Differential geometry three dimensional space

1
Differential Geometry
Three Dimensional Euclidian
Space
SOLO HERMELIN
Updated: 1.04.07
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http://www.solohermelin.com

2
SOLO Vectors & Tensors in a 3D Space
TABLE OF CONTENT
Introduction
Curvilinear Coordinates in a Three Dimensional Space
Theory of Curves
Unit Tangent Vector of path C at a point P
Curvature of curve C at P
Osculating Plane of C at P
Osculating Circle of C at P
Binormal
Torsion
Seret-Frenet Equations
Involute
Evolute
Surfaces in the Three Dimensional Spaces
First Fundamental Form:
Arc Length on a Path on the Surface
Surface Area
Change of Coordinates
Second Fundamental Form

3
TABLE OF CONTENT (continue)
Surfaces in the Three Dimensional Spaces (continue)
Normal Curvature
Principal Curvatures and Directions
Planar Curves
References
Conjugate Directions
Asymptotic Lines

4
Differential Geometry is the study of geometric figures using the methods of Calculus.
Here we present the curves and surfaces embedded in a three dimensional space.
Properties of curves and surfaces which depend only upon points close to a particular
point of the figure are called local properties.. The study of local properties is called
differential geometry in the small.
Those properties which involve the entire geometric figure are called global properties.
The study of global properties is called differential geometry in the large.
Hyperboloid
of RotationToroyd
Mobius
Movement
Introduction
Return to Table of Contents

5
Curvilinear Coordinates in a Three Dimensional Space
Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space
as a function of three curvilinear coordinates (u1, u2, u3), where:
→
dr
constu =3
i

j

k

1
1
ud
u
r
∂
∂
→
2
2
ud
u
r
∂
∂
→
3
3
ud
u
r
∂
∂
→
constu =1
constu =2
curveu1
curveu2
curveu3
( ) ( ) ( )
( ) ( ) ( )


===
=
zyxuuzyxuuzyxuu
uuuzuuuyuuuxx
,,,,,,,,
,,,,,,,,
332211
321321321
Those functions are single valued with continuous
derivatives and the correspondence between (x,y,z)
and (u1,u2,u3) is unique (isomorphism).
kzjyixr

++=
3
3
2
2
1
1
3
333
2
222
1
111
3
3
12
2
1
1
3
3
12
2
1
1
3
3
12
2
1
1
ud
u
r
ud
u
r
ud
u
r
udk
u
z
j
u
y
i
u
x
udk
u
z
j
u
y
i
u
x
udk
u
z
j
u
y
i
u
x
kud
u
z
d
u
z
ud
u
z
jud
u
y
d
u
y
ud
u
y
iud
u
x
d
u
x
ud
u
x
kzdjydixdrd
∂
∂
+
∂
∂
+
∂
∂
=








∂
∂
+
∂
∂
+
∂
∂
+





∂
∂
+
∂
∂
+
∂
∂
+





∂
∂
+
∂
∂
+
∂
∂
=








∂
∂
+
∂
∂
+
∂
∂
+







∂
∂
+
∂
∂
+
∂
∂
+







∂
∂
+
∂
∂
+
∂
∂
=
++=




or
3
3
2
2
1
1
ud
u
r
ud
u
r
ud
u
r
rd
∂
∂
+
∂
∂
+
∂
∂
=



6
SOLO
Differential Geometry in the 3D Euclidean Space
A curve C in a three dimensional space is
defined by one parameter t, ( )tr

( )ur

rd
P
O
a
b
C
Theory of Curves
Regular Parametric Representation of a Vector Function:
parameter t, defined in the interval I and:
( ) Ittrr ∈= ,

( )tr

(i) is of class C1
(continuous and 1st
order differentiable) in I
(ii)
( ) It
td
trd
∈∀≠ 0

Arc length differential: ( ) ( )[ ] td
td
rd
td
td
rd
td
rd
trdtrdsd


=





⋅=⋅=
2/1
2/1
:
We also can define ( ) ( )[ ] sdtrdtrdsd −=⋅−=
2/1*
:

( ) Iinconstantnottr

→
Arc length as a parameter: ∫=
t
t
td
td
rd
s
0

Regular Curves:
A real valued function t = t (θ), on an interval Iθ, is an allowable change of parameter if:
(i) t (θ) is of class C1
in Iθ (ii) d t/ d θ ≠ 0 for all θ in Iθ
A representation on Is is a
representation in terms of arc length or a
natural representation
( )srr

=

7
SOLO
A curve C in a three dimensional space is defined by one parameter t, ( )tr

( )ur

rd
P
O
a
b
C
- arc length differential( ) ( )[ ] td
td
rd
td
rd
trdtrdsd
2/1
2/1
: 





⋅=⋅=


td
rd
td
rd
r
sd
rd
t /:: ===  - unit tangent vector of path C at P
(tangent to C at P)
1
x
2x
3
x
td
rd
r ='
 - tangent vector of path C at P
(tangent to C at P)
0,0,sincos 321
≠>++= baetbetaetar

Example: Circular Helix
0,0,cossin' 321 ≠>++−== baebetaeta
td
rd
r


( ) 2/122
2/1
ba
td
rd
td
rd
td
rd
+=





⋅=

( ) ( )321
2/122
cossin/: ebetaetaba
td
rd
td
rd
t ++−+==
−
Theory of Curves (continue – 1)
We also can define ( ) ( )[ ] sdtrdtrdsd −=⋅−=
2/1*
:

t
sd
rd
sd
rd 
−=−=*
Unit Tangent Vector of path C at a point P

8
SOLO
The earliest investigations by means of analysis were made
by René Descartes in 1637.
( )tr

( )ur

rd
P
O
a
b
C
René Descartes
1596 - 1650
Pierre Fermat
1601 - 1665
Christian Huyghens
1629 - 1695
Gottfried Leibniz
1646 - 1716
The general concept of tangent was introduced in
seventeenth century, in connexion with the basic concepts of
calculus. Fermat, Descartes and Huyghens made important
contributions to the tangent problem, and a complete
solution was given by Leibniz in 1677.
The first analytical representation of a tangent was given
by Monge in 1785.
Gaspard Monge
1746 - 1818

9
SOLO
A curve C in a three dimensional space is defined by one parameter t, ( )tr

- arc length differential( ) ( )[ ] td
td
rd
td
rd
trdtrdsd
2/1
2/1
: 





⋅=⋅=


'/'/:: rr
td
rd
td
rd
r
sd
rd
t

====
- unit tangent vector of path C at P
(tangent to C at P)
Normal Plane to at P: t

( ) 00
=⋅− trr

We also can define - arc length differential( ) ( )[ ] sdtrdtrdsd −=⋅−=
2/1*
:

t
sd
rd
sd
rd 
−=−=*
O
a
C
t

P
r

b
0r

NormalPlane ( ) 00 =⋅− trr


10
SOLO
O
a
C
t

P
r

b
NormalPlane ( ) 00 =⋅− trr

0
r

Curvature of curve C at P: rt
sd
td
k 

===:
Since 01 =⋅=⋅=⋅→=⋅ tkttt
sd
td
tt



Define nnkkkkkkn

ρ
ρ
1
/1:&/: ==→===
ρ – radius of curvature of C at P
k – curvature of C at P
A point on C where k = 0 is called a point of inflection and the radius of curvature
ρ is infinite.
'' st
td
sd
sd
rd
td
rd
r



===
( ) ( ) ( ) "'"'"'
'
'''''
22
stskstststs
td
sd
sd
td
td
sd
ts
td
td
st
td
d
r
td
d
r





+=+=+=+===
( )( ) ( )32
'"''''' skntstskstrr

×=+×=×
'' sr =

3
1
'''' skntrr


×=× 3
'
'''
r
rr
k 

×
=
Let compute k as a function of and :'r

''r


11
SOLO
1
x
2x
3
x
t

k

0,0,sincos 321

Example 2: Circular Helix
0,0,cossin' 321 ≠>++−== baebetaeta
td
rd
r


( ) ( ) 2/1222/122
2/1
bardsdba
td
rd
td
rd
td
rd
+=+=





⋅=


( ) ( )321
2/122
cossin/: ebetaetaba
td
rd
td
rd
t ++−+==
−
( )2122
sincos/ etet
ba
a
td
sd
td
rd
t
sd
td
k +
+
−====



1
x
2
x
3
x
t

k

0,sincos 21
>+= aetaetar

Example 1: Circular Curve
0,cossin' 21 >+−== aetaeta
td
rd
r


( ) ( ) 2/1222/122
2/1
bardsdba
td
rd
td
rd
td
rd
+=+=





⋅=


( )21 cossin/: etaetaa
td
rd
td
rd
t +−==

( )21
sincos
1
/ etet
atd
sd
td
rd
t
sd
td
k +−====




12
SOLO
O
a
C
t

P
b
ntk

ρ
1
==
NormalPlane
Osculating
Plane
( ) 00
=⋅− trr

0
r

( ) 00 =×⋅− ktrr

Osculating Plane of C at P is the plane that contains
and P: ( ) ( ) 00
=×⋅− ktrr
kt

,
The name “osculating plane” was
introduced by D’Amondans
Charles de Tinseau (1748-1822) in
1780.
O
a
C
t

P
b
ntk

ρ
1
==
NormalPlane
Osculating
Plane
( ) 00 =⋅− trr

0r

( ) 00 =×⋅− ktrr

The osculating plane can be also defined as the limiting position of a plane passing
through three neighboring points on the curve as the points approach the given point.
If the curvature k is zero along a curve C then:
tarrconstartt

+=→===→≡ 00
The curve C is a straight line. Conversely if C is a straight line:
0//0
==→==→+= tkaa
td
rd
td
rd
ttarr 
C a regular curve of class ≥2 (Cclass
) is a straight line if and only if k = 0 on C

13
SOLO
Osculating Circleof C at P is the plane that contains
and P: kt

,
The osculating circle of a curve C at a given point P is the circle that has the same
tangent as C at point P as well as the same curvature.
Just as the tangent line is the line best approximating a curve at a point P,
the osculating circle is the best circle that approximates the curve at P.
http://mathworld.wolfram.com/OsculatingCircle.html
Osculating Circles on the Deltoid
The word "osculate" means "to kiss."

14
SOLO
Osculating Circleof C at P is the plane that contains
and P: kt

,
Theory of Curves (continue – 6a)
3
xy =
xy /1=
xy cos= xy sin= http://curvebank.calstatela.edu/osculating/osculating.htm
xy tan=

15
SOLO
O
a
C
t

P
b
ntk

ρ
1
==
NormalPlane
Osculating
Plane
( ) 00 =⋅− trr

0r

( ) 00 =×⋅− ktrr
b

Rectifying
Plane
( ) 00 =⋅− krr

Binormal ntb

×=:
Tangent Line:
Principal Normal Line:
Binormal Line:
Normal Plane:
Rectifying Plane:
Osculating Plane:
tmrr

+= 0
nmrr

+= 0
bmrr

+= 0
( ) 00
=⋅− trr

( ) 00
=⋅− nrr

( ) 00
=⋅− brr

The name binormal was introduced by
Saint-Venant
Jean Claude Saint-Venant
1797 - 1886
Fundamental Planes:Fundamental Lines:

16
SOLO
Torsion
Suppose that is a regular curve of class ≥ 3 (Cclass
) along which is of
class C1
. then let differentiate to obtain:
( )srr

= ( )sn

( ) ( ) ( )snstsb

×=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )snstsnstsnsnksnstsnstsb 
×=×+×=×+×=
Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 001 =⋅→=⋅+⋅→=⋅ snsnsnsnsnsnsnsn 
Therefore is normal to , meaning that is in the rectifying plane,
or that is a linear combination of and .
n
n
t

b

( ) ( ) ( ) ( ) ( )sbsstssn

τµ +=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )snssbsstsstsnstsb

ττµ −=+×=×=
O
a
C
t

P
b
n

0
r

b

The continuous function τ (s) is called the second curvature
or torsion of C at P.
( ) ( ) ( )snsbs

⋅−=τ

17
SOLO
Torsion (continue – 1)
Suppose that the torsion vanishes identically (τ ≡0) along a curve , then( )srr

=
( ) ( ) ( ) ( ) 00 bsbsnssb

=→=−= τ
O
a
C
t

P
b
n

0
r

0
b

Since and are orthogonal( )st

( )sb

( )[ ] ( )[ ] ( ) constbsrbtbsr
sd
d
bsr
sd
d
=⋅→≡⋅=⋅=⋅ 0000
0

Therefore is a planar curve confined to the plane( )srr

= ( ) constbsr =⋅ 0

C a regular curve of class ≥3 (Cclass
) is a planar curve if and only if τ = 0 on C
1
x
2x
3
x
t

k

0,0,sincos 321

Example 2: Circular Helix
( ) ( )321
2/122
cossin ebetaetabat ++−+=
−
( )21 sincos etetn +−=

( ) ( ) ( )
( ) ( )321
2/122
21321
2/122
cossin
sincoscossin
eaetbetbba
etetebetaetabantb
+−+=
+×++−+−=×=
−
−
( ) ( ) ( )21
1222/122
sincos etbetbbaba
td
bd
sd
td
td
bd
sd
bd
b ++=+===
−−


( ) 122 −
+=⋅−= babnb

τ

18
SOLO
Torsion (continue – 2)
Let compute τ as a function of and :'',' rr

'''r

ttr
sd
td
td
rd
sd
rd
r




==== ' ( ) ( ) ( )tbknkttrtrtr
sd
d
trtr
sd
d
r


×===+=+== 2
"''''
( )
( ) ( ) tkbkbknkbktnkbktbktbkbk
trttrtrtrttrttrtrtrtr
sd
d
r





2
332
'''"3''''"2"'"'
−−=×+×−+=×+×+=
++=+++=+=
ττ
( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) τττ 2
0
3
1
2
0
26
6
0
3
0
4
0
22
5232
32
,,,,,,'''",'
'''",'',",'''',','",','3
'''"'"''''"'3'
'''"3'"'',,
ktntkbntknntkktkbknknktrrrt
rrrtrrrttrrrttrrrtt
rrtrrttrrttrrtttr
trttrtrtrtrtrrrr
−=−−=−−×⋅=×=
×+++=
×+×+×+×⋅=
++×+⋅=















'
1
/
1
rtdsdsd
td
t  ===
3
'
'''
r
rr
k 

×
=We also found:
( ) ( ) ττ 6
2
2
6
'
'''
'
'''",'
,,
r
rr
k
r
rrr
rrr 



 ×
−=−=
×
=
( )
2
'''
'''",'
rr
rrr


×
×
=τ

19
SOLO
Seret-Frenet Equations
We found and ( ) ( ) ( )snssb

τ−=( ) ( ) ( )snskst

=
Let differentiate ( ) ( ) ( )stsbsn

×=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )stsksbssnsbskstsnsstsbstsbsn


−=×+×−=×+×= ττ
We obtain
( ) ( ) ( ) ( ) ( )sbsnskstst

00 ++=
( ) ( ) ( ) ( ) ( ) ( )sbsbsnstsksn

τ++−= 0
( ) ( ) ( ) ( ) ( )sbsnsstsb

00 +−= τ
or
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )



















−
+−
+
=












sb
sn
st
s
ssk
sk
sb
sn
st






00
0
00
τ
τ
Jean Frédéric Frenet
1816-1900
Those are the Serret – Frenet Equations of a curve.
Joseph Alfred Serret
1819-1885

20
SOLO
Let compute:
Seret-Frenet Equations (continue – 1)
Let show that if two curves C and C* have the same curvature k (s) = k* (s) and
torsion τ (s) = τ*(s) for all s then C and C* are the same except for they position in
space. Assume that at some s0 the triads and
coincide.
( ) ( ) ( )999 ,, sbsnst

( ) ( ) ( )999 *,*,* sbsnst

( ) ( )********
*
nttnknkttnktttttt
sd
d kk 
⋅+⋅=⋅+⋅=⋅+⋅=⋅
=
( ) ( ) ( ) ( ) ( )************
*
*
bnnbnttnkbtknnbtknnnnnn
sd
d kk 
⋅+⋅+⋅+⋅−=+−⋅+⋅+−=⋅+⋅=⋅
=
=
τττ ττ
( ) ( )********
*
nbbnnbbnbbbbbb
sd
d 
⋅+⋅−=⋅+⋅−=⋅+⋅=⋅
=
τττ
ττ
Adding the equations, we obtain: ( ) 0***
=⋅+⋅+⋅ bbnntt
sd
d 
Integrating we obtain: ( ) ( ) 30
******
=⋅+⋅+⋅==⋅+⋅+⋅ sbbnnttconstbbnntt

Since: and1,,1 ***
≤⋅⋅⋅≤− bbnntt

( ) 3***
=⋅+⋅+⋅ bbnntt

we obtain: 1***
=⋅=⋅=⋅ bbnntt

Finally since: ( ) ( ) ( ) ( ) constsrsr
sd
rd
stst
sd
rd
+=→=== *
*
* 




21
SOLO
Existence Theorem for Curves
Let k (s) and τ (s) be continuous functions of a real variable s for s0 ≤ s ≤ sf.
Then there exists a curve , s0 ≤ s ≤ sf, of class C2 for which k is the curvature,
τ is the torsion and s is a natural parameter.
( )srr

=
332211332211332211 ,, ebebebbenenennetetett

++=++=++=
( ) ( ) ( ) ( )tnktttttt
sd
d 
⋅=⋅+⋅=⋅ 2 ( ) ( ) ( ) ( ) ( )nbntknnnnnn
sd
d 
⋅+⋅−=⋅+⋅=⋅ τ22
( ) ( )bnbbbbbb
sd
d 
⋅−=




 ⋅+




 ⋅=⋅ τ2
with:
Proof: Consider the system of nine scalar differential equations:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3,2,1,,, =−=+−== isnssbsbsstsksnsnskst iiiiiii

ττ
and initial conditions: ( ) ( ) ( ) 302010 ,, esbesnest

===
( ) ( ) ( ) ( ) ( ) ( )btttktnkntntnt
sd
d 
⋅+⋅−⋅=⋅+⋅=⋅ τ ( ) ( ) ( ) ( ) ( )nnbbbtkbnbnbn
sd
d 

⋅−⋅+⋅−=




 ⋅+⋅=⋅ ττ
( ) ( ) ( ) ( )ntbnkbtbtbt
sd
d 
⋅−⋅=




 ⋅+⋅=⋅ τ
and initial conditions:
( ) ( ) ( ) ( ) ( ) ( ) 1,0,1,0,0,1 000000
=⋅=⋅=⋅=⋅=⋅=⋅ ssssss bbbnnnbtnttt


22
SOLO
Existence Theorem for Curves (continue – 1)
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )bnbb
sd
d
ntbnkbt
sd
d
nnbbbtkbn
sd
d
btttktnknt
sd
d
nbntknn
sd
d
tnktt
sd
d



⋅−=⋅⋅−⋅=⋅
⋅−⋅+⋅−=⋅⋅+⋅−⋅=⋅
⋅+⋅−=⋅⋅=⋅
ττ
τττ
τ
2
222
Proof (continue – 1):
and initial conditions: ( ) ( ) ( ) ( ) ( ) ( ) 1,0,1,0,0,1 000000
=⋅=⋅=⋅=⋅=⋅=⋅ ssssss bbbnnnbtnttt

We obtain:
The solution of this type of differential equations with given initial conditions has
a unique solution and since
is a solution, it is unique.
( ) ( ) ( ) ( ) ( ) ( ) 1,0,1,0,0,1 =⋅=⋅=⋅=⋅=⋅=⋅ bbbnnnbtnttt

The solution is an orthonormal triad.bnt

,,
We now define the curve: ( ) ( )∫==
s
s
dtsrr
0
: σσ

We have: and , therefore k (s) is the curvature.1== tr

( ) ( ) ( ) ( ) 1& == snsnskst

Finally since: ( ) ( ) ( ) nbtttknnkntntbntb

ττ −=×+×−×=×+×=→×=
Therefore τ (s) is the torsion of ( )srr

= q.e.d.

23
SOLO
From the previous development we can state the following theorems:
A curve is defined uniquely by the curvature and torsion
as functions of a natural parameter.
The equations k = k (s), τ = τ (s), which give the curvature
and torsion of a curve as functions of s are called the natural
or intrinsec equations of a curve, for they completely define
the curve. O
0s
C
t

P
n

0r

b

k
1
=ρ fs
Fundamental Existence and Uniqueness Theorem of Space
Curves
Let k (s) and τ (s) be arbitrary continuous functions on
s0≤s≤sf. Then there exists, for position in space, one and only
one space curve C for which k (s) is the curvature, τ (s) is the
torsion and s is a natural parameter along C. O
0s
C
t

P
n

b

f
s
*
C
0r

*
0
r


24
SOLO
Let consider a space curve C. We construct the tangent
lines to every point on C and define an involute Ci
as any
curve which is normal to every tangent of C.
Involute
From the Figure we can see that the equation of the
Involute is given by:
turr

+=1
Differentiating this equation we obtain:
11
1
1
1
sd
sd
t
sd
ud
nkut
sd
sd
t
sd
ud
sd
td
u
sd
rd
t
sd
rd






++=





++==




Scalar multiply this equation by and use the fact that and from the
definition of involute :
t

0=⋅nt

01
=⋅tt

  
1101
1
0
sd
sd
tt
sd
ud
ntkutttt 





⋅+⋅+⋅=⋅=

01 =+
sd
ud
scu −=
( ) ( ) ( ) ( )stscsrsr

−+=1
C
i
C
O
r
 1
r

t

1t
(
)s
c −
Involute
Curve

25
SOLO
Involute (continue – 1)
C
i
C
O
r
 1
r

t

1t
(
)s
c −
Involute
Curve
and are collinear unit vectors, therefore:
( ) ( ) ( ) ( )stscsrsr

−+=1

( ) ( ) n
sd
sd
ksc
sd
sd
t
sd
td
sc
sd
rd
sd
rd
t
t




111
1
1
−=










−−+==
1
t

n

( )
( ) kscsd
sd
sd
sd
ksc
−
=→=−
1
1
11
The curvature of the involute, k1, is obtained from:
( )
( ) ( ) ksc
btk
kscsd
nd
sd
sd
sd
td
nk
sd
td nt
kscsd
sd
−
+−
=
−
===
=
−
=


 
τ11
1
1
1
1
11
1
1
Hence:
( ) 22
22
2
1
ksc
k
k
−
+
=
τ
For a planar curve (τ=0) we have: ( )
t
sc
nk

−
−
==
1
011 τ

26
SOLO
Involute (continue – 3)
C
i
C
O
r
 1
r

t

1t
(
)s
c −
Involute
Curve
http://mathworld.wolfram.com/Involute.html

27
SOLO
The curve Ce
whose tangents are perpendicular to a
given curve C is called the evolute of the curve.
Evolute
11 twrbvnurr

+=++=
Differentiating this equation we obtain:
( )
11
1
1
1
sd
sd
b
sd
vd
n
sd
ud
nvbtkut
sd
sd
b
sd
vd
n
sd
ud
sd
bd
v
sd
nd
u
sd
rd
t
sd
rd






++−+−+=








++++==




ττ
Scalar multiply this equation by and use the fact that and from the
definition of evolute :
t

0=⋅=⋅ btnt

01
=⋅tt

 
111
10
sd
sd
ttkutttt 





⋅−⋅=⋅=

01 =− ku ρ==
k
u
1
C
e
C
O
r

1
r

t
1t

Evolute
Curve
The tangent to Ce
, , must lie in the plane of
and since it is perpendicular to . Therefore:
n

b

t
1
t

1
1
sd
sd
n
sd
ud
vb
sd
vd
ut














+−+





+=

ττ

28
SOLO
Evolute (continue – 1)
We obtain:
( ) ( )ccuv −=−= ϕρϕ tantan
ρ==
k
u
1
C
e
C
O
r

1
r

t
1t

Evolute
Curve
1
1
sd
sd
n
sd
ud
vb
sd
vd
ut














+−+





+=

ττ
( ) ( ) 111 // wbvnuwrrt

+=−=But:
Therefore:
v
v
sd
ud
u
u
sd
vd
ττ −
=
+
or:












=
+
−
= −
u
v
sd
d
vu
sd
ud
v
sd
vd
u
1
22
tanτ
c
u
v
ds
s
s
+





== −
∫
1
tan
0
τϕ
and: ( ) bcnrr

−++= ϕρρ tan1
We have one parameter family that describes the evolutes to the curve C.

29
SOLO
Evolute (continue – 2)
C
e
C
O
r

1
r

t
1t

Evolute
Curve
http://math.la.asu.edu/~rich/MAT272/evolute/ellipselute.html
Evolute of Ellipse
Evolute of Logarithmic Spiral
also a Logarithmic Spiral
Evolute of Parabola

30
SOLO
The vector defines a surface in E3
( )vur ,

vu
vu
rr
rr
N 

×
×
=
( )vur ,

( )vdvudur ++ ,

rd 2
rd
r∆
udru

vdrv

Σd
Nd
P
O
( )vudur ,+

( ) ( ) ( ) ( )
( ) ( ) ( )22
2
22
22
2
2
,2
2
1
,
2
1
,,,
vdudOvdrvdudrudrvdrudr
vdudOrdrdvurvdvudurvur
vvvuuuvu /+++++=
/++=−++=∆


The vectors and define the
tangent plane to the surface at point P.
P
u
u
r
r
∂
∂
=


P
v
v
r
r
∂
∂
=


Define: Unit Normal Vector to the surface at P
vu
vu
rr
rr
N 

×
×
=:
First Fundamental Form:
( ) ( ) ( ) 2222
22: vdGvdudFudEvdrrvdudrrudrrrdrdI vvvuuu ++=⋅+⋅+⋅=⋅=

[ ]   

0
2
0,0,00:
>



>−>>→>











=⋅=
GF
FEforConditionSylvester
FEGGE
vd
ud
GF
FE
vdudrdrdI
Surfaces in the Three Dimensional Spaces

31
SOLO
Arc Length on a Path on the Surface:
( ) ( )[ ]∫ ∫∫ +⋅+=








⋅==
b
a
b
a
vuvu
b
a
tdvdrudrvdrudrtd
td
rd
td
rd
td
td
rd
L
2/1
2/1

∫∫






































=














+











+





=
b
a
b
a
td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
vd
G
td
vd
td
ud
F
td
ud
EL
2/1
2/122
2
Surface Area:
( )vur ,

rd
udru

vdrv

Σd
P
O
( )
( )[ ]
vdudFGEvdud
GE
F
GE
vdud
rr
rr
rrvdudrrrr
vdudrrrrvdudrrvdrudrd
vu
vu
vuvuvu
vuvuvuvu
2
2/1
2
2/1
2
2/12
1
1,cos1
,sin
−=





−=















 ⋅
−=−=
=×=×=Σ




∫∫∫∫ −=Σ=Σ vdudFGEd 2
( )vur ,

rd
udru

vdrv
P
O
a
b

32
SOLO
Change of Coordinates
( )vur ,

rd
udru

vdrv

Σd
P
O
vdrv

udru

vdrudrvdrudrd vuvu

×=×=Σ
( ) ( )vurvurr ,,

==
vdudFGEvdud
vu
vu
JFGEvdudFGEd 222
,
,
−=





−=−=Σ
Change of coordinates from u,v to θ,φ
( )
( )vuvv
vuuu
,
,
=
=
The coordinates are related by












=





v
u
vv
uu
vd
ud
vu
vu
[ ] [ ] [ ] I
vd
ud
GF
FE
vdud
vd
ud
vv
uu
GF
FE
vu
vu
vdud
vd
ud
GF
FE
vdudI
vu
vu
vv
uu
=











=

























=













=
td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
vd
td
ud
GF
FE
td
vd
td
ud
td
td
rd
td
rd
Ld
2/12/1
2/1






































=








































=








⋅=






−=





−=

















==−
vu
vu
JFGE
vv
uu
FGE
vv
uu
GF
FE
vu
vu
GF
FE
FGE
vu
vu
vu
vu
vv
uu
,
,
detdetdetdetdet 22
**
**
2
Arc Length on a Path on the Surface and Surface Area are Invariant of the Coordinates:
First Fundamental Form is Invariant to Coordinate Transformation

33
SOLO
vu
vu
rr
rr
N 

×
×
=
( )vur ,

( )vdvudur ++ ,

rd 2
rd
r∆
udru

vdrv

Σd
Nd
P
O
( )vudur ,+

Second Fundamental Form: NdrdII ⋅−=:
( ) ( )
( ) ( ) ( )
22
2
2
2
2
:
vdNvdudMudL
vdNrvdudNrNrudNr
vdNudNvdrudrNdrdII
N
vv
M
uvvu
L
uu
vuvu
++=
⋅−+⋅−⋅−+⋅−=
+⋅+−=⋅−=


  




vdNudNNdNNdNN vu

+=→=⋅→=⋅ 01
( )
( )
( )






⋅−=⋅→=⋅+⋅=⋅
⋅=⋅−→=⋅+⋅=⋅
→=⋅
NrNrNrNrNr
vd
d
NrNrNrNrNr
ud
d
Nr
vuvuvuvuu
uuuuuuuuu
u



0
0
0
( )
( )
( )






⋅−=⋅→=⋅+⋅=⋅
⋅−=⋅→=⋅+⋅=⋅
→=⋅
NrNrNrNrNr
vd
d
NrNrNrNrNr
ud
d
Nr
vvvvvvvvv
vuuvuvvuv
v



0
0
0

34
SOLO
vu
vu
rr
rr
N 

×
×
=
( )vur ,

( )vdvudur ++ ,

rd 2
rd
r∆
udru

vdrv

Σd
Nd
P
O
( )vudur ,+

( ) ( ) ( ) 2
2
2
: vdNrvdudNrNrudNrNdrdII
N
vv
M
uvvu
L
uu


  



⋅−+⋅−⋅−+⋅−=⋅−=
NrNr uuuu

⋅=⋅− NrNr vuuv

⋅−=⋅




⋅=⋅−
⋅−=
NrNr
NrL
uuuu
uu








⋅−=⋅−=
⋅−=⋅
⋅−=⋅
uvvu
vuuv
vuvu
NrNrM
NrNr
NrNr







⋅−=⋅
⋅−=
NrNr
NrN
vvvv
vv


NrNr vuvu

⋅−=⋅ NrNr vvvv

⋅−=⋅
22
2: vdNvdudMudLNdrdII ++=⋅−=
NrL uu

⋅=
NrM vu

⋅=
NrN vv

⋅=

35
SOLO
vu
vu
rr
rr
N 

×
×
=
( )vur ,

O
( )vdvudur ++ ,
udru

vdrv

rd
( ) ( ) ( ) ( )
( ) ( )
( ) ( )33
3
3223
22
33
3
32
,33
6
1
2
2
1
,
6
1
2
1
,,,
vdudOvdrvdudrvdudrudr
vdrvdudrudrvdrudr
vdudOrdrdrdvurvdvudurvur
vvvvuvvuuuuu
vvvuuuvu
/+++++
++++=
/+++=−++=∆



( )
( ) ( )
( ) ( ) IINvdudOvdNvdudMudL
NvdudOvdNrvdudNrudNr
NvdudONrdNrdNrdNr
vvvuuu
2
1
,2
2
1
,2
2
1
,
6
1
2
1
22
2
22
22
2
22
33
3
32
0
≈⋅/+++=
⋅/+⋅+⋅+⋅=
⋅/+⋅+⋅+⋅=⋅∆






36
SOLO
N

N

N

(i) Elliptic Case (ii) Hyperbolic Case (iii) Parabolic Case
02
>− MNL 02
<− MNL
0
&0
222
2
≠++
=−
MNL
MNL

37
SOLO
( )vur ,

vdrv

P
O
N

1nr

2nr

udru

2
M
1
M
02
>− MNL
Dupin’s Indicatrix
N

1nr

2n
r

P
2
M
1
M
02
<− MNL
N

1n
r
2nr

P
1M
2M
0
0
222
2
≠++
=−
MNL
MNL
http://www.mathcurve.com/surfaces/inicatrixdedupin/indicatrixdedupin.html
Pierre Charles François
Dupin
1784 - 1873
We want to investigate the curvature propertiesat a point P.
( ) ( ) IINvdudOvdNvdudMudLNr
2
1
,2
2
1 22
2
22
≈⋅/+++=⋅∆

The expression
12
2
221
2
1
=++ xNxxMxL
was introduced by Charles Dupin in 1813 in “Développments
de géométrie”, to describe the local properties of a surface.
http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html

38
SOLO
N

N

(iv) Planar Case
0=== MNL
( ) ( )
( )3223
33
3
3223
6
1
,33
6
1
vdDvdudCvdudBudA
vdudOvdrNvdudrNvdudrNudrNNr vvvvuvvuuuuu
+++≈
/+⋅+⋅+⋅+⋅=⋅∆

DxCxBxA +++ 23
has 3 real roots
Monkey Saddle
DxCxBxA +++ 23
has one real root

39
SOLO
( ) ( )vurvurr ,,

==Change of coordinates from u,v to θ,φ
( )
( )vuvv
vuuu
,
,
=
=












=





v
u
vv
uu
vd
ud
vu
vu
( ) ( )[ ] ( ) ( )[ ]2222
22 uuuuuvvuuvuuuuuu vNvuMuLNvrvururNrL ++±=⋅++=⋅=

[ ] ( )[ ]vuvuvuvuvuvvvuvuvuuvvuuuvu
vvNvuuvMuuLNvvrvuruvruurNrM +++±=⋅+++=⋅=

( ) ( )[ ] ( ) ( )[ ]2222
2 vvvvvvvvvuvvvvuvuuvv
vNvuMuLNvruvrvururNrN ++±=⋅+++=⋅=

Unit Normal Vector to the surface at P
vu
vu
vu
vu
rr
rr
rr
rr
N 



×
×
±=
×
×
=:
uvuuvuu
vrur
u
v
r
u
u
rr

+=
∂
∂
+
∂
∂
= vvvuvuv
vrur
v
v
r
v
u
rr

+=
∂
∂
+
∂
∂
=
[ ] [ ] [ ] II
vd
ud
NM
ML
vdud
vd
ud
vv
uu
NM
ML
vu
vu
vdud
vd
ud
NM
ML
vdudII
vu
vu
vv
uu
±=











±=

























=













=
Second Fundamental Form is Invariant (unless the sign) to Coordinate Transformation

40
SOLO
N

Osculating
Plane of C
at P
Principal Normal
Line of C at P
Surface
t

P
k

n1
( )vur ,

Normal Curvature
( ) ( )( )tvturr ,

=
- Length differential( ) 2/1
rdrdrdsd ⋅==
Given a path on a surface of class
Ck
( k ≥ 2) we define:
td
rd
td
rd
sd
rd
t /: ==
 - unit vector of path C at P
(tangent to C at P)
td
rd
td
td
sd
td
k /: ==

- curvature vector of path C at P



 =⋅
===
curvatureofradius
nn
nnk
sd
td
k
ρρ
111
1
1
1

( )NNkkn

⋅=: - normal curvature vector to C at P
( )
( ) ραα /coscos1
:
==⋅=
⋅=
kNnk
Nkkn


- normal curvature to C at P

41
SOLO
N

Osculating
Plane of C
at P
Principal Normal
Line of C at P
Surface
t

P
k

n1
( )vur ,

Normal Curvature (continue – 1)
N
Because C is on the surface, is on the tangent
plan normal to .
t

( ) td
Nd
tN
td
td
td
Nd
tN
td
td
Nt
td
d
Nt






⋅−=⋅→=⋅+⋅=⋅→=⋅ 00
and
( ) ( ) ( ) ( )[ ]vdrudrvdrudrvdNudNvdrudr
td
rd
td
rd
td
Nd
td
rd
td
rd
td
Nd
td
rd
td
rd
td
Nd
t
td
rd
N
td
td
N
sd
td
Nkk
vuvuvuvu
n









+⋅++⋅+−=






⋅⋅−=
⋅−=⋅−=⋅=⋅=⋅=
/
/
///
2
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
td
vd
G
td
vd
td
ud
F
td
ud
E
td
vd
N
td
vd
td
ud
M
td
ud
L
kn
+





+





+





+





==
++
++
=






+











+











+











+





=
2
2
2
2
2
2
2
2
22
22
22
22

42
SOLO
Normal Curvature (continue – 2)
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
td
vd
G
td
vd
td
ud
F
td
ud
E
td
vd
N
td
vd
td
ud
M
td
ud
L
kn
+





+





+





+





==
++
++
=






+











+











+











+





=
2
2
2
2
2
2
2
2
22
22
22
22
- kn is independent on dt therefore on C.
- kn is a function of the surface parameters L, M, N, E, F, G
and of the direction .vd
ud
- Because I = E du2
+ 2 F du dv + G dv2
> 0 → sign kn=sign II
- kn is independent on coordinates since I and II are independent.
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C

43
SOLO
Principal Curvatures and Directions
G
vd
ud
F
vd
ud
E
N
vd
ud
M
vd
ud
L
I
II
vdGvdudFudE
vdNvdudMudL
kn
+





+





+





+





==
++
++
=
2
2
2
2
2
2
22
22
- kn is a function of the surface parameters L, M, N, E, F, Gand of the direction .vd
ud
Let find the maximum and minimum of kn as functions of the directions d u/ d v.
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
If this occurs for d u0/ d v0 we must have:
( ) ( ) ( ) ( )
0&0
00
00
0000
00
00 ,
2
,,
2
,
=
−
=
∂
∂
=
−
=
∂
∂
vdud
vdvd
vdud
n
vdud
udud
vdud
n
I
IIIIII
v
k
I
IIIIII
u
k
Multiply by I and use
( ) ( )
( )( )
( ) ( )
( )( )
0&0
00
00
00
00
00
00
00
00
00
00
,
,,
0
,
,,
0
=−=





−=
∂
∂
=−=





−=
∂
∂
vdud
vdnvd
vdud
vdvd
vdud
n
vdud
udnud
vdud
udud
vdud
n
IkII
I
II
III
v
k
IkIII
I
II
II
u
k
( )00 ,
0
vdud
n
I
II
k =

44
SOLO
Principal Curvatures and Directions (continue – 1)
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
( )
( )( )
( )
( )( )
0&0
00
00
00
00
00
00
,
,
0
,
,
0
=−=
∂
∂
=−=
∂
∂
vdud
vdnvd
vdud
n
vdud
udnud
vdud
n
IkII
v
k
IkII
u
k
22
2: vdNvdudMudLNdrdII ++=⋅−=
22
2: vdGvdudFudErdrdI ++=⋅=
00 220
vdFudEI ud += 00 220
vdGudFI vd +=
00
220
vdMudLII ud
+= 00 220
vdNudMII vd +=
( )
( )( )
0
00
00
00
,
,
0
=−=
∂
∂
vdud
udnud
vdud
n
IkII
u
k
( )
( )( )
0
00
00
00
,
,
0
=−=
∂
∂
vdud
vdnvd
vdud
n
IkII
v
k
( ) ( ) 00000 0
=+−+ vdFudEkvdMudL n
( ) ( ) 00000 0
=+−+ vdGudFkvdNudM n

45
SOLO
We found:
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
( ) ( )
( ) ( )



=+−+
=+−+
0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
or:






=













−−
−−
0
0
0
0
00
00
vd
ud
GkNFkM
FkMEkL
nn
nn
This equation has non-trivial solution if:
0det
00
00
=








−−
−−
GkNFkM
FkMEkL
nn
nn
or expending: ( ) ( ) ( ) 02 222
00
=−+−+−− MNLkMFLGNEkFGE nn

46
SOLO
Study of the quadratic equation:
( )vur ,

rd
udru

vdrv
P
O
N
1C
k
2Ck

1C
2C
The discriminant of this equation is:
( ) ( ) ( ) 02 222
00
( ) ( ) ( )222
42 MNLFGEMFLGNE −−−−+=∆
( ) ( )
2
22
222
2
2
2222
2
2
22222424 





−++−−++−




 −
−−+




 −
=
E
LF
LG
E
LF
MFLGNEENLLFMELF
E
FGE
LFMELFME
E
FGE
( ) ( ) NLFNLGE
E
MLF
LMGF
E
LF
E
LGF
LFME
E
F
LGNELFME
E
FGE 2
3
2
24222
2
2
2
44884424 +−−++−





−−−+−




 −
=
E
LGF
LG
E
LGF
LMGFLG
NLGE
E
LF
E
MLF
E
LGF
NLF
E
LF
22
22
22
22
2
24322
2
2
24
84884
488444
−++−−
+−++−+
( ) ( ) ( )
( ) ( ) 024
42
2
2
2
0
2
222
≥





−−−+−










−
≡
−−−−+=∆
≥
LFME
E
F
LGNELFME
E
FGE
MNLFGEMFLGNE


47
SOLO
Study of the quadratic equation (continue – 1):
( )vur ,

rd
udru

vdrv
P
O
N
1C
k
2Ck

1C
2C
The discriminant of this equation is:
( ) ( ) ( ) 02 222
00
( ) ( ) ( )
( ) ( ) 024
42
2
2
2
0
2
222
≥





−−−+−










−
≡
−−−−+=∆
≥
LFME
E
F
LGNELFME
E
FGE
MNLFGEMFLGNE

The discriminant is greater or equal to zero, therefore we always obtain two real solutions
that give extremum for kn: 21
, nn
kk
Those two solutions are called Principal Curvatures and the corresponding two directions
are called Principal Directions ( ) ( )2211
,,, vdudvdud
The discriminant can be zero if: ( ) ( ) 02&0 =−−−=− LFME
E
F
LGNELFME
0&0 =−=− LGNELFME
G
N
F
M
E
L
==
In this case:
G
N
F
M
E
L
vdGvdudFudE
vdNvdudMudL
kn ===
++
++
= 22
22
2
2
This point in which kn is constant
in all directions is called an
Umbilical Point.

48
SOLO
Gaussian and Mean Curvatures
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
Rewrite the equation:
( ) ( ) ( ) 02 222
00
as:
( )
( )
( )
( )
0
2
2
2
2
2
00
=
−
−
+
−
−+
−
FGE
MNL
k
FGE
MFLGNE
k nn
We define:
( )
( )2
2
: 21
FGE
MFLGNE
kkH nn
−
−+
−=+=
( )
( )2
2
21
:
FGE
MNL
kkK nn
−
−
==
Mean Curvature
Gaussian Curvature
Karl Friederich Gauss
1777-1855

49
SOLO
Gaussian and Mean Curvatures (continue – 1)
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
We found:
( )
( )2
2
21
:
FGE
MNL
kkK nn
−
−
== Gaussian Curvature
( ) ( )vurvurr ,,

==Change of coordinates from u,v to θ,φ
( )
( )vuvv
vuuu
,
,
=
=












=





v
u
vv
uu
vd
ud
vu
vu
[ ] [ ] II
vd
ud
NM
ML
vdud
vd
ud
vv
uu
NM
ML
vu
vu
vdudII
vu
vu
vv
uu
±=











±=

























=
[ ] [ ] I
vd
ud
GF
FE
vdud
vd
ud
vv
uu
GF
FE
vu
vu
vdudI
vu
vu
vv
uu
=











=

























=




















=





vu
vu
vv
uu
vv
uu
GF
FE
vu
vu
GF
FE




















±=





vu
vu
vv
uu
vv
uu
NM
ML
vu
vu
NM
ML
( )
2
2
2
2
detdetdetdet














−=






















=





=−
vu
vu
vu
vu
vv
uu
FGE
vv
uu
GF
FE
GF
FE
FGE
( )
2
2
2
2
detdetdetdet














−=






















=





=−
vu
vu
vu
vu
vv
uu
MNL
vv
uu
NM
ML
NM
ML
MNL
Therefore: invariant to coordinate changes
( )
( )
( )
( )2
2
2
2
21
:
FGE
MNL
FGE
MNL
kkK nn
−
−
=
−
−
==

50
SOLO
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2CStart with: ( ) ( )
( ) ( )



=+−+
=+−+
0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
rewritten as :






=







−





++
++
0
01
00000
0000
nkvdGudFvdNudM
vdFudEvdMudL
that has a nontrivial solution (1,-kn0) only if:
0det
0000
0000
=





++
++
vdGudFvdNudM
vdFudEvdMudL
or: ( ) ( ) ( ) 0
2
000
2
0
=−+−+− vdNFMGvdudNEGLudMEFL
or:
( ) ( ) ( ) 0
0
0
2
0
0
=−+







−+







− NFMG
vd
ud
NEGL
vd
ud
MEFL

51
SOLO
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
We obtained:
This equation will define the two Principal Directions 2211 21
& vdrudrrvdrudrr vunvun

+=+=
( ) ( ) ( ) ( )
( )
( )
( )
( )
021
21
2
2
1
1
2
2
1
1
2112212121
≡





+
−
−
−
−
−
=








+





++=
⋅++⋅+⋅=⋅
vdvdG
MEFL
NEGL
F
MEFL
NFMG
E
vdvdG
vd
ud
vd
ud
F
vd
ud
vd
ud
E
vdvdrrvdudvdudrrududrrrr vVvuuunn

( ) ( ) ( ) 0
0
0
2
0
0
=−+







−+







− NFMG
vd
ud
NEGL
vd
ud
MEFL
From the equation above we have:
( )
( )
( )
( )MEFL
NFMG
vd
ud
vd
ud
MEFL
NEGL
vd
ud
vd
ud
−
−
=
−
−
−=+
2
2
1
1
2
2
1
1
Let compute the scalar product of the Principal Direction Vectors:
The Principal Direction Vectors
are perpendicular.

52
SOLO
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
Let perform a coordinate transformation to the Principal
Direction:
Since the two Principal Directions are orthogonal
21 21
& vdrrudrr vnun

==
They must satisfy the equation:
( )vu,
( ) ( ) ( ) 0
2
000
2
0
=−+−+− vdNFMGvdudNEGLudMEFL
( ) ( )21
,0&0, vdud
or:
( ) 0
2
1
=− udMEFL
( ) 0
2
2
=− vdNFMG 0=− NFMG
0
1
≠ud
0=− MEFL
02
≠vd
0≠E
0≠G
0
0
=⋅=
=⋅=
NrM
rrF
vu
vu


at P
Definition:
A Line of Curvature is a curve whose tangent at any point has a direction
coinciding with a principal direction at that point. The lines of curvature
are obtained by solving the previous differential equation

53
SOLO
( )vur ,

rd
udru

vdrv
P
O
N
1Ck
2Ck

1C
2C
Suppose (du0,dv0) is a Principal Direction, then they must satisfy the equations:
Rodriguez Formula
NrNrL uuuu

⋅=⋅−=
NrNrNrM vuuvvu

⋅=⋅−=⋅−=
NrNrN vvvv

⋅=⋅−=
( ) ( )
( ) ( )



=+−+
=+−+
0
0
0000
0000
0
0
vdGudFkvdNudM
vdFudEkvdMudL
n
n
( ) ( )
( ) ( )



=⋅+⋅−⋅−⋅−
=⋅+⋅−⋅−⋅−
0
0
0000
0000
0
0
vdrrudrrkvdNrudNr
vdrrudrrkvdNrudNr
vvvunvvuv
vuuunvuuu


uu
rrE

⋅=
vu
rrF

⋅=
vv
rrG

⋅=
( ) ( )[ ]
( ) ( )[ ]



=⋅+−−−
=⋅+−−−
0
0
0000
0000
0
0
vvunvu
uvunvu
rvdrudrkvdNudN
rvdrudrkvdNudN


( )
( )



=⋅+
=⋅+
0
0
0
0
vn
un
rrdkNd
rrdkNd


But are in the tangent plane at P since and are, and the
vectors and are independent, therefore:
rdkNd n

0
+ Nd

rd

vr
 ur

00

=+ rdkNd n
The direction (du0,dv0) is a Principal Direction on a point on a surface if and only if
from some scalar k, and satisfy:00 vdNudNNd vu

+= 00 vdrudrrd vu

+=
rdkNd

−= Rodriguez Formula
We found:

54
SOLO
Conjugate Directions
( )vur ,

rd
udru

vdrv
P
O
N

Q
NdN

+
l
Let P (u,v) and Q (u+du,v+dv) neighboring points on a
surface. The tangent planes to the surface at p and Q
intersect along a straight line L. Now let Q approach P
along a given direction (du/ dv=const= PQ), then the line l
will approach a limit LC. The directions PQ and LC are
called Conjugate Directions.
Let be the normal at P and the normal at Q.N

NdN

+
Let the direction of LC be given by: vrurr vu
δδδ

+=
Since LC is in both tangential planes at P and at Q we have:
( ) 0&0 =+⋅=⋅ NdNrNr

δδ ( ) ( ) 0=+⋅+=⋅ vdNudNvrurNdr vuvu

δδδ
( ) ( ) ( ) ( ) 0=⋅+⋅+⋅+⋅ vdvNrvduNrudvNruduNr vvvuuvuu
δδδδ

We found vvuvvuuu
NrNNrNrMNrL

⋅−=⋅−=⋅−=⋅−= &&
The previous relation becomes: ( ) 0=+++ vdvNvduudvMuduL δδδδ
Given (du,dv) there is only one conjugate direction (δu,δv) given by the previous
equation.

55
SOLO
Asymptotic Lines
The directions which are self-conjugate are called asymptotic directions.
becomes:
( ) 0=+++ vdvNvduudvMuduL δδδδ
We see that the asymptotic directions are those for which the second fundamental
form vanishes. Moreover, the normal curvature kn vanishes for this direction.
Those curves whose tangents are asymptotic directions are called asymptotic lines.
v
u
vd
ud
δ
δ
=If a direction (du,dv) is self-conjugate than and the equation of
conjugate lines
02 22
=++ vdNvdudMudL
The conjugat and asymptotic lines were introduced by Charles
Dupin in 1813 in “Dévelopments de Géométrie”.
Pierre Charles François
Dupin
1784 - 1873
http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html

56
SOLO
T J Freeth (1819-1904) was an
English mathematician. In a
paper published by the London
Mathematical Society in 1879
he described various strophoids,
including the strophoid of a
trisectrix.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Freeths.html
Planar Curves
Nephroid (meaning 'kidney shaped')
http://mathworld.wolfram.com/Nephroid.html
( )
( )





−=
−=
θθ
θθ
3sinsin3
2
3coscos3
2
a
y
a
x
Freeth’s Nephroid ( )2/sin1 θbr +=
http://www.2dcurves.com/derived/strophoidn.html
http://curvebank.calstatela.edu/nephroid/nephroid.htm

57
SOLO
Planar Curves
Bow Tie
( )θ2sin21+=r ( )θ3sin31+=r
Double Rose
http://curvebank.calstatela.edu/index/bowtie2.gif

58
SOLO
Planar Curves






−−=
−=
θθ
θθ
3sin
2
1
sin
2
3
3cos
2
1
cos
2
3
y
x
Parabolic Spiral
bar += θ

59
SOLO
Planar Curves
Fermat Spiral θar =
http://www.wco.com/~ejia/eduframe.htm
( )
( ) 2/32
2
41
432
θ
θθ
+
+
=
a
k
Pierre Fermat
1601 - 1665
This spiral was discussed by Fermat in 1636
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Fermats.html

60
SOLO
Planar Curves
http://en.wikipedia.org/wiki/Hyperbolic_spiral
θ
a
r =
Archimedean Spiral θbar +=
http://en.wikipedia.org/wiki/Archimedean_spiral
2/πθ −
= ar
Hyperbolic Spiral

61
SOLO
Planar Curves
Equiangular Spiral
Equiangular spiral (also known as logarithmic spiral,
Bernoulli spiral, and logistique) describe a family of
spirals. It is defined as a curve that cuts all radii vectors
at a constant angle.
The famous Equiangular Spiral was discovered by
Descartes, its properties of self-reproduction by
James (Jacob) Bernoulli (1654-1705) who requested that
the curve be engraved upon his tomb with the phrase
"Eadem mutata resurgo" ("I shall arise the same,
though changed.")
http://xahlee.org/SpecialPlaneCurves_dir/EquiangularSpiral_dir/equiangularSpiral.html
Logarithmic Spiral
θb
ear =
http://en.wikipedia.org/wiki/Logarithmic_spiral
Nautilus Shells
René Descartes
1596 - 1650
Jacob Bernoulli
1654-1705
http://mathworld.wolfram.com/LogarithmicSpiral.html

62
SOLO
Planar Curves
Equiangular (Logarithmic, Bernoulli) Spiral θb
ear =
Evolute of Logarithmic Spiral
also a Logarithmic Spiral

63
SOLO
Planar Curves
Equiangular (Logarithmic, Bernoulli) Spiral θb
ear =
http://www2.mat.dtu.dk/people/V.L.Hansen/nautilus/naustory.html
If the circles of curvature of the logarithmic spiral are placed so that they are
centered on the curve and are othogonal both to the curve and to the plane of the
curve, then a Nautilus shell appears.

64
SOLO
Planar Curves
Cycloid
http://mathworld.wolfram.com/Cycloid.html
( )
( )


−=
−=
tay
ttax
2cos1
2sin
http://xahlee.org/SpecialPlaneCurves_dir/Cycloid_dir/cycloid.html
Cycloid is defined as the trace of a point on
the circumsference of a circle rolling on
a line without slipping.
Curtate Cycloid
http://mathworld.wolfram.com/CurtateCycloid.html



−=
−=
tbay
tbtax
cos
sin
Curtate Cycloid is defined as the trace of
a fixed point at a distance b from the center
of a circle of radius a rolling on a line
without slipping.

65
SOLO
Planar Curves
Prolate Cycloid



−=
−=
tbay
tbtax
cos
sin
The path traced out by a fixed point at a
radius b>a , where a is the radius of a rolling
circle, also sometimes called an extended
cycloid. The prolate cycloid contains loops,
and has parametric equations:
http://mathworld.wolfram.com/ProlateCycloid.html

66
SOLO
Planar Curves
Epicycloid
( ) ( ) ( )( )
( ) ( )( )













+
+
−+=






+
+
−+=
1
1sin
sin1
1
1cos
cos1
k
k
kry
k
k
krx
θ
θ
θ
θθ
http://en.wikipedia.org/wiki/Epicycloid
Epicycloid is defined as the trace of a point P on the
circumsference of a circle rolling on a circle without slipping.
http://mathworld.wolfram.com/Epicycloid.html
bak /=

67
SOLO
Planar Curves
Epicycloid Involute
( ) ( )
( )

















 +
−+=











 +
−+=
θθ
θθθ
b
ba
bbay
b
ba
bbax
sinsin
coscos
http://mathworld.wolfram.com/Epicycloid.html
( ) ( )
( )

























 +
++
+
=



















 +
++
+
=
θθ
θθθ
b
ba
bba
a
ba
y
b
ba
bba
a
ba
x
sinsin
2
coscos
2
Epicycloid Involute
Epicycloid
http://mathworld.wolfram.com/EpicycloidInvolute.html

68
SOLO
Planar Curves
Cardioid
( )θcos1−= ar
http://en.wikipedia.org/wiki/Cardioid













−=






−=
ttry
ttrx
2sin
2
1
sin2
2cos
2
1
cos2
Studied by Roemer (1674) in an investigation for the best form of gear teeth.
The name cardioid (heart-shaped; from Greek root cardi, meaning heart) was first used
by de Castillon in the Philosophical Transactions of the Royal Society of 1741.
Its length is found by La Hire in 1708.
Cardioid is a special case of Limaçon (or Snail) of Pascal and is an
Epicycloid (k = 1).
Cardioid is defined as the trace of a point on the
circumsference of a circle rolling on a circle with equal radius
without slipping.
Philippe de la Hire
1640 - 1718
Johann Castillon
1704 - 1791

69
SOLO
Planar Curves
Ranunculoid
An epicycloid with n = 5 cusps, named after the buttercup genus Ranunculus
(Madachy 1979).
http://mathworld.wolfram.com/Ranunculoid.html

70
SOLO
Planar Curves
Limaçon of Pascal or Snail of Pascal
http://xahlee.org/SpecialPlaneCurves_dir/LimaconOfPascal_dir/limaconOfPascal.html
http://mathworld.wolfram.com/Limacon.html
θcosabr +=
Blaise Pascal
1623 - 1662
Discovered and named after Etienne Pascal (1588-1640) , father of Blaise Pascal.
Also discussed by Roberval in 1650.

71
SOLO
Planar Curves
Hypotrohoid
( ) ( )
( )











 −
−−=





 −
+−=
θθ
θθθ
b
ba
bbay
b
ba
bbax
sinsin
coscos
The curve produced by fixed point P at a distance h from the center of a small circle
of radius b rolling without slipping around the inside of a large circle of radius a > b.
http://mathworld.wolfram.com/Hypotrochoid.html

72
SOLO
Planar Curves
Hypocycloid
( ) ( )
( )











 −
−−=





 −
+−=
θθ
θθθ
b
ba
bbay
b
ba
bbax
sinsin
coscos
The curve produced by fixed point P on the circumference of a small circle of radius b
rolling without slipping around the inside of a large circle
of radius a > b. A hypocycloid is a hypotrochoid with h=b .
http://mathworld.wolfram.com/Hypocycloid.html

73
SOLO
Planar Curves
Hypocycloid Involute
( ) ( )
( )

















 −
−−
−
=











 −
+−
−
=
θθ
θθθ
b
ba
bba
ba
a
y
b
ba
bba
ba
a
x
sinsin
2
coscos
2
http://mathworld.wolfram.com/Hypocycloid.html
( ) ( )
( )

















 −
−−
−
=











 −
+−
−
=
θθ
θθθ
b
ba
bba
a
ba
y
b
ba
bba
a
ba
x
sinsin
2
coscos
2
Hypocycloid Involute
Hypocycloid
http://mathworld.wolfram.com/HypocycloidInvolute.html

74
SOLO
Planar Curves
Steiner’s Hypocycloid (Deltoid)
http://mathworld.wolfram.com/Deltoid.html
http://mathworld.wolfram.com/SteinersHypocycloid.html
( )
( )





−=
+=
θθ
θθ
2sinsin2
3
2coscos2
3
a
y
a
x
The deltoid was first considered by Euler in 1745 in
connection with an optical problem. It was also
investigated by Steiner in 1856 and is sometimes called
Steiner's hypocycloid
Jacob Steiner
1796 - 1863
Leonhard Euler
1707 - 1783

75
SOLO
Planar Curves
Astroid
The Astroid was tudied by Johan Bernoulli (1667 – 1748) ,by D’Alembert
in 1748. The name was given by Littrow in 1838.
Johann Bernoulli
1667-1748
Jean Le Rond D’Alembert
1717 - 1783



=
=
θ
θ
3
3
sin
cos
ay
ax
The Astoid can be obtained as a hypocycloid with b/a=1/4 or 3/4
The astroid is the curve performed by
a bus door
The astroid is the curve obtained by the
intersection of two circles rolling without
slipping inside a bigger circle.

76
SOLO
Planar Curves
Lituus
θ
1
=r
Roger Cotes
1682 - 1716
The Lituus curve was studied by Roger Cotes
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lituus.html http://curvebank.calstatela.edu/hyperbolafermat

77
SOLO
Planar Curves
Cisoid of Diocles
http://en.wikipedia.org/wiki/Cissoid_of_Diocles
( ) 2/2/tansin2cossec2 πθπθθθθ ≤≤−=−= aar
http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/cissoiddiocles/
The cissoid of Diocles is named after the Greek geometer Diocles who used it in
180 B.C. to solve the Delian problem: how much must the length of a cube be increased
in order to double the volume of the cube?
The name "cissoid" first appears in the work of Geminus about 100 years later.
Fermat and Roberval constructed the tangent in 1634. Huygens and Wallis found,
in 1658, that the area between the curve and its asymptote was (MacTutor Archive).
From a given point there are either one or three tangents to the cissoid.

78
SOLO
Planar Curves
Sinusoid Spiral
A sinusoidal spiral is a curve of the form: with n rational, which is
not a true spiral.
( )θnar nn
cos=
n curve
-2 hyperbola
-1 line
parabola
Tschirnhausen
cubic
Cayley;s sextic
cardioid
1 circle
2 lemniscate
The curvature is: ( )
( )θ
θ
na
n
k n 1/1
cos
1
−
+
=

79
SOLO
Planar Curves
Witch of Agnesi
This was studied and named versiera by Maria Agnesi in 1748 in her book
Istituzioni Analitiche. It is also known as Cubique d'Agnesi or Agnésienne.
Maria Gaetana Agnesi
1718 - 1799
http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Witch.html
The curve had been studied earlier by Fermat and Guido Grandi in 1703
Luigi Guido Grandi
1671 - 1742
Pierre Fermat
1601 - 1665
The curve is obtained by drawing a line
from the origin through the circle of radius
a and center (0,a), then picking the point
with the y coordinate of the intersection with
the circle and the x coordinate of the
intersection of the extension of line OB with
the line y = 2 a .
http://mathworld.wolfram.com/WitchofAgnesi.html
( )


−=
=
tay
tax
2cos1
cot2
The name "witch" derives from a mistranslation of the term averisera ("versed sine curve,"
from the Latin vertere, "to turn") in the original work as avversiera ("witch" or "wife of the
devil") in an 1801 translation of the work by Cambridge Lucasian Professor of Mathematics
John Colson (Gray).

80
SOLO
Planar Curves
Cassini Ovals
The curve was first investigated by Cassini in 1680 when he was
studying the relative motions of the Earth and the Sun. Cassini
believed that the Sun traveled around the Earth on one of these ovals,
with the Earth at one focus of the oval.
The Cassini ovals are a family of quartic curves, also called Cassini ellipses,
described by a point such that the product of its distances from two fixed points a
distance 2 a apart is a constant b2
. The shape of the curve depends on b/a. If a < b ,
the curve is a single loop with an oval (left figure above) or dog bone (second figure)
shape. The case a = b produces a lemniscate (third figure). If a > b, then the curve
consists of two loops (right figure).
( ) 4222222
4 bxaayx =−++
( ) 42244
2cos2 braar =−+ θ
( )[ ] ( )[ ] 42222
byaxyax =+++−
Giovanni Domenico Cassini
1625 - 1712

81
SOLO
In polar form:
Planar Curves
Lemniscate
The Cartesian equation: ( ) yxyx 2
222
=+
( )θ2sin2
=r

82
SOLO
In polar form:
Planar Curves
Lemniscate of Bernoulli
The lemniscate, also called the lemniscate of Bernoulli, is a polar curve whose most
common form is the locus of points the product of whose distances from two fixed points
(called the foci) a distance 2c away is the constant c2
. This gives the Cartesian equation:
( ) ( )222222
2 yxcyx −=+
http://mathworld.wolfram.com/Lemniscate.html
( )θ2cos2 22
cr =
Jakob Bernoulli published an article in Acta Eruditorum in 1694 in
which he called this curve the lemniscus (Latin for "a pendant
ribbon"). Bernoulli was not aware that the curve he was describing
was a special case of Cassini Ovals which had been described by
Cassini in 1680. The general properties of the lemniscate were
discovered by G. Fagnano in 1750 (MacTutor Archive)
Jacob Bernoulli
1654-1705
The most general form of the lemniscate is a toric section of a torus.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lemniscate.html
( )[ ] ( )[ ] 42222
cycxycx =+++−

83
SOLO
In polar form:
Planar Curves
Ovales &Lemniscate of Booth
http://www.mathcurve.com/courbes2d/booth/booth.shtml
( )
( )
Lemniscate
Ovaleab
ybxayx
1
01
2222222
−=
<≤=
+=+
ε
ε
ε
θεθ 22222
sincos bar +=
J. Booth (1810 -1878 ) : ????
The Ovales of Booth are the locus of the center of an ellipse rolling without slipping around
an identical Ellipse.

84
SOLO
In polar form:
Planar Curves
( )
( )
Lemniscate
Ovaleab
ybxayx
1
01
2222222
−=
<≤=
+=+
ε
ε
ε
θεθ 22222
sincos bar +=
J. Booth (1810 -1878 ) : ????
On en déduit que les lemniscates de Booth sont les lieux du centre d'une hyperbole
roulant sans glisser sur une hyperbole égale, avec des sommets coïncidants.
The Lemniscate of Booth are the locus of the center of a hyperbole rolling without
slipping on an equal hyperbole.

85
SOLO
In polar form:
Planar Curves
( )
( )
Lemniscate
Ovaleab
ybxayx
1
01
2222222
−=
<≤=
+=+
ε
ε
ε
θεθ 22222
sincos bar +=
J. Booth (1810 -1878 ) : ????
Ce sont donc les enveloppes de cercle
de diamètre joignant le centre d'une
conique à un point de cette conique
The envelopes of the circle
with a diameter that joints the
center of a conic with a point
on the circle.

86
SOLO
Planar Curves
Concoid (cochloid) of Nicomedes
http://nvizx.typepad.com/nvizx_weblog/2005/08/conchoid_of_nic.html
Nicomedes (circa 280 BCE - 210 BCE) is best known for his treatise on conchoids,
a family of curves of one parameter. This family, now known as the Conchoid of
Nicomedes, has a number of interesting properties and uses. Classical applications
included the trisection of an angle and a means of solving the Greek cube doubling
problem. In two dimensions, a family of these curves can be generated by varying
the parameters a and b while plotting the
http://www.oberonplace.com/products/plotter/gallery/page2.htm
Nicomedeshttp://mathworld.wolfram.com/ConchoidofNicomedes.html
θsecbar +=
http://curvebank.calstatela.edu/conchoid/conchoidforever.gif

87
SOLO
Planar Curves
Roses






+= 0
cos ϕθ
q
p
ar
http://xahlee.org/SpecialPlaneCurves_dir/Rose_dir/rose.html

88
SOLO
Planar Curves
Roses






+= 0
cos ϕθ
q
p
ar

89
SOLO
Planar Curves
Roses






+= 0
cos ϕθ
q
p
ar
p = 1, q =-5 p = 1, q = 5
http://www.mathcurve.comcourbes2drosacerosace.shtml

90
SOLO
The case n = 2/3 is theAstroid, while the case n = 3 is the (so-called) Witch of Agnesi.
Planar Curves
Lamé Curves (Super-ellipses)
Gabriel Lamé
1795 - 1870
1=+
nn
b
y
a
x
In 1818 Lamé discussed the curves with equation given above. He considered more general curves
than just those where n is an integer. If n is a rational then the curve is algebraic but, for irrational
n, the curve is transcendental.
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lame.html
http://en.wikipedia.org/wiki/Super_ellipse
http://mathworld.wolfram.com/Superellipse.html
Families of curves generated by the "superformula" with a = b = 1 and n varying from 0 to 2
are illustrated above for values of n=n1=n2=n3 varying from 1 to 8.
( )
1
32
/1
4
1
sin
4
1
cos
n
nn
b
m
a
m
r
−
































+


















=
θθ
θ
A polar generalization of
Lamé’s formula.

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Planar Curves
Lissajous or Bowditch Curves
Nathaniel Bowditch (1773 - 1838) was the first American to receive international
recognition as a mathematician. Moreover, he was the first to investigate a family
of curves now usually named for the French physicist, Jules-Antoine Lissajous.
Lissajous independently published his work much later in 1857
( )



=
+=
tby
ctnax
sin
sin
Nathaniel Bowditch
(1773 - 1838)
Jules-Antoine Lissajous
1822 - 1880
http://curvebank.calstatela.edulissajous

92
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Planar Curves







−
=
−
−
=
tt
tt
y
tt
tt
x
3coscos2
3coscos3
3coscos2
13coscos2
( )
( )



−=
+=
1cos2sin
1cos2cos
θθ
θθ
y
x

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Space Curves
Spherical Spiral











+
=
+
=
+
=
2
2
2
1
1
sin
1
cos
t
t
z
t
t
y
t
t
x
α
α
α
α
http://en.wikipedia.org/wiki/Spiral

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References
H. Lass, “Vector and Tensor Analysis”, McGraw Hill, 1950, Ch. 3: “Differential Geometry”
M.R. Spiegel, “Vector Analysis and an Introduction to Tensor Analysis”, Schaum’s
Outline Series, McGraw Hill, 1959
E. Kreyszig, “Introduction to Differential Geometry and Riemannian Geometry”,
University of Toronto Press, 1968
M.M. Lipschutz, “Differential Geometry”, Schaum’s Outline Series, McGraw Hill,
1969
Bo-Yu Hou & Bo-Yuan Hou, “Differential Geometry for Physicists”, World Scientific,
1997
http://www.mathcurve.com
http://www-groups.dcs.st-and.ac.uk/~history/Curves
http://mathworld.wolfram.com
http://en.wikipedia.org/wiki/List_of_differential_geometry_topics
http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html

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References
Tensors
H. Lass, “Vector and Tensor Analysis”, McGraw Hill, 1950
M.R. Spiegel, “Vector Analysis and an Introduction to Tensor Analysis”, Schaum’s
Outline Series, McGraw Hill, 1959
D. Lovelock & H. Rund, “Tensor, Differential Forms, and Variational Principles”,
Dover Publications, 1975, 1989
J.A. Schouten, “Tensor Analysis for Physicists”, 2nd
Ed., Dover Publications, 1989 (1959)
A.I. Borisenko & I.E. Tarapov, “Vector and Tensor Analysis with Applications”,
Dover Publications, 1968
A.J. McConnell, “Applications of Tensor Analysis ”, Dover Publications, 1957
Bishop, R. and Goldberg, S., “Tensor Analysis on Manifolds”, New York: Dover, 1980.
Aris, R., “Vectors, Tensors and the Basic Equations of Fluid Mechanics”,
New York: Dover, 1989.

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References
Tensors
Table of Content

97
Camille Jordan
1838 - 1922
http://www.mathcurve.com/surfaces/mobius/mobius.shtml
http://curvebank.calstatela.edu/arearev

January 6, 2015 98
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Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –2013
Stanford University
1983 – 1986 PhD AA

Differential geometry three dimensional space

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