Curves - University of British Columbiasheffa/dgp/ppts/DifGeom1.pdf · 3 University of British Columbia Curvature Normal curvatureof surface is defined for each tangential direction

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University ofUniversity ofBritish ColumbiaBritish Columbia

Differential Geometry & Discrete Operators


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

Unit length tangent vector

Curvature

Curves

( ) ( ) ( )[ ]T C tdC t

dtx t y t= ′ = = ′ ′

( ),

( ) ( ) ( )[ ]( ) ( )

T C tx t y t

x t y t

→ →= =

′ ′

′ + ′

,2 2

( )T dC tdt

=

( )C t

2/322

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

⎟⎠⎞⎜

⎝⎛ ′+′

−=

⎟⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛ tytx

txtytytxtk

2


( )tC

Curvature (Curves)

Curvature is independentof parameterization

C(t), C(t+5), C(2t) have same curvature (at corresponding locations)

Corresponds to radius of osculating circle R=1/k

Measure curve bending

R=1/k


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

Normal to surface

perpendicular to tangent planeAny vector in tangent plane is tangential to S(u,v)

Surfaces

( )∂∂

S u vv,

∂∂

S u vu

( , ) ∂∂

S u vv

( , )

nSu

Sv

→= ×

∂∂

∂∂

( )S u v,( )∂∂

S u vu,

rn

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Curvature

Normal curvature of surface is defined for each tangential direction

Principal curvatures Kmin & Kmax: maximum and minimum of normal curvature

Correspond to two orthogonal tangent directions

Principal directions Not necessarily partial derivative directionsIndependent of parameterization


3D Curvature

IsotropicIsotropic

Equal in all directionsEqual in all directions

spherical planar

kkminmin==kkmaxmax > 0> 0kkminmin==kkmaxmax = 0= 0

AnisotropicAnisotropic

2 distinct principal directions

elliptic parabolic hyperbolic

kkmaxmax > 0> 0

kkminmin > 0> 0

kmin = 0

kkmaxmax > 0> 0

kkminmin < 0< 0

kkmaxmax > 0> 0

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Principal Directions

min curvaturemin curvature max curvaturemax curvature


Curvature

Typical measures:Gaussian curvature

Mean curvature

maxminkkK =

2maxmin kkH +

=

5


Curvature on Mesh

Approximate curvature of (unknown) underlying surface

Continuous approximationApproximate the surface & compute continuous differential measures

Discrete approximationApproximate differential measures for mesh


Normal Estimation

Need surface normal to construct approximate surface

Defined for each face Solution 1: average face normals

Does not reflect face “influence”Solution 2: weighed average of face normals

Weights:Face areasAngles at vertex

What happens at creases?

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Curvature Estimate: Polyhedral Curvature

Treat mesh as polyhedron with rounded corners with infinitesimally small radiusDerive discrete properties from integrals across vertex region

Convention – associate half of each edge & 1/3 of triangle with vertex


Mean Curvature

Integral of curvature on circular arc β - central angle

On cylindrical parts H= kmax/2 (kmin=0)On planar faces H=0

βππβ

===∫ RR

arclengthR

k 22

11β R

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Mean Curvature

For entire vertex region

Mean curvature at vertex (Ai triangle area)

||||412/||||2/ i

iii

ii eeH ∑∑∫ == ββ

||||4

3i

ii

i

eA

H ∑∑= β

β R


Gaussian Curvature

Use Gauss-Bonnet Theorem

sum of exterior (jump) angles of polygon around v = sum of face angles at v

Curvature at vertex

Note (Gauss-Bonnet for closed surfaces) –Integral Gaussian curvature ≈ π genus

∑∫∑∫ −=−−=∂

∂i

iT

Ti

iT

kK γπγπ 22

∑∑−

=

i

ii

AK

)2(3 απ

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Mean Curvature – Another View

Mean Curvature Flow

Area gradient (per triangle):

Integrate (sum) on all triangles

To get K (and normal) divide by “control”area

AAknxK ∇

≈=)(


Curvature – Practicalities Example: Gaussian

Approximation always results in some noiseSolution

Truncate extreme valuesCan come for instance from division by very small area

Smooth More later

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Examples: Mean Curvature


Some Applications

Remeshing

Viewpoint selection

Ridges and Valleys

Documents

Curves - University of British Columbiasheffa/dgp/ppts/DifGeom1.pdf · 3 University of British Columbia Curvature Normal curvatureof surface is defined for each tangential direction