introduction to differential geometry ron kimmel ron computer science department technion-israel...

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