Discrete Differential Geometry, Technion - Spring 2008 1 Discrete Differential Geometry Discrete Differential Geometry Discrete Differential Geometry, Technion - Spring 2008 2 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) = (t 2 ,t 2 ), t∈[0,∞) C(t) = (cos(t),sin(t)), t∈[0,2π). C(t) = ((t 2 -1)/(t 2 +1),2t/(t 2 +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) Discrete Differential Geometry, Technion - Spring 2008 3 Tangent vector to curve C(t)=(x(t),y(t)) is T=C’(t): In arc-length parameterization: Differential Properties of Curves Tangent Vector () () () ( , ) dC t T C t dt x t y t ′ ′ ′ = = =⎡ ⎤ ⎣ ⎦ () ' 1 T C t = = () dC t T dt = ( ) C t Discrete Differential Geometry, Technion - Spring 2008 4 Curvature is an intrinsic property Independent of parameterization Corresponds to radius of osculating circle R=1/k Measures curve bend Differential Properties of Curves Curvature
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Discrete Differential Geometry, Technion - Spring 2008 1
Discrete Differential Geometry, Technion - Spring 2008 3
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
Discrete Differential Geometry, Technion - Spring 2008 4
Curvature is an intrinsic propertyIndependent of parameterization
Corresponds to radius of osculating circle R=1/k
Measures curve bend
Differential Properties of CurvesCurvature
Discrete Differential Geometry, Technion - Spring 2008 5
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
Discrete Differential Geometry, Technion - Spring 2008 6
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
Discrete Differential Geometry, Technion - Spring 2008 7
Average face normals around vertex
Problem: does not reflect face “influence”
Normal Estimation on MeshOption 1
Discrete Differential Geometry, Technion - Spring 2008 8
Weighed average of face normals around vertex Using face area or angles at vertex
Normal Estimation on MeshOption 2
Discrete Differential Geometry, Technion - Spring 2008 9
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 Estimation on MeshOption 3
Discrete Differential Geometry, Technion - Spring 2008 10
Normal curvature of surface Defined for each tangential direction Normal direction + tangent direction = normal planeIntersection Normal plane + surface = curve Curvature
Differential Properties of SurfacesCurvature
Discrete Differential Geometry, Technion - Spring 2008 11
Principal curvatures Kmin & KmaxMaximum and minimum of normal curvature
Differential Properties of SurfacesCurvature
Discrete Differential Geometry, Technion - Spring 2008 12
Principal curvatures Kmin & KmaxMaximum and minimum of normal curvature
Correspond to two orthogonal tangent directionsPrincipal directions
Not necessarily partial derivative directions
Independent of parameterization
Differential Properties of SurfacesCurvature
Discrete Differential Geometry, Technion - Spring 2008 13
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
Discrete Differential Geometry, Technion - Spring 2008 14
Principal Directions
min min curvaturecurvature
max max curvaturecurvature
Discrete Differential Geometry, Technion - Spring 2008 15
Curvature
Typical measures:Gaussian curvature
Mean curvature
min maxK k k=
2maxmin kkH +
=
Discrete Differential Geometry, Technion - Spring 2008 16
Gauss-Bonnet Theorem
For ANY closed manifold surface with Euler number χ=2-2g
2K =∫ πχ
( )K∫ ( )K=∫ ( )K=∫ 4= π
Discrete Differential Geometry, Technion - Spring 2008 17