cotangent formulaCan be derived using linear Finite Elements

Nice property: gives zero for planar 1-rings!

32Slide33

Discrete Laplace-Beltrami

Uniform

Laplacian

L

u

(

vi)Cotangent Laplacian Lc(vi

)Mean curvature normal

v

i

v

j

a

b

33Slide34

Discrete Laplace-Beltrami

v

i

v

j

a

b

Uniform

Laplacian

L

u

(

v

i

)

Cotangent

Laplacian

L

c

(vi)Mean curvature normalFor nearly equal edge lengthsUniform ≈ Cotangent34Slide35

Discrete Laplace-Beltrami

v

i

v

j

a

b

Uniform

Laplacian

L

u

(

v

i

)

Cotangent

Laplacian

L

c

(vi)Mean curvature normalFor nearly equal edge lengthsUniform ≈ CotangentCotan Laplacian allows computing discrete normal35Slide36

Discrete Curvatures

Mean curvature (sign defined according to normal)Gaussian curvature

Principal curvatures

A

i



j

36Slide37

Discrete Gauss-Bonnet Theorem

Total Gaussian curvature is fixed for a given topology

37Slide38

Example: Discrete Mean Curvature

38Slide39

Links and Literature

M. Meyer, M. Desbrun, P. Schroeder, A. Barr

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, VisMath, 2002

39Slide40

P.

Alliez, Estimating Curvature Tensors on Triangle Meshes, Source Codehttp://www-sop.inria.fr/geometrica/team/Pierre.Alliez/demos/curvature/

principal directions

Links and Literature

40Slide41

Measuring Surface Smoothness

41Slide42

Links and Literature

Grinspun et al.:Computing

discrete shape operators on general meshes, Eurographics 2006

42Slide43

Reflection Lines as an Inspection Tool

Shape optimization using reflection lines

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

43Slide44

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

44Slide45

Thank You