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University ofUniversity ofBritish ColumbiaBritish Columbia
Differential Geometry & Discrete Operators
University ofUniversity ofBritish ColumbiaBritish Columbia
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
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University ofUniversity ofBritish ColumbiaBritish Columbia
( )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
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
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
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
Principal Directions
min curvaturemin curvature max curvaturemax curvature
University ofUniversity ofBritish ColumbiaBritish Columbia
Curvature
Typical measures:Gaussian curvature
Mean curvature
maxminkkK =
2maxmin kkH +
=
5
University ofUniversity ofBritish ColumbiaBritish Columbia
Curvature on Mesh
Approximate curvature of (unknown) underlying surface
Continuous approximationApproximate the surface & compute continuous differential measures
Discrete approximationApproximate differential measures for mesh
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
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
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
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
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
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 ∇
≈=)(
University ofUniversity ofBritish ColumbiaBritish Columbia
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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University ofUniversity ofBritish ColumbiaBritish Columbia
Examples: Mean Curvature
University ofUniversity ofBritish ColumbiaBritish Columbia
Some Applications
Remeshing
Viewpoint selection
Ridges and Valleys