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Introduction to Differential Geometry
Ron Kimmel
www.cs.technion.ac.il/~ron
Computer Science Department Technion-Israel Institute of Technology
Geometric Image Processing Lab
Planar Curves C(p)={x(p),y(p)}, p [0,1]
y
x
C(0)
C(0.1) C(0.2)
C(0.4)
C(0.7)
C(0.95)
C(0.9)
C(0.8)
pC =tangent
Arc-length and Curvature
s(p)= | |dp 0
p
| | 1,sC pC
1
C
nCss
nCss
|| p
ps C
CCt
Linear Transformations
Euclidean:
Affine:
.1, and 0, where,,
,},{}~,~{
2121
ii
TT
uuuuuuA
byxAyx
Linear Transformations
Equi-Affine: .1)det( ,},{}~,~{ AbyxAyx TT
Differential Signatures
Euclidean invariant signature )}(,{ ss
s
Differential Signatures
Euclidean invariant signature )}(,{ ss
s
Differential Signatures
Euclidean invariant signature
s
)}(),({ ss s
Cartan Theorem
Differential Signatures
~Affine
~Affine
Image transformation
)),(),,((),( 2112 yxTyxTIyxI
Affine:
Equi-affine:
f
e
y
x
dc
ba
yxT
yxT
),(
),(
2
1
1det
dc
ba
Invariant arclength should be
1. Re-parameterization invariant
2. Invariant under the group of transformations
drCCCFdpCCCFw rrrppp ,...,,,...,,
Geometric measure
Transform
Euclidean arclength
Length is preserved, thus ,
dpCdpdydxdp
dpdydxds pdp
dydp
dx 222222
ds dy
dx
1sC
dpCs p
L
ppp dsdpCCdpCL0
1
0
1
0
21
,Length Total
Euclidean arclength
Length is preserved, thus
pC
dpCCs pp
21
,
1, ss CC
dpCds p
re-parameterizationinvariance
1Area
Equi-affine arclength
Area is preserved, thus
vC
vvC
dpCCv ppp
31
,
1, vvv CC
dsdsCCv sss
31
31
,
dsdv 31
re-parameterizationinvariance
Equi-affine curvature
is the affine invariant curvature
vvvvvvvv
vvvv
vvvvvvvv
vvvdvd
vvv
CCCC
CC
CCCC
CCCC
0,
0,,
0, 1,
Differential Signatures
Equi-affine invariant signature )}(,{ vv
v
From curves to surfaces
Its all about invariant measures…
Surfaces
Topology (Klein Bottle)
Surface
A surface, For example, in 3D
Normal
Area element Total area
2 M: 2 nS nR
),(),,(),,(),( vuzvuyvuxvuS
vu
vu
SS
SSN
N
uS
vS
u vdA S S dudv
dudvSSA vu
Example: Surface as graph of function
A surface, 32: RR S
),(,,),( vuzvyuxvuS
z
x
y
N
xS
yS
Curves on Surfaces: The Geodesic Curvature
NNCCN ssssg
,ˆ
ssC
N
Curves on Surfaces: The Geodesic Curvature
ssC
N
n)(min
)(max
2
1
Principle Curvatures
221
HMean Curvature
21KGaussian Curvature
NCssn
,
Normal Curvature
Gauss
Geometric measures
Curvature , normal , tangent , arc-length s Mean curvature H Gaussian curvature K principle curvatures geodesic curvature normal curvature tangent plane
www.cs.technion.ac.il/~ron
21,
g
pT
t
n
n
t
n
ssCn