Differential geometry I

Lectu

re 1

©A

lexander &

Mic

hael B

ronste

in

tosca.c

s.technio

n.a

c.il/book

Num

erical geom

etry o

f non-rig

id s

hapes

Sta

nfo

rd U

niv

ers

ity, W

inte

r 2009

2N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Manifolds 2-manifold

Not a manifold

A topolo

gic

al space in w

hic

h e

very

poin

t has a

neig

hborh

ood h

om

eom

orp

hic

to (topological disc) is

calle

d a

n n-dimensional (o

r n-)manifold

Earth is a

n e

xam

ple

of a 2

-manifold

3N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Charts and atlases

Chart

A h

om

eom

orp

his

m

from

a n

eig

hborh

ood of

to is c

alle

d a

chart

A c

olle

ction o

f charts w

hose d

om

ain

s

cover th

e m

anifold

is c

alle

d a

n atlas

4N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Charts and atlases

5N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Smooth manifolds

Giv

en tw

o c

harts and

with o

verlappin

g

dom

ain

s change o

f

coord

inate

s is d

one b

y transition

function

If a

ll transitio

n functions a

re , th

e

manifold

is s

aid

to b

e

A m

anifold

is c

alle

d smooth

6N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Manifolds with boundary

A topolo

gic

al space in w

hic

h e

very

poin

t has a

n o

pen n

eig

hborh

ood

hom

eom

orp

hic

to e

ither

�to

polo

gic

al dis

c ; o

r

�to

polo

gic

al half-d

isc

is c

alle

d a

manifold with boundary

Poin

ts w

ith d

isc-lik

e n

eig

hborh

ood a

re

calle

d interior, d

enote

d b

y

Poin

ts w

ith h

alf-d

isc-lik

e n

eig

hborh

ood

are

calle

d boundary

, denote

d b

y

7N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Embedded surfaces

�Boundaries o

f tangible physical objects

are

tw

o-d

imensio

nal manifolds.

�They resid

e in (are

embedded

into

, are

subspaces

of) the ambient

thre

e-d

imensio

nal Euclidean space.

�Such m

anifold

s a

re c

alle

d embedded surfaces

(or sim

ply

surfaces).

�C

an o

ften b

e d

escribed b

y the m

ap

�is

a parametrizationdomain

.

�th

e m

ap

is a

global parametrization

(embedding) of .

�Sm

ooth

glo

bal para

metriz

ationdoes not always existor is

easy to fin

d.

�Som

etim

es it is

more

convenie

nt to

work

with multiple charts.

8N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Parametrizationof the Earth

9N

um

eric

al g

eom

etry

of

no

n-r

igid

sh

apes

D

iffe

ren

tial

geo

met

ry I

Tangent plane & normal

�At each p

oin

t , w

e d

efine

local system of coordinates

�A p

ara

metriz

ation

is regularif

and are

linearly independent.

�The p

lane

is tangent plane

at .

�Local Euclidean approximation

of th

e s

urface.

�is

the normalto

surface.

10

Nu

mer

ical

geo

met

ry o

f n

on

-rig

id s

hap

es

Dif

fere

nti

al g

eom

etry

I

Orientability

�N

orm

al is

defined u

p to a

sign.

�Partitio

ns a

mbie

nt space into

inside

and outside.

�A s

urface is orientable

, if n

orm

al

depends s

mooth

ly o

n .

August Ferd

inand M

öbiu

s

(1790-1

868)

Felix

Christian K

lein

(1849-1

925)

Möbiusstripe

Klein bottle

(3D

section)

11

Nu

mer

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geo

met

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on

-rig

id s

hap

es

Dif

fere

nti

al g

eom

etry

I

First fundamental form

�Infinitesimal displacementon the

chart .

�D

ispla

ces on the surface

by

�is

the Jacobainmatrix

, w

hose

colu

mns a

re and .

12

Nu

mer

ical

geo

met

ry o

f n

on

-rig

id s

hap

es

Dif

fere

nti

al g

eom

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I

First fundamental form

�Length

of th

e d

ispla

cem

ent

�is

a symmetric positive

definite

2×2 m

atrix

.

�Ele

ments

of are

inner products

�Q

uadra

tic form

is the first fundamental form

.

13

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Dif

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I

First fundamental form of the Earth

�Parametrization

�Jacobian

�First fundamental form

14

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Dif

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I

First fundamental form of the Earth

15

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Dif

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I

First fundamental form

�Sm

ooth

curve

on the c

hart:

�Its im

age o

n the s

urface:

�D

ispla

cem

ent on the c

urv

e:

�D

ispla

cem

ent in

the c

hart:

�Length

of dis

pla

cem

ent on the

surface:

16

Nu

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geo

met

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on

-rig

id s

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Dif

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nti

al g

eom

etry

I

�Length

of th

e c

urv

e

�First fu

ndam

enta

l fo

rm induces a

length metric (intrinsic metric

)

�Intrinsic geometry

of th

e s

hape is completely described

by the first

fundam

enta

l fo

rm.

�First fu

ndam

enta

l fo

rm is invariant to isometries.

Intrinsic geometry

17

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id s

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Dif

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nti

al g

eom

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I

Area

�Differential area elementon the

chart: rectangle

�C

opie

d b

y to a

parallelogram

in tangent space.

�D

iffe

rential are

a e

lem

ent on the

surface:

18

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mer

ical

geo

met

ry o

f n

on

-rig

id s

hap

es

Dif

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nti

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Area

�Area

or a regio

n charted a

s

�Relative area

�Probability

of a p

oin

t on p

icked a

t random

(with u

niform

dis

trib

ution) to

fall

into

.

Formally

�are

measures

on .

19

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mer

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geo

met

ry o

f n

on

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id s

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es

Dif

fere

nti

al g

eom

etry

I

Curvature in a plane

�Let be a

smooth curve

para

mete

rized b

y arclength

�trajectory

of a race c

ar drivin

g a

t consta

nt velo

city.

�velocity

vecto

r (rate

of change o

f positio

n), tangent to

path

.

�acceleration

(curvature

) vecto

r, p

erp

endic

ula

r to

path

.

�curvature

, m

easuring rate

of ro

tation o

f velo

city v

ecto

r.

20

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met

ry o

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on

-rig

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Dif

fere

nti

al g

eom

etry

I

�N

ow

the c

ar drives o

n terrain

.

�Tra

jecto

ry d

escribed b

y .

�Curvaturevector

decom

poses into

�geodesic curvature

vecto

r.

�normal curvature

vecto

r.

�Normal curvature

�C

urv

es p

assin

g in d

iffe

rent directions

have d

iffe

rent valu

es o

f .

Said differently:

�A p

oin

t h

as multiple curvatures!

Curvature on surface

21

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met

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es

Dif

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nti

al g

eom

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I

�For each d

irection , a c

urv

e

passin

g thro

ugh in the

direction m

ay h

ave

a d

iffe

rent norm

al curv

atu

re .

�Principal curvatures

�Principal directions

Principal curvatures

22

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Dif

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�Sign of normal curvature

= d

irection o

f ro

tation o

f norm

al to

surface.

�a s

tep in d

irection ro

tate

s in same direction.

�a s

tep in d

irection ro

tate

s in opposite direction.

Curvature

23

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geo

met

ry o

f n

on

-rig

id s

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es

Dif

fere

nti

al g

eom

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I

Curvature: a different view

�A plane

has a

constant normalvecto

r, e

.g. .

�W

e w

ant to

quantify

how

a c

urv

ed s

urface is d

iffe

rent from

a p

lane.

�R

ate

of change o

f i.e., how fast the normal rotates.

�Directional derivative

of at poin

t in the d

irection

is a

n a

rbitra

ry s

mooth

curv

e w

ith

and .

24

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mer

ical

geo

met

ry o

f n

on

-rig

id s

hap

es

Dif

fere

nti

al g

eom

etry

I

Curvature

�is

a v

ecto

r in

m

easuring the

change in a

s w

e m

ake d

iffe

rential ste

ps

in the d

irection .

�D

iffe

rentiate

w

.r.t.

�H

ence o

r .

�Shape operator (a

.k.a

. Weingarten map):

is the m

ap d

efined b

y

Juliu

s W

ein

garten

(1836-1

910)

25

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Dif

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eom

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I

Shape operator

�C

an b

e e

xpre

ssed in parametrizationcoordinates

as

is a

2×2 m

atrix

satisfy

ing

�M

ultip

ly b

y

where

26

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Second fundamental form

�The m

atrix

g

ives ris

e to the quadratic form

calle

d the second fundamental form

.

�R

ela

ted to shape operatorand first fundamental form

by identity

27

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eom

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�Let b

e a

curv

e o

n the s

urface.

�Sin

ce , .

�D

iffe

rentiate

w.r.t. to

� �is

the smallesteigenvalue

of .

�is

the largest eigenvalue

of .

�are

the c

orrespondin

g eigenvectors

.

Principal curvatures encore

28

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�Parametrization

�Normal

�Second fundamental form

Second fundamental form of the Earth

29

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�First fundamental form

�Shape operator

�C

onsta

nt at every

poin

t.

�Is

there

connection b

etw

een algebraic invariants

of shape

opera

tor (trace, dete

rmin

ant) w

ith geometric invariants

of th

e

shape?

Shape operator of the Earth

�Second fundamental form

30

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�Mean curvature

�Gaussian curvature

Mean and Gaussian curvatures

hyperbolic point

elliptic point

31

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Extrinsic & intrinsic geometry

�First fundamentalfo

rm d

escribes c

om

ple

tely

the intrinsic geometry.

�Second fundamentalfo

rm d

escribes c

om

ple

tely

the extrinsic

geometry

–th

e “la

yout”

of th

e s

hape in a

mbie

nt space.

�First fundamental form

is invariant to

isometry.

�Second fundamental form

is invariant to

rigid motion

(congruence).

�If and are

congruent(i.e

., ), then

they h

ave identical in

trin

sic

and e

xtrin

sic

geom

etrie

s.

�Fundamental theorem: a m

ap p

reserv

ing the first and the s

econd

fundam

enta

l fo

rms is a

congru

ence.

Said differently: an isom

etry

pre

serv

ing s

econd fundam

enta

l fo

rm is a

restric

tion o

f Euclid

ean isom

etry.