Parameteric Curves Discrete Differential Geometry

Discrete Differential Geometry, Technion - Spring 2008 1

Discrete Differential GeometryDiscrete Differential Geometry


Parameteric Curves

C(t)=(x(t),y(t))Trajectory in the plane over “time”.

Examples:C(t) = (t,t), t∈[0,∞) C(t) = (2t,2t), t∈[0,∞) C(t) = (t2,t2), t∈[0,∞)

C(t) = (cos(t),sin(t)), t∈[0,2π).C(t) = ((t2-1)/(t2+1),2t/(t2+1)), t∈(-∞,∞)

x(t)x(t)

y(t)y(t) C(t)C(t)

x(t)x(t)

y(t)y(t)

C(t)C(t)


Tangent vector to curve C(t)=(x(t),y(t)) is T=C’(t):

In arc-length parameterization:

Differential Properties of CurvesTangent Vector

( ) ( ) ( )( ,)dC tT C tdt

x t y t′ ′′= = = ⎡ ⎤⎣ ⎦

( )' 1T C t= =

( )dC tT

dt=

( )C t


Curvature is an intrinsic propertyIndependent of parameterization

Corresponds to radius of osculating circle R=1/k

Measures curve bend

Differential Properties of CurvesCurvature


Curvature of C(t)=(x(t),y(t))

In arc-length parameterization:

( ) ( )( )'( ) ''( ) '( ) ''( )

( ) 3/ 22 2

x t y t y t x tk t

x t y t

−=

′ ′+

( ) ''( )k t C t=

Differential Properties of CurvesCurvature

1( )( )

k tR t

=

( )C t


Tangent plane to S(u,v) spanned by two partials of S

Normal to surfacePerpendicular to tangent plane

Any vector in tangent plane is tangent to S(u,v)

( , ) ( , ),S u v S u vu v

∂ ∂∂ ∂

S Snu v

→ ∂ ∂= ×∂ ∂

( , )S u v

( , )S u vu

∂∂

( , )S u vv

∂∂

n̂

Differential Properties of SurfacesTangent Plane


Average face normals around vertex

Problem: does not reflect face “influence”

Normal Estimation on MeshOption 1


Weighed average of face normals around vertex Using face area or angles at vertex



Estimate tangent plane and take normal to that

Center the dataVertex + neighbors

Compute covariance matrix

Tangent plane Spanned by two eigenvectors of A with largest eigenvalues

Normal Eigenvector with smallest eigenvalueOrientation?

2

2

2

i i i i ii i i

i i i i ii i i

i i i i ii i i

x x y x z

A x y y y z

x z y z z

⎛ ⎞⎜ ⎟⎜ ⎟

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑

← −

← −

← −

( )

( )

( )

i i i

i i i

i i i

x x Avg x

y y Avg y

z z Avg z



Normal curvature of surface Defined for each tangential direction Normal direction + tangent direction = normal planeIntersection Normal plane + surface = curve Curvature

Differential Properties of SurfacesCurvature


Principal curvatures Kmin & KmaxMaximum and minimum of normal curvature



Principal curvatures Kmin & KmaxMaximum and minimum of normal curvature

Correspond to two orthogonal tangent directionsPrincipal directions

Not necessarily partial derivative directions

Independent of parameterization



Surface Curvature Examples

IsotropicIsotropic

Equal in all directionsEqual in all directions

spherical planar

kkminmin=k=kmaxmax > 0> 0kkminmin=k=kmax max = 0= 0

AnisotropicAnisotropic

2 distinct principal directions

elliptic parabolic hyperbolic

kkmax max > 0> 0

kkmin min > 0> 0

kmin = 0

kkmax max > 0> 0

kkmin min < 0< 0

kkmax max > 0> 0


Principal Directions

min min curvaturecurvature

max max curvaturecurvature


Curvature

Typical measures:Gaussian curvature

Mean curvature

min maxK k k=

2maxmin kkH +

=


Gauss-Bonnet Theorem

For ANY closed manifold surface with Euler number χ=2-2g

2K =∫ πχ

( )K∫ ( )K=∫ ( )K=∫ 4= π


Gauss-Bonnet TheoremExample

Spherekmin = kmax = 1/rK = kminkmax = 1/r 2

Manipulate sphereNew positive + negative curvatureCancel out!

22

14 4K rr

= ⋅ =∫ π π


Curvature on Mesh

Approximate curvature of (unknown) underlying surface

Continuous approximationApproximate surfaceCompute continuous differential measures

Discrete approximationApproximate differential measures for mesh


Approximate surface by quadratic

At each mesh vertex - use surrounding trianglesCompute normal at vertex

Typically average face normals

Compute tangent plane

Compute local coordinate system vertex = (0,0,0)

For each neighbor vertex Compute location in local systemRelative to vertex and tangent plane

(0,0,0)(x1,y1,z1)

z1

Quadratic ApproximationLocal coordinates


Find quadratic function approximating vertices

To find coefficients use least squares fit

2 2( , , ) 0F x y z ax bxy cy z= + + − =

2min ( , , )i i ii

F x y z∑

Quadratic ApproximationLeast squares fit


To find coefficients use least squares fit

Linear system

Least-squares approximation is:

( )22 2min i i i i ii

ax bx y cy z+ + −∑

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

nnnnn z...z

cba

yyxx.........yyxx 1

22

2111

21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

nnnnn z...z

b ,cba

X ,yyxx.........yyxx

A1

22

2111

21

bA)AA(X TT 1−≈

=

Quadratic ApproximationLeast squares fit


Given quadratic surface F, its principal curvatures kmin and kmax are real roots of:

Mean curvature: H = (kmin + kmax)/2

Gaussian curvature: K = kmin kmax

2 2( ) 0k a c k ac b− + + − =

If If bb=0, then =0, then k k = = aa, , cc..

Quadratic ApproximationCurvature


Discrete ApproximationPolyhedral Curvature

Treat mesh as polyhedron with rounded corners with infinitesimally small radius

Derive discrete properties from integrals across vertex region

Convention – associate half of edge and third of triangle with vertexNormalize surface area to unit


Mean Curvature

Supported on edges only.Integral of curvature on circular arc is central angle

For entire vertex region

Mean curvature at vertex (Ai triangle area)

1 1 22

k arclength RR R

β π βπ

= = =∫

1 1|| || || ||2 2i i i i

i iH e eβ β= =∑ ∑∫

3 || ||2 i i

ii

H eA

β= ∑∑

β R

dihedral angledihedral angle


Gaussian Curvature

Supported on vertices only.Simple estimate:

Weighted estimate:

Gauss-Bonnet theorem holds:

2= −∑v ii

K π α

3 (2 )2

= −∑∑v iii

i

KA

π α

2 2⎡ ⎤= − ==⎢ ⎥⎣ ⎦∑ ∑ ∑v i

v v iK π α πχ


Example: Gaussian Curvature

Approximation always results in some noiseSolution

Truncate extreme valuesCan result from division by very small area

Smooth More later


Example: Mean Curvature

Parameteric Curves Discrete Differential Geometry

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