Lamarle Formula in 3-Dimensional Lorentz Space

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by  Soley Ersoy

Lamarle Formula in 3-Dimensional Lorentz Space
Soley ERSOY, Murat TOSUN
Department of Mathematics, Faculty of Arts and Sciences
Sakarya University, 54187 Sakarya/TURKEY
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1 January 5, 2010
0
2

n Abstract
a
J TheLamarleFormula,givenbyKruppain[8],isknownasarelation-
shipbetweentheGaussiancurvatureandthedistributionparameterofa
5
ruled surface in the surface theory. The ruled surfaces were investigated

] in3differentclasseswithrespecttothecharacterofbasecurvesandrul-
G ings,[14],[15]. Inthispaperonaccountofthesestudies,therelationships
D between the Gaussian curvatures and distribution parameters of space-
like ruled surface, timelike ruled surface with spacelike ruling and time-
.
h like ruled surface with timelike ruling are obtained, respectively. These
at relationshipsarecalledasLorentzianLamarleformulas. Finallysomeex-
m amples concerning with these relations are given.
Subject Classification: 53B30, 53C50, 14J26
[
Key Words: Ruledsurface,distributionparameter,Gaussiancurvature,

1 Lamarle formula.
v
9
9
6 1 Introduction
0
1. The study of ruled surface in R3 is classical subject in differential geometry. It
0 hasagainbeenstudiedinsomeareas(i.e. Projectivegeometry, [13], Computer-
0
aideddesign,[11],etc.) Also,itiswellknownthatthegeometryofruledsurface
1
is very important of kinematics or spatial mechanisms in R3, [4], [7]. A ruled
:
v surface is one which can be generated by sweeping a line through space. Devel-
i
X opable surfaces are special cases of ruled surfaces, [10]. Cylindrical surfaces are
examples of developable surfaces. On a developable surface at least one of the
r
a two principal curvatures is zero at all points.Consequently the Gaussian curva-
ture is zero everywhere too. So it is meaningful for us to study non-cylindrical
ruled surfaces.
Lorentz metrics in 3 dimensional Lorentz space R3 is indefinite. In the theory
− 1
of relativity, geometry of indefinite metric is very crucial. Hence, the theory of
ruled surface in Lorentz space R3, which has the metric ds2 =dx2+dx2 dx2,
1 1 2− 3
attracted much attention. The situation is much more complicated than the
Euclidean case, since the ruled surfaces may have a definite metric (spacelike
surfaces), Lorentz metric (timelike surfaces) or mixed metric. Some character-
izations for ruled surfaces are obtained by [9]. Timelike and spacelike ruled
surfaces are defined and the characterizations of timelike and spacelike ruled
surfaces are found in [2], [5], [6], [14] and [15].
1
2 Preliminaries
Let R3 denote the 3 dimensional Lorentz space, i.e. the Euclidean space E3
1 −
with standard flat metric given by
g =dx2+dx2 dx2 (2.1)
1 2− 3
where (x ,x ,x ) is rectangular coordinate system of R3. Since g is indefinite
1 2 3 1
metric, recall that a vector ~v in R3 can have one of three casual characters:
1
it can be space-like if g(~v,~v) > 0 or ~v = 0, time-like if g(~v,~v) < 0 and null
g(~v,~v)=0 and~v =0. Similarly, an arbitrary curve α~ =α~(s) R3 can locally
6 ⊂ 1
be space-like, time-like or null (light-like), if all of its velocity vectors α~′(s) are
respectively space-like, time-like or null (light-like). The norm of a vector ~v
is given by ~v = g(~v,~v). Therefore, ~v is a unit vector if g(~v,~v) = 1.
k k | | ∓
Furthermore, vectorps~v and w~ are said to be orthogonal if g(~v,w~)=0, [10].
Let the set of all timelike vectors in R3 be Γ. For ~u Γ, we call
1 ∈
C(~u)= ~v Γ ~v,~u <0
{ ∈ | h i }
as time-conic of Lorentz space R3 including vector ~u, [10].
1
Let ~v and w~ be two time-like vectors in Lorentz space R3. In this case there
1
exists the following inequality
g(~v,w~) ~v . w~ .
| |≥k k k k
In this inequality if one wishes the equality condition, then it is necessary for~v
and w~ be linear dependent.
If time-like vectors ~v and w~ stay inside the same time-conic then there is a
unique non-negative real number of θ 0 such that
≥
g(~v,w~)= ~v . w~ .coshθ (2.2)
−k k k k
where the number θ is called an angle between the timelike vectors, [10].
Let~v andw~ bespacelikevectorsinR3 thatspanaspacelikesubspace. Wehave
1
that
g(~v,w~) ~v . w~
| |≤k k k k
with equality if and only if ~v and w~ are linearly dependent. Hence, there is a
unique angle 0 θ π such that
≤ ≤
g(~v,w~)= ~v . w~ .cosθ (2.3)
k k k k
where the number θ is called the Lorentzian spacelike angle between spacelike
vectors~v and w~, [12].
Let~v and w~ be spacelike vectors in R3 that span a timelike subspace. We have
1
that
g(~v,w~) > ~v . w~ .
| | k k k k
Hence, there is a unique real number θ >0 such that
g(~v,w~)= ~v . w~ .coshθ. (2.4)
k k k k
The Lorentzian timelike angle between spacelike vectors ~v and w~ is defined to
be θ, [12].
2
Let ~v be a spacelike vector and w~ be a timelike vector in R3. Then there is a
1
unique real number θ 0 such that
≥
g(~v,w~)= ~v . w~ .sinhθ. (2.5)
k k k k
The Lorentzian timelike angle between~v and w~ is defined to be θ, [12].
For any vectors~v =(v ,v ,v ), w~ =(w ,w ,w ) R3 , the Lorentzian product
1 2 3 1 2 3 ∈ 1
~v w~ of~v and w~ is defined as [1]
∧
~v w~ =(v w v w ,v w v w ,v w v w ). (2.6)
3 2 2 3 1 3 3 1 1 2 2 1
∧ − − −
3 Ruled Surface in R3
1
A ruled surface M R3 is a regular surface that has a parametrization ϕ :
(I R) R3 of the∈form1
× → 1
ϕ(u,v)=α~(u)+v~γ(u) (3.1)
whereα~ and~γ arecurvesinR3 withα~′ nevervanishes. Thecurveαiscalledthe
1
basecurve. Therulingsofruledsurfacearethestraightlinesv α~(u)+v~γ(u).
IfconsecutiverulingsofaruledsurfaceinR3intersect,thenthe→surfaceissaidto
1
bedevelopable. Allotherruledsurfacesarecalledskewsurfaces. Ifthereexistsa
commonperpendiculartotwoconstructiverulingsintheskewsurface, thenthe
foot ofthecommon perpendicularonthemain rulingiscalled astriction point.
The set of striction points on a ruled surface defines the striction curve,[15].
The striction curve, β(u) can be written in terms of the base curve α(u) as
g(α~′(u),~γ′(u))
β~(u)=α~(u) ~γ(u). (3.2)
− ~γ′,~γ′
h i
A ruled surface given by (3.1) is called non-cylindrical if ~γ ~γ′ is nowhere
∧
zero. Thus, the rulings are always changing directions on a non-cylindrical
ruled surface. A non-cylindrical ruled surface always has a parameterization of
the form
ϕ˜(u,v)=β~(u)+v~e(u) (3.3)
where ~e(u) = ~γ(u) = 1, β~′(u),~e′(u) = 0 and β~(u) is striction curve of
k k k~γ(u)k D E
ϕ˜, [3].
Thedistributionparameter(ordrall)ofanon-cylindricalruledsurfacegivenby
equation (3.3), is a function P defined by
det β~′,~e,~e′
P = (cid:16) (cid:17). (3.4)
~e′,~e′
h i
where β~ is the striction curve and ~e is the director curve. Moreover, Gaussian
curvature of non-cylindrical ruled surface ϕ˜(u,v) is
LM N2
K = − (3.5)
EG F2
−
3
whereE, F andGarethecoefficientsofthefirstfundamentalform, whereasL,
N andM arethecoefficientsofthesecondfundamentalform,ofnon-cylindrical
ruled surface, [3].
The unit normal vector of non-cylindrical ruled surface ϕ˜ is given by
ϕ˜ ϕ˜
η(u,v)= u∧ v . (3.6)
ϕ˜ ϕ˜
u v
k ∧ k
A surface in the 3 dimensional Minkowski space-time R3 is called a time-like
− 1
surface if induced metric on the surface is a Lorentzian metric i.e. the normal
on the surface is a space-like vector, [15].
In R3, according to the character of the non-null base curve and the non-null
1
ruling, ruled surfaces are classified into three different groups. As a spacelike
ruling moves along a spacelike curve it generates a spacelike ruled surface, that
will be denoted by M . Furthermore, the movement of a timelike ruling along
1
a spacelike curve and the movement of a spacelike ruling along a timelike curve
generate timelike ruled surfaces. Let us denote these timelike ruled surfaces
by M and M , respectively. Now, we will establish Lamarle formula for these
2 3
ruled surfaces M , M , M separately.
1 2 3
4 Lamarle Formula for the Spacelike Ruled Sur-
face
Let M be a spacelike ruled surface parametrized by
1
ϕ :I R R3
1 × → 1
(u,v) ϕ (u,v)=α~ (u)+v~e (u).
1 1 1
→
Ifwechoose ~e =1,~n = ~e′1 andξ~ = ~e1∧~e′1 , weobtain the orthonormal
k 1k 1 ~e′1 1 ~e1∧~e′1
k k k k
framefield ~e ,~n ,ξ~ . Supposethattheseorthonormalframefieldformsright
1 1 1
n o
handed system and is space,time,space type. In this case we may write
{ }
~e ,~e =1, ~n ,~n = 1, ξ~ ,ξ~ =1,
1 1 1 1 1 1
h i h i − D E (4.1)
~e ,~n = ~n ,ξ~ = ξ~ ,~e =0
1 1 1 1 1 1
h i D E D E
and
~e ~n = ξ~ , ~n ξ~ = ~e , ξ~ ~e =~n . (4.2)
1 1 1 1 1 1 1 1 1
∧ − ∧ − ∧
The Frenet formulae of this orthonormal frame along e become
1
~e′ =κ ~n , ~n′ =κ ~e +τ ξ~ , ξ~′ =τ ~n . (4.3)
1 1 1 1 1 1 1 1 1 1 1
Let β~ (u) be a striction curve of spacelike ruled surface M given by equation
1 1
(3.2) in R3. In this case the tangent vector β~′ of this curve stays in spacelike
1 1
plane. Taking the angle σ to be the angle between β~′ and~e since the tangent
1 1 1
vector of striction curve of M is
1
β~′ =~e cosσ +ξ~ sinσ
1 1 1 1 1
4
we find the striction curve of M to be
1
β~ = cosσ ~e + sinσ ξ~ du.
1 Z (cid:16) 1 1 1 1(cid:17)
The spacelike non-cylindrical ruled surface M is parametrized by
1
ϕ˜ (u,v)= cosσ ~e +sinσ ξ~ du+v~e .
1 Z (cid:16) 1 1 1 1(cid:17) 1
From the equation (3.4) the distribution parameter of M is found to be
1
det cosσ ~e +sinσ ξ~ ,~e ,κ ~n
1 1 1 1 1 1 1 sinσ
P = (cid:16) (cid:17) = 1.
κ ~n ,κ ~n κ
1 1 1 1 1
h i
Adopting κ = 1 we get the distribution parameter as follows
1 ρ1
P =ρ sinσ . (4.4)
1 1
Consideringequation(4.2)fromequation(3.6)wewritetheunitnormaltangent
vector of spacelike non-cylindrical ruled surface M
1
sinσ ~n +vκ ξ~
~η = 1 1 1 1 . (4.5)
1
sin2σ +v2κ2
q − 1 1
(cid:12) (cid:12)
Taking into consideration that κ (cid:12)= 1 and equatio(cid:12)n (4.4) we obtain
1 ρ1
P~n + vξ~
~η = 1 1 . (4.6)
1
P2+v2
|− |
p
Furthermore, since the unit normal tangent vector η of a spacelike surface M
1 1
is timelike we find that P2+v2 <0, that is v < P .
− | | | |
The partial differentiation of M with respect to u and v from equation (4.3)
1
are as follows
ϕ˜ =cosσ ~e +sinσ ξ~ + vκ ~n ,
1u 1 1 1 1 1 1 (4.7)
ϕ˜ =~e .
1v 1
Therefore, we find the first fundamental form’s coefficients of M to be
1
E = ϕ˜ ,ϕ˜ =cos2σ +sin2σ v2κ2 =1 v2κ2,
h 1u 1ui 1 1− 1 − 1
F = ϕ˜ ,ϕ˜ =cosσ , (4.8)
1u 1v 1
h i
G= ϕ˜ ,ϕ˜ =1.
1v 1v
h i
Inadditiontothese, thesecondorderpartialdifferentialsofM arefoundtobe
1
ϕ˜ = σ′ sinσ +vκ2 ~e +(κ cosσ +τ sinσ +vκ′)~n +(σ′ cosσ +vκ τ )ξ~ ,
1uu − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ϕ˜ =κ(cid:0) ~n , (cid:1)
1uv 1 1
ϕ˜ =0.
1vv
From equation (4.5) and the last equations we get the coefficients of second
fundamental of M as
1
5
L= ϕ˜ ,~η = −κ1cosσ1sinσ1−τ1sin2σ1−vκ′1sinσ1+σ1′cosσ1vκ1+v2κ21τ1,
1uu
h i q−sin2σ1+v2κ21
| |
N = ϕ˜ ,~η = −κ1sinσ1 , (4.9)
1uv
h i q−sin2σ1+v2κ21
| |
M = ϕ˜ ,~η =0.
1vv
h i
Consideringequation(4.8)and(4.9)together,wegivethefollowingtheoremfor
the Gaussian curvature of spacelike ruled surface M .
1
Theorem 4.1 Let M be spacelike non-cylindrical ruled surface in R3. The
1 1
Gaussian curvature of spacelike non-cylindrical ruled surface M is given in
1
terms of its distribution parameter P by
P2
K = (4.10)
−(P2 v2)2
−
where v < P .
| | | |
Proof. Substituting equations (4.8) and (4.9) into equation (3.5) and making
appropriate simplifications we find the Gaussian curvature of M to be
1
κ2sin2σ
K = 1 1 .
− sin2σ v2κ2 2
1− 1
(cid:0) (cid:1)
Considering κ = 1 and equation (4.4) completes the proof.
1 ρ1
The relation between Gaussian curvature and the distribution parameter of
M given by equation (4.10) is called Lorentzian Lamarle formula for the
1
spacelike non-cylindrical ruled surface M .
1
The Lorentzian Lamarle formula for the spacelike ruled surface in R3 is non-
1
positive. Therefore we give the following corollary.
Corollary 4.1 Let M be a spacelike non-cylindrical ruled surface with distri-
1
bution parameter P and Gaussian curvature K in R3.
1
1. Along a ruling the Gaussian curvature K(u,v) 0 as v .
→ →∓∞
2. K(u,v)=0 if and only if P =0.
3. If the distribution parameter is P never vanishes then K(u,v) is contin-
uous and when v = 0 i.e. at the central point on each ruling, K(u,v)
assumes its maximum value.
Example 4.1 In3 dimensionalLorentzspaceR3 letusdefineanon-cylindrical
− 1
ruled surface as
ϕ(u,v)=( vcoshu, u, vsinhu)
− −
that is a 2nd type helicoid and a spacelike surface where 1<v <1, see Figure
−
4.1.
6
The Gaussian curvature of this 2nd type helicoid is K = 1 , v < 1,
−(1−v2)2 | |
see Figure 4.2.
5 Lamarle Formula for the Timelike Ruled Sur-
face with Spacelike Base Curve and Timelike
Ruling
Let M be timelike ruled surface with spacelike base curve and timelike ruling
2
in 3 dimensional Lorentz space, R3. Thus, this ruled surface is parametrized
− 1
as follows
ϕ :I R R3
2 × → 1
(u,v) ϕ (u,v)=α~ (u)+v~e (u).
2 2 2
→
Here, taking ~e = 1 , ~n = ~e′2 and ξ~ = ~e2∧~e′2 , we reach the or-
k 2k 2 ~e′2 2 ~e2∧~e′2
k k k k
thonormal frame field ~e ,~n ,ξ~ .This forms a right handed system and in
2 2 2
n o
time,space,space type. Therefore,
{ }
~e ,~e = ~n ,~n = ξ~ ,ξ~ =1
2 2 2 2 2 2
−h i h i D E (5.1)
~e ,~n = ~n ,ξ~ = ξ~ ,~e =0
2 2 2 2 2 2
h i D E D E
and
~e ~n = ξ~ , ~n ξ~ =~e , ξ~ ~e = ~n . (5.2)
2 2 2 2 2 2 2 2 2
∧ − ∧ ∧ −
The differential formulae of this orthonormal system are
~e′ =κ ~n , ~n′ =κ ~e τ ξ~ , ξ~′ =τ ~n . (5.3)
2 2 2 2 2 2− 2 2 2 2 2
Now,letthestrictioncurvegivenbyequation(3.2)oftimelikeruledsurfaceM
2
be β~ (u). β~ (u) is a spacelike curve and the tangent vector of this curve β~′
2 2 2
stays in the timelike plane ~e ,ξ~ . Adopting the hyperbolic angle σ between
2 2 2
(cid:16) (cid:17)
β~′ and~e we write
2 2
β~′ = sinhσ ~e +coshσ ξ~ .
2 2 2 2 2
7
From the last equation we write for the striction curve of M
2
β~ = (sinhσ ~e +coshσ ξ )du.
2 Z 2 2 2 2
Let M be timelike non-cylindrical ruled surface with spacelike base curve and
2
timelike ruling in R3. In this case we reparametrize M such as
1 2
ϕ˜ (u,v)= sinhσ ~e + coshσ ξ~ du+v~e .
2 Z (cid:16) 2 2 2 2(cid:17) 2
Considering equation (3.4) we find the distribution parameter of timelike non-
cylindrical ruled surface M to be
2
det sinhσ ~e +coshσ ξ~ ,~e ,κ ~n
2 2 2 2 2 2 2 coshσ
P = (cid:16) (cid:17) = 2.
κ ~n ,κ ~n κ
2 2 2 2 2
h i
Taking κ = 1 we rewrite the distribution parameter of M as
2 ρ2 2
P =ρ coshσ . (5.4)
2 2
If we consider equation (5.2), from equation (3.6) we see that the timelike non-
cylindrical ruled surface’s unit normal vector becomes
coshσ ~n +vκ ξ~
~η = − 2 2 2 2. (5.5)
2
cosh2σ +v2κ2
2 2
q
(cid:12) (cid:12)
(cid:12) (cid:12)
Since κ = 1 , from equation (5.4) we find
2 ρ2
P~n +vξ~
~η = − 2 2. (5.6)
2
√P2+v2
From equation (5.3), partial differential of M with respect to u and v are
2
ϕ˜ = sinhσ ~e + coshσ ξ~ + vκ ~n ,
2u 2 2 2 2 2 2 (5.7)
ϕ˜ =~e .
2v 2
Considering the last equations with equation (5.1) we find the coefficients of
first fundamental form of M to be
2
E = ϕ˜ ,ϕ˜ = sinh2σ +cosh2σ +v2κ2 =1+v2κ2,
h 2u 2ui − 2 2 2 2
F = ϕ˜ ,ϕ˜ = sinhσ , (5.8)
2u 2v 2
h i −
G= ϕ˜ ,ϕ˜ = 1.
2v 2v
h i −
Furthermore, if we consider equation, (5.7) we reach that the second order
partial differentials of M
2
ϕ˜ = σ′ coshσ +vκ2 ~e +(κ sinhσ +τ coshσ +vκ′)~n +(σ′ sinhσ vκ τ )ξ~ ,
2uu 2 2 2 2 2 2 2 2 2 2 2 2− 2 2 2
ϕ˜ =κ(cid:0) ~n , (cid:1)
2uv 2 2
ϕ =0.
2vv
8
From equation (5.5) and the last equations we find the second fundamentals
form’s coefficients as follows
L= ϕ˜ ,~η = −κ2sinhσ2coshσ2−τ2cosh2σ2−vκ′2coshσ2−vκ2σ2′ sinhσ2+v2κ22τ2,
h 2uu 2i √cosh2σ2+v2κ22
N = ϕ˜ ,~η = κ2coshσ2 ,
h 2uv 2i √cosh2σ2+vκ22
M = ϕ˜ ,~η =0.
2vv 2
h i
(5.9)
Therefore, for the Gaussian curvature of timelike ruled surface M , we give the
2
following theorem.
Theorem 5.1 Let M be a timelike non-cylindrical ruled surface with spacelike
2
base curve and timelike ruling in R3. Taking P to be the distribution parameter
1
of M we see that the Gaussian curvature of M is
2 2
P2
K = . (5.10)
(P2+v2)2
Proof. Substituting equations (5.8) and (5.9) into equation (3.5) we find the
Gaussian curvature of M to be
2
κ2 cosh2σ
K = 2 2 .
cosh2σ +κ2v2 2
2 2
(cid:0) (cid:1)
Here considering κ = 1 and equation (5.4) completes the proof.
2 ρ2
The relation between Gaussian curvature and the distribution parameter of
M given by equation (5.10) is called Lorentzian Lamarle formula for the
2
timelike non-cylindrical ruled surface with spacelike base curve and timelike
ruling.
The Lamarle formula for the timelike ruled surface in R3 is non–negative. So,
1
we give the following corollary.
Corollary 5.1 Let P be a distribution parameter and K be a Gaussian curva-
ture of a timelike non-cylindrical ruled surface M with spacelike base curve and
2
timelike ruling in R3. In this case
1
1. Along ruling as v , K(u,v) 0.
→∓∞ →
2. K(u,v)=0 if and only if P =0.
3. If the distribution parameter of M never vanishes, then K(u,v) is con-
2
tinuous and as v = 0 i.e. at the central point on each ruling, K(u,v)
takes its minimum value.
Example 5.1 In 3 dimensional Lorentz space R3.
− 1
ϕ(u,v)=( vsinhu, u, vcoshu)
− −
is a 3rd type helicoid and a timelike non-cylindrical ruled surface with spacelike
base curve and timelike ruling, see: Figure 5.1.
9
The Gaussian curvature of this 3rd type helicoid is K = 1 , see Figure
−(1+v2)2
5.2.
6 Lamarle Formula for Timelike Ruled Surface
with Timelike Base Curve and Spacelike Rul-
ing
Suppose that the timelike ruled surface M with timelike base curve and space-
3
like ruling in three dimensional Lorentz space R3 is parametrized as follows
1
ϕ :I R R3
3 × → 1
(u,v) ϕ (u,v)=α~ (u)+v~e (u).
3 3 3
→
Consideringthat ~e =1,~n = ~e′3 andξ~ = ~e3∧~e′3 ,wereachtheorthonor-
k 3k 3 ~e′3 3 ~e3∧~e′3
k k k k
mal frame field ~e ,~n ,ξ~ .This forms a right handed system which is in type
3 3 3
n o
space,space,time . Thus we write
{ }
~e ,~e = ~n ,~n = ξ~ ,ξ~ =1
3 3 3 3 3 3
h i h i −D E (6.1)
~e ,~n = ~n ,ξ~ = ξ~ ,~e =0
3 3 3 3 3 3
h i D E D E
and cross product is defined to be
~e ~n =ξ~ , ~n ξ~ = ~e , ξ~ ~e = ~n . (6.2)
3 3 3 3 3 3 3 3 3
∧ ∧ − ∧ −
Differential formulae for this orthonormal system is expressed by
~e′ =κ ~n , ~n′ = κ ~e +τ ξ~ , ξ~′ =τ ~n . (6.3)
3 3 3 3 − 3 3 3 3 3 3 3
Letthestrictioncurveoftimelikeruledsurfacegivenbyequation(3.2)beβ~ (u).
3
This curve is a timelike curve and the tangent vector of this curve stays within
the timelike plane ~e ,ξ~ . Adopting the hyperbolic angle σ to be the angle
3 3 3
(cid:16) (cid:17)
between β~′ and~e we may write
3 3
β~′ = sinhσ ~e + coshσ ξ~
3 3 3 3 3
10