copyright: dietmar hildenbrand, tu darmstadt, nov. 2002 fundamentals of differential geometry ( part...

Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002

Fundamentals of Differential Geometry

( Part 2 )

2Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002

What do the fundamental forms

mean ?1. Length, angle, surface area2. curvatures ( deviation between the

surface and the tangent plane )


Literature

• Manfredo P. do Carmo : Differentialgeometrie von Kurven und Flächen. Vieweg, 1998

• http://mathworld.wolfram.com/topics/DifferentialGeometry.html

• http://www.mpi-sb.mpg.de/~belyaev/Math4CG/Math4CG.html


Curves on surfaces

)(

constantwith curves describe

,

..

)(),(

bydefinedare),(surfaceaonCurves

linesparametriccalledso

v

constvtu

ge

tvvtuu

vuX


Curves on surfaces e.g. cylinder

),sin,cos()(,.3

:

),,0()(,2

.2

),0,()(,0.1

:

2,0);,sin,cos(),(

ctrtrtXcvtu

constv

trtXtvu

trtXtvu

constu

uvururvuX


Curves on surfaces e.g. cylinder

),sin,cos()(,.4

2,0);,sin,cos(),(

ttrtrtXtvtu

helix

uvururvuX


tangent vector of curves on surfaces

)()()( tangent

),(

))(),(()(

tvXtuXdt

XdtXvector

rulechaingeneralthetoaccording

vuXsurfacetheoncurveabe

tvtuXtXlet

vu


Arc length of the curves on surfaces

dttvXtvtuXXtuXslengtharc

tvXtvtuXXtuXtX

dsdttXslengtharc

b

a

vvuu

vvuu

b

a

b

a

2222

2222

)()()(2)(

)()()(2)()(

)(

Arc length mean the length of a parametric curve between two points defined by its parameter values t=a and t=b


first fundamental form

22222

2222

2

2

2

dt

du(t)u,

dt

dv(t)vsince

)()()(2)(

dvXdudvXXduXds

tvXtvtuXXtuXdt

ds

dtdt

dss

vvuu

vvuu

b

a



)(XX

)(XX

)(XX

2

22vv

12vu

11uu

222

gG

gF

gE

with

dvGdudvFduEdsI

I determines the arc length of a curve on the surface



arc length

Angle of parametric lines

surface area

GE

F

cos

dudvFEGO 2

dttvGtvtuFtuEsb

a 22 )()()(2)(


Length of curves on the cylinder

1. Calculation of the coefficients

1,0

,cossin

)1,0,0(

)0,cos,sin(

),sin,cos(

22222

vvvu

uu

v

u

XXGXXF

rururXXE

X

ururX

vururX


Length of curves on the cylinder

2. Calculation of the arc length according to the curve definition

222

0

22

2

0

2

0

2

222

2

)(,1)()(,)(.2

2

0)(,1)(0)(,)(.1

2,0..,

hrdthrs

htvtuhttvttu

rdtrdtrs

tvtutvttu

bagedtvursb

a

cylinder


First fundamental form of the sphere

22222

2222222

22222222

22222222

cos

cos)sin(cossin

cossinsincossin

0

coscoscossincos

)cos,sinsin,cossin(

)0,coscos,sincos(

2,

2,2,0

)sin,sincos,coscos(

dvrduvrI

rvruuvr

vruvruvrXXG

XXF

vruvruvrXXE

vruvruvrX

uvruvrX

vu

vruvruvrX

sphere

vv

vu

uu

v

u


Length of curves on the sphere

crdtcrs

dtcrs

tvtu

ba

constctvttuge

dtvuvrsb

a

sphere

cos2cos

cos

0)(,1)(

2,0

)(,)(..

,cos

2

0

2

0

2

222


Surface area of the sphere

25.00

25.0

0

2

222

0

5.0

0

2

0

2

242

4sin4cos22

cos2cos

)11(cos2

)14(cos

rvrvdvrO

vrvdur

pagevdudvrO

pagevrFEG

v

Sphere

u

v u

Sphere


The curvature vector of the curves on

surfaces

vectortheissvXsuXsX

and

vectortheissvXsuXsX

vuXsurfacetheonparameterlengtharcwithcurveabe

svsuXsXLet

vu

vu

curvature)()()(

tangent)()()(

),(

))(),(()(



surfaces

22 2

)(

)(

vXvuXuX

vXuX

vXvvXuXuXuvXuXsX

vXuXX

vXuXX

vXuXsX

vvuvuu

vu

vvvvuuuvuu

vvvuv

uvuuu

vu



surfaces

22

2

2

1since0

since0,0

:

vNXvuNXuNXNX

XXX

XX

XXNNXNX

vectorsotherwithrelation

vvuvuu

vu

vuvu


Second fundamental form

Ng

Nf

Ne

with

dvgdudvfdueII

vv

uv

uu

22

X

X

X

2

II measures how far the surface is from being a plane


Second fundamental form

0sinceX

0sinceX

0sinceX

vv

uv

uu

vvvv

vuvu

uuuu

NXNXNg

NXNXNf

NXNXNe

Alternative notation for the coefficients :


Second fundamental form of the sphere

)sin,sincos,cos(cos

cossinsincoscoscoscos

)sin,sincos,cos(coscos

)sincos,sincos,coscos(

)cossincossinsincos,sincos,coscos(

)14(

cossinsincossin

0coscossincos

ˆˆˆ

det

2222222

2

22222

22222222

vuvuvXX

XXN

vrvuvuvrvXX

vuvuvrv

vvruvruvr

uvvruvvruvruvr

page

vruvruvr

uvruvr

zyx

XX

vu

vu

vu

vu

1. Compute the normal vector


Second fundamental form of the sphere

222

2

cos

1

01

cos1

1

dvrduvrII

rr

GXX

rNXg

XXr

NXf

vrr

EXX

rNXe

rXN

sphere

vvvv

vuvu

uuuu

spheresphere

2. Compute the coefficients


Normal curvature of surfaces

22 2 vgvufue

NXcurvatureNormal

n

n

Note :

Cut the surface with the plane spanned by the tangent vector and the normal vector

->the curvature of this curve equals the normal curvature of the surface



22

22

0000

000

2

2

tangent

),(),(

point),(

GbFabEa

gbfabea

istofdirectiontheincurvatureNormal

planetheinsidevectordirectiona

vuXbvuXatand

XsurfacetheofabevuXPlet

n

vu



G

g

bavuXt

constulinesparametricfor

E

e

bavuXt

constvlinesparametricfor

n

v

n

u

1,0),,(

0,1),,(

00

00


Normal curvature of the sphere

rr

r

G

g

constulinesparametricthealong

rvr

vr

E

e

constvlinesparametricthealong

n

n

1

1

cos

cos

2

22

2


Principal curvatures of surfaces and

principal directions

21,

are the maximum and the minimum of the normal curvature ( so-called principal curvatures ).

Principal directions are the directions of a surface in which the principal curvatures occur.


Elliptic Points

021

e. g. Ellipsoid :


Parabolic points

021

e. g. cylinder :

Note. : zero principal curvatures ->

planar point of the surface

( e.g. All points of the plane )


Hyperbolic points

021

e. g. Torus :


curvature definitions

KHH

KHH

curvature

FEG

gEfFeGH

curvature

FEG

fegK

curvature

22

21

221

2

2

21

:Principal

)(2

2

2

1

:Mean

:Gaussian

copyright: dietmar hildenbrand, tu darmstadt, nov. 2002 fundamentals of differential geometry ( part...

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