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 )
3Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
4Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Curves on surfaces
)(
constantwith curves describe
,
..
)(),(
bydefinedare),(surfaceaonCurves
linesparametriccalledso
v
constvtu
ge
tvvtuu
vuX
5Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
6Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Curves on surfaces e.g. cylinder
),sin,cos()(,.4
2,0);,sin,cos(),(
ttrtrtXtvtu
helix
uvururvuX
7Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
tangent vector of curves on surfaces
)()()( tangent
),(
))(),(()(
tvXtuXdt
XdtXvector
rulechaingeneralthetoaccording
vuXsurfacetheoncurveabe
tvtuXtXlet
vu
8Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
9Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
10Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
first fundamental form
)(XX
)(XX
)(XX
2
22vv
12vu
11uu
222
gG
gF
gE
with
dvGdudvFduEdsI
I determines the arc length of a curve on the surface
11Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
first fundamental form
arc length
Angle of parametric lines
surface area
GE
F
cos
dudvFEGO 2
dttvGtvtuFtuEsb
a 22 )()()(2)(
12Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
13Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
14Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
15Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
16Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
17Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
The curvature vector of the curves on
surfaces
vectortheissvXsuXsX
and
vectortheissvXsuXsX
vuXsurfacetheonparameterlengtharcwithcurveabe
svsuXsXLet
vu
vu
curvature)()()(
tangent)()()(
),(
))(),(()(
18Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
The curvature vector of the curves on
surfaces
22 2
)(
)(
vXvuXuX
vXuX
vXvvXuXuXuvXuXsX
vXuXX
vXuXX
vXuXsX
vvuvuu
vu
vvvvuuuvuu
vvvuv
uvuuu
vu
19Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
The curvature vector of the curves on
surfaces
22
2
2
1since0
since0,0
:
vNXvuNXuNXNX
XXX
XX
XXNNXNX
vectorsotherwithrelation
vvuvuu
vu
vuvu
20Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
21Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Second fundamental form
0sinceX
0sinceX
0sinceX
vv
uv
uu
vvvv
vuvu
uuuu
NXNXNg
NXNXNf
NXNXNe
Alternative notation for the coefficients :
22Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
23Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
24Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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
25Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Normal curvature of surfaces
22
22
0000
000
2
2
tangent
),(),(
point),(
GbFabEa
gbfabea
istofdirectiontheincurvatureNormal
planetheinsidevectordirectiona
vuXbvuXatand
XsurfacetheofabevuXPlet
n
vu
26Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Normal curvature of surfaces
G
g
bavuXt
constulinesparametricfor
E
e
bavuXt
constvlinesparametricfor
n
v
n
u
1,0),,(
0,1),,(
00
00
27Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Normal curvature of the sphere
rr
r
G
g
constulinesparametricthealong
rvr
vr
E
e
constvlinesparametricthealong
n
n
1
1
cos
cos
2
22
2
28Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
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.
29Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Elliptic Points
021
e. g. Ellipsoid :
30Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Parabolic points
021
e. g. cylinder :
Note. : zero principal curvatures ->
planar point of the surface
( e.g. All points of the plane )
31Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
Hyperbolic points
021
e. g. Torus :
32Copyright: Dietmar Hildenbrand, TU Darmstadt, Nov. 2002
curvature definitions
KHH
KHH
curvature
FEG
gEfFeGH
curvature
FEG
fegK
curvature
22
21
221
2
2
21
:Principal
)(2
2
2
1
:Mean
:Gaussian