On Spacelike Constant Slope Surfaces and Bertrand Curves in Minkowski 3-Space

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On spacelike constant slope surfaces and Bertrand curves in Minkowski 3-space

Murat Babaarslana,* Yusuf Yaylib

aDepartment of Mathematics, Bozok University, 66100, Yozgat-Turkey

bDepartment of Mathematics, Ankara University, 06100, Ankara-Turkey

ABSTRACT

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position

vector makes a constant angle with the normal at each point on the surface. These surfaces were

completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this work, we show

that timelike Bertrand curves and spacelike Bertrand curves can be constructed from unit speed

spacelike curves on the hyperbolic plane 2 and from unit speed spacelike curves on the de

Sitter 2-space 21 ,S respectively. Furthermore, we obtain the relations between Bertrand curves

and helices. We define the notion of de Sitter evolutes of curves on 2 and 21S and demonstrate

that the Darboux indicatrices of Bertrand curves are equal to these evolutes. Also, we investigate

the relations between Bertrand curves and spacelike constant slope surfaces in 31 .E Moreover we

give some examples of our main results and the corresponding pictures are drawn by using

Mathematica.

Key words: Bertrand curve, helix, Darboux indicatrix, Lorentzian Sabban frame, de Sitter

evolute, spacelike constant slope surface

*Corresponding Author: E-mail: [email protected]

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1. Introduction

Lorentzian geometry helps to bridge the gap between modern differential geometry and

mathematical physics of general relativity by giving an invariant treatment of Lorentzian

geometry. The fact that relativity theory is expressed in terms of Lorentzian geometry is attractive

for geometers who can penetrate surprisingly into cosmology (red-shift, expanding universe and

big bang) and a topic no less interesting geometrically, the gravitation of a single star (perihelion

procession, bending of light and black holes).

Several classical, well-known geometric objects are defined in terms of making a constant

angle with a given, distinguished direction. Firstly, classical helices are curves making a constant

angle with a fixed direction. A second example is the logarithmic spiral, the spira mirabilis

studied by Jacob Bernoulli, which makes a constant angle with the radial direction.

There are a lot of interesting applications of helices (e.g., DNA double and collagen triple

helix, nano-springs, carbon nano-tubes, K-helices, helical staircases, helical structures in fractal

geometry and so on).

Another interesting and important notion is Bertrand curve. It is defined as a special curve

which shares its principal normals with another special curve (called Bertrand mate). Bertrand

mates represent particular examples of offset curves [13] which are used in computer-aided

design (CAD) and computer-aided manufacture (CAM). Furthermore, Bertrand curves may be

regarded as 1-dimensional analogue of Weingarten surfaces.

A constant angle surface in 3E is a surface whose tangent planes make a constant angle

with a fixed vector field of ambient space. These surfaces are the generalizations of the concept

of helix. These kinds of surfaces are models to describe some phenomena in physics of interfaces

in liquid crystals and of layered fluids. Constant angle surfaces were studied for arbitrary

dimensions in Euclidean space nE [4, 14] and in different ambient spaces, e.g. 2 ,S R 2 H R

and 3Nil [5, 6, 7]. Moreover these surfaces were studied in Minkowski 3-space and classified in

[11].

Another nice paper in this direction is [12], where Munteanu made a review of some

applications of constant angle surfaces and gave a complete classification of the so-called

constant slope surfaces in 3.E Such surfaces are those whose position vectors make a constant

3

angle with the normals at each point on the surfaces. Recently [2], we gave some

characterizations of constant slope surfaces and Bertrand curves in Euclidean 3-space. We found

parametrization of constant slope surfaces for spherical indicatrices of a space curve.

Furthermore, we investigated Bertrand curves corresponding to parameter curves of constant

slope surfaces. More recently, Fu and Yang [8] studied spacelike constant slope surfaces in

Minkowski 3-space and classified these surfaces in the same space.

Constant slope surfaces have nice shapes and they are interesting in terms of differential

geometry. The study of these surfaces is similar to that of the logarithmic spirals and classical

helices.

In this study, we will show that timelike Bertrand curves and spacelike Bertrand curves

can be constructed from unit speed spacelike curves on the hyperbolic plane 2Η and from unit

speed spacelike curves on the de Sitter 2-space 21 ,S respectively. Furthermore, we will obtain the

relations between Bertrand curves and helices. We will define the notion of de Sitter evolutes of

curves on 2 and 21S and demonstrate that the Darboux indicatrices of Bertrand curves are equal

to these evolutes. Afterwards, we will investigate the relations between Bertrand curves and

spacelike constant slope surfaces in 31 .E Moreover we will give some examples of our main

results and the corresponding pictures will be drawn by using Mathematica.

2. Basic Notations, Definitions and Formulas

In this section, we give the basic notations and some results in Minkowski 3-space. Let 31E be the Minkowski 3-space, that is, 3

1E is the real vector space 3R endowed with the standard

flat Lorentzian metric

2 2 21 2 3, ,dx dx dx (2.1)

where 1 2 3( , , )x x x is a rectangular coordinate system of 31 .E An arbitrary vector 3

1Ex is said

spacelike if , 0x x or 0,x timelike if , 0x x and lightlike (null) if , 0x x and

0.x The norm of a vector x is given by , .x x x

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Given a regular (smooth) curve 31: ,I R E we say that is spacelike (resp.

timelike, lightlike) if all of its velocity vectors ( )t are spacelike (resp. timelike, lightlike).

Minkowski space is originally from the relativity in physics. In fact, a timelike curve

corresponds to path of an observer moving at less than the speed of light, a lightlike curve

corresponds to the moving at a speed of light and a spacelike curve corresponds to the moving

faster than light.

If is spacelike or timelike we say that is a non-null curve. In such case, there exist a

change of parameter ,t namely, ( ),s s t such that ( ) 1.s Then we say that is

parametrized by the arc-length parameter. In this case, we say that is a unit speed curve.

The angle between two vectors in Minkowski 3-space is defined by [1, 3]:

Definition 2.1. Let x and y be spacelike vectors in 31E that span a spacelike vector subspace.

Then we have ( , )g x y x y and hence, there is a unique positive real number such that

( , ) cos .g x y x y (2.2)

The real number is called the Lorentzian spacelike angle between x and .y

Definition 2.2. Let x and y be spacelike vectors in 31E that span a timelike vector subspace.

Then we have ( , )g x y x y and hence, there is a unique positive real number such that

( , ) cosh .g x y x y (2.3)

The real number is called the Lorentzian timelike angle between x and .y

Definition 2.3. Let x and y be positive (negative) timelike vectors in 31 .E Then there is a unique

non-negative real number such that

( , ) cosh .g x y x y (2.4)


Definition 2.4. Let x be spacelike vector and y be positive timelike vector in 31 .E Then there is

a unique non-negative real number such that

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( , ) sinh .g x y x y (2.5)


We also need to recall the notion of the Lorentzian cross-product:

3 3 31 1 1

1 2 3 1 2 3 1 2 3 2 3 3 2 3 1 1 3 2 1 1 2

1 2 3

:

( , , ), ( , , ) ( , , ).x x x y y y x x x x y x y x y x y x y x yy y y

E E E

1 2 3e e e (2.6)

As the cross-product in Euclidean 3-space, the Lorentzian cross-product has similar algebraic and

geometric properties:

(i) det ;, , , x y z x y z

(ii) ; x y y x

(iii) ;, + , x y z x z y y z x

(iv) 0, x y x and 0;, x y y

(v) 2,, , , , x y x y x x y y x y for all , , x y z in 3

1 .E

Given a unit speed curve in 31 ,E it is possible to define a Frenet frame

( ), ( ), ( )s s sT N B associated for each point .s Here ,T N and B are the tangent, normal and

binormal vector fields, respectively. Depending on the causal character of the curve , we have

the following Frenet equations, the Darboux indicatrices and the curvatures:

Let be a unit speed timelike curve in 31 .E The Frenet frame ( ), ( ), ( )s s sT N B of is

given by

( ) ( ),s s T ( ) ( ) / ( ) ,s s s N ( ) ( ) ( ).s s s B T N (2.7)

We define the curvature of at s as ( ) ( ) .s s T Moreover ( ) ( ), ( ) .s s s T N We define

the torsion of at s as ( ) ( ), ( ) .s s s N B The Frenet equations are

( ) 0 ( ) 0 ( )( ) ( ) 0 ( ) ( ) .( ) 0 ( ) 0 ( )

s s ss s s ss s s

T TN NB B

(2.8)

The Darboux vector for the timelike curve is given by ( ) ( ) ( ) ( ) ( ).s s s s s D T B We consider

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the normalization of the Darboux vector ( ) ( ) / ( ) ,s s sC D D which is called the Darboux

indicatrix of the timelike curve in 31 .E

For a general parameter t of a timelike space curve , we can calculate the curvature and the

torsion as follows:

3/ 2 2

( ) ( ) det( ( ), ( ), ( ))( ) , ( ) .( ) ( )( ), ( )

t t t t tt tt tt t

(2.9)

Let be a unit speed spacelike curve in 31 .E We assume that ( )sT is spacelike. Again

we write ( ) ( ) ,s s T ( ) ( ) / ( )s s s N and ( ) ( ) ( ).s s s B N T The Frenet equations are

( ) 0 ( ) 0 ( )( ) ( ) 0 ( ) ( ) .( ) 0 ( ) 0 ( )

s s ss s s ss s s

T TN NB B

(2.10)

The Darboux vector for this spacelike curve is given by ( ) ( ) ( ) ( ) ( )s s s s s C T B and the

Darboux indicatrix is ( ) ( ) / ( ) .s s sC D D The torsion of is defined by ( ) ( ), ( ) .s s s N B

Similarly, let be a unit speed spacelike curve in 31 .E We assume that ( )sT is timelike.

Then we write ( ) ( ), ( ) ,s s s T T ( ) ( ) / ( )s s s N and ( ) ( ) ( ).s s s B T N The Frenet

equations are

( ) 0 ( ) 0 ( )( ) ( ) 0 ( ) ( ) .( ) 0 ( ) 0 ( )

s s ss s s ss s s

T TN NB B

(2.11)

The Darboux vector for this spacelike curve is given by ( ) ( ) ( ) ( ) ( )s s s s s D T B and the

Darboux indicatrix is ( ) ( ) / ( ) .s s sC D D The torsion of is defined by ( ) ( ), ( ) .s s s N B

For a general parameter t of a spacelike space curve , we can calculate the curvature and the

torsion as follows:

3/ 2 2

( ) ( ) det( ( ), ( ), ( ))( ) , ( ) .( ) ( )( ), ( )

t t t t tt tt tt t

(2.12)

A helix in 31E is a regular curve such that (s),T w is a constant function for some fixed

vector 0.w Any line parallel this direction w is called the axis of the helix. If a (timelike or

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spacelike) curve in 31E is a helix, then / is a constant function. Conversely, let be a

timelike or a spacelike curve with non-null normal vector. If / is constant, then is a helix.

On the other hand, a (timelike or spacelike) curve is a Bertrand curve if and only if there are

non-zero real constants ,A B such that ( ) ( ) 1A s B s for any s in 31E (see [10] for details).

We now define “spheres” in 31E as follows:

2 3 2 2 21 1 2 3 1 1 2 3( , , ) : 1 ,x x x x x x S E (2.13)

2 3 2 2 21 2 3 1 1 2 3( , , ) : 1 .x x x x x x E (2.14)

We call 21S a de Sitter 2-space and 2 a hyperbolic plane.

3. Spacelike constant slope surfaces lying in the timelike cone and timelike Bertrand curves

In this section, firstly we give characterizations of spacelike constant slope surfaces lying

in the timelike cone and timelike Bertrand curves in Minkowski 3-space, respectively. Also, we

define the notion of de Sitter evolute of a curve on 2. Secondly, we explore relations among

timelike Bertrand curves, helices, Darboux indicatrices, de Sitter evolutes and these surfaces in 31 .E

Now we give this theorem:

Theorem 3.1. Let 31:x M E be a spacelike surface immersed in the 3-dimensional Minkowski

space 31 .E If x lies in the timelike cone, then M is a constant slope surface if and only if one of

the following five statements holds:

(1) the immersion ( )x M is an open portion of the hyperbolic plane 2 centered at the origin;

(2) the immersion ( )x M is a surface of revolution with a lightlike axis, given by

2 coth coth 2 coth 2 coth coth 2( , ) sinh ( 1),2 ,sinh ( 1) ,2ux u v u u v u v u u v

where is a positive constant angle function;


2 coth coth 2 coth 2 coth coth 2( , ) sinh ( 1),2 ,sinh ( 1) ,2ux u v u u v u v u u v

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(4) the immersion ( )x M is a surface of revolution with a timelike axis, given by

( , ) sinh sinh(coth ln )cos ,sinh(coth ln )sin ,cosh(coth ln ) ,x u v u u v u v u


(5) the immersion ( )x M is given by

( , ) sinh cosh(coth ln ) ( ) sinh(coth ln ) ( ) ( ) ,x u v u u f v u f v f v

where f is a unit speed spacelike curve on 2 and is a positive constant angle function [8].

Let 2:f I be a unit speed spacelike curve. We denote v as the arc-length

parameter of .f Let us denote ( ) ( )v f vt and we call ( )vt the unit tangent vector of f at .v

We now set a vector ( ) ( ) ( )v f v v s t and as a consequence ( ) ( ) ( ),v v f v s t where f denotes

the position vector of the curve. By definition of the spacelike curve ,f we have a Lorentzian

Sabban frame ( ), ( ), ( )f v v vt s along .f Then we have the following hyperbolic Frenet-Serret

formulae of :f

( ) ( ),( ) ( ) ( ) ( ),

( ) ( ) ( ),g

g

f v vv f v v vv v v

tt ss t

(3.1)

where ( )g v is the geodesic curvature of the curve f on 2 which is given by

( ) det( ( ), ( ), ( ))g v f v v v t t (see [9]).

Now we can express the following lemma:

Lemma 3.2. Let 2:f I be a unit speed spacelike curve. Then

0 0( ) ( ) tanh ( ) ( )

v vv a f t dt a f t f t dt (3.2)

is a timelike Bertrand curve, where a and ( ) coth lnu u are constant numbers, and is a

positive constant angle function. Moreover, all timelike Bertrand curves can be constructed by

this method.

Proof. We now calculate the curvature and the torsion of ( ).v Taking the derivative of Eq. (3.2)

with respect to ,v we have

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2

( ) ( ( ) tanh ( ))( ) (1 tanh ( )) ( )

( ) (1 tanh ( )) ( ) tanh ( ) ( ) ( ( ) tanh ( )) ( ).g

g g g g

v a f v vv a v v

v a v f v a v v a v v v

st

t s

(3.3)

Therefore, by Eq. (2.9), we can calculate ( )v and ( )v as follows: 2 2cosh (1 tanh ( )) cosh ( ( ) tanh )

( ) and ( ) ,g gv vv v

a a

(3.4)

where 1. It follows from these formulae that ( ( ) tanh ( )) 1,a v v so ( )v is a timelike

Bertrand curve.

Conversely, let be a timelike Bertrand curve. There exist real constants ,A B different from 0

such that ( ) ( ) 1.A s B s Here we put ,a A tanh / .B a We assume that 0a and choose

1 with cosh / 0.a We consider the Frenet frame ( ), ( ), ( )s s sT N B for the timelike

curve ( ).s In this trihedron ( )sT is timelike vector, ( )sN and ( )sB are spacelike vectors. For

these vectors, we can write ( ) ( ) ( )s s s T N B and ( ) ( ) ( ).s s s B N T Now we define a

spacelike curve on 2 as

( ) cosh ( ) sinh ( ) .f s s s T B (3.5)

Thus we have

( ) cosh ( ( ) tanh ( )) ( ) cosh ( ).f s s s s sa N N (3.6)

Let v be the arc-length parameter of ,f then we have / cosh / .dv ds a Moreover, we have

( ) cosh cosh ( ) sinh ( )dvaf s s sds

T B (3.7)

and

tanh ( ) tanh (cosh ( ) sinh ( )) cosh ( )

sinh (cosh ( ) sinh ( )).

df dva f s a s s sdv ds a

s s

T B N

B T (3.8)

By using Eqs. (3.7) and (3.8), we obtain

0

0

0

0 0( ) tanh ( ) ( ) cosh (cosh ( ) sinh ( ))

sinh (cosh ( ) sinh ( ))

( ) ( ).

v v s

s

s

s

s

s

a f t dt a f t f t dt t t dt

t t dt

t dt s

T B

B T

T

(3.9)

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This completes the proof.

As a consequence of this lemma, we can make a connection between timelike Bertrand

curves and helices.

Corollary 3.3. The unit speed spacelike curve f on 2 is a part of a pseudo-circle if and only if

the corresponding timelike Bertrand curve is a helix.

Proof. By using Eq. (3.4), we have 2sinh 2 ( ) cosh ( )

( ) and ( ) .2

g gv vv v

a a

(3.10)

The unit speed spacelike curve f on 2 is a part of a pseudo-circle if and only if

( )g v constant. This condition is equivalent to the condition that both ( )v and ( )v are non-

zero constants. The proof is completed.

Under the assumption that 2 ( ) 1,g v we define a curve in 31 :E

2

( ) ( ) ( )( ) .

1 ( )g

f

g

v f v vd v

v

s (3.11)

We remark that ( )fd v is located in 21S if and only if 2 ( ) 1,g v otherwise it is in 2. We call fd

the de Sitter evolute of f or the pseudo-spherical evolute of f (see [9]).

Then we have the following proposition:

Proposition 3.4. Let 2:f I be a unit speed spacelike curve and 31: I E be a timelike

Bertrand curve corresponding to .f Then the Darboux indicatrix of is equal to the de Sitter

evolute of .f

Proof. By Eq. (3.4), we have 2 2cosh (1 tanh ( )) cosh ( ( ) tanh )

( ) and ( ) .g gv vv v

a a

For the timelike curve , we have

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( ) ( ) tanh ( ) dvv a f v vds

T s and ( ) ( ).v vN t (3.12)

Then we get

( ) ( ) ( ) ( ) tanh ( ) .dvv v v a v f vds

B T N s (3.13)

We can easily show that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .gdvv v v v v v f v vds

D T B s (3.14)

Therefore we have ( ) = ( ) / ( ) ( ).fv v v d vC D D This completes the proof.

We have the following theorem:

Theorem 3.5. Let 2:f I be a unit speed spacelike curve and 31: I E be a timelike

Bertrand curve corresponding to .f Then ( )v lies on the spacelike constant slope surface

( , )x u v lying in the timelike cone.

Proof. By Lemma 3.2, taking the derivative of Eq. (3.2) with respect to v we obtain

( ) ( ) tanh ( ) ( ).v af v a f v f v (3.15)

We can take a as sinh cosha u and so tanh sinh sinh ,a u where ,u are constants.

Thus by the Statement (5) in Theorem 3.1, ( )v is v parameter curve of spacelike constant

slope surface ( , )x u v lying in the timelike cone and ( )v lies on it. This completes the proof.

We now state the relation between timelike Bertrand curves and spacelike constant slope

surfaces lying in the timelike cone.


space 31E and x lies in the timelike cone. If ( )x v is v parameter curve of spacelike constant

slope surface ( , )x u v lying in the timelike cone, then 0

( )vx v dv is a timelike Bertrand curve.

Proof. From the Statement (5) in Theorem 3.1, we get

( ) sinh cosh ( ) sinh sinh ( ) ( )x v u f v u f v f v (3.16)

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for u constant, where ( ) coth ln .u u By integrating ( ),x v we have the equation as

0 0 0

( ) sinh cosh ( ) sinh sinh ( ) ( ) .v v vx v dv u f v dv u f v f v dv (3.17)

Since coefficients of ( )f v and ( ) ( )f v f v are constants, we can take sinh coshu as

sinh coshu a and so sinh sinh tanh .u a Therefore we obtain

0 0 0

( ) ( ) tanh ( ) ( ) .v v vx v dv a f v dv a f v f v dv (3.18)

By Lemma 3.2, 0

( )vx v dv is a timelike Bertrand curve. This completes the proof.

4. Spacelike constant slope surfaces lying in the spacelike cone and spacelike Bertrand

curves

In this section, as in Section 3, we give characterizations of spacelike constant slope

surfaces lying in the spacelike cone and spacelike Bertrand curves in 31 .E We define the notion of

de Sitter evolute of a curve on 21 .S Moreover we explore relations among spacelike Bertrand

curves, helices, Darboux indicatrices, de Sitter evolutes and these surfaces in the same space.

Now we give this theorem:


space 31E . If x lies in the spacelike cone, then M is a constant slope surface if and only if one of

the following five statements holds:

(1) the immersion ( )x M is an open portion of a cone with the vertex at the origin, which can be

parametrized as

( , ) ( ),x u v uf v

where f is a unit speed spacelike curve on 21 ;S


2 tanh tanh 2 tanh 2 tanh tanh 2( , ) cosh ( 1),2 , cosh ( 1) ,2ux u v u u v u v u u v



13

2 tanh tanh 2 tanh 2 tanh tanh 2( , ) cosh ( 1),2 , cosh ( 1) ,2ux u v u u v u v u u v


(4) the immersion ( )x M is a surface of revolution with a spacelike axis, given by

( , ) cosh cosh(tanh ln ),sinh(tanh ln )sinh ,sinh(tanh ln )cosh ,x u v u u u v u v


(5) the immersion ( )x M is given by

( , ) cosh cosh(tanh ln ) ( ) sinh(tanh ln ) ( ) ( ) ,x u v u u f v u f v f v

where f is a unit speed spacelike curve on 21S and is a positive constant angle function [8].

Similar to the Section 3, we define a pseudo-orthonormal frame for a spacelike curve on 21S . Let 2

1:f I S be unit speed spacelike curve with the tangent vector ( ) ( ).v f vt We now

set a vector ( ) ( ) ( )v f v v s t and as a consequence ( ) ( ) ( ),v v f v s t where f denotes the

position vector of the curve. By definition of the spacelike curve ,f we have a Lorentzian

Sabban frame ( ), ( ), ( )f v v vt s along .f Then we have the following spherical Frenet-Serret

formulae of :f

( ) ( ),( ) ( ) ( ) ( ),

( ) ( ) ( ),g

g

f v vv f v v vv v v

tt ss t

(4.1)

where ( )g v is the geodesic curvature of the curve f on 21S which is given by

( ) det( ( ), ( ), ( )).g v f v v v t t

Now we can express the following lemma:

Lemma 4.2. Let 21:f I S be a unit speed spacelike curve. Then

0 0( ) ( ) tanh ( ) ( )

v vv a f t dt a f t f t dt (4.2)

is a spacelike Bertrand curve, where ,a ( ) tanh lnu u are constant numbers and is a

positive constant angle function. Moreover, all spacelike Bertrand curves can be constructed by

this method.

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Proof. We now calculate the curvature and the torsion of ( ).v Taking the derivative of Eq. (4.2)

with respect to ,v we have

2

( ) ( ( ) tanh ( ))( ) (1 tanh ( )) ( )

( ) (1 tanh ( )) ( ) tanh ( ) ( ) ( ( ) tanh ( )) ( ).g

g g g g

v a f v vv a v v

v a v f v a v v a v v v

st

t s

(4.3)

Therefore, by Eq. (2.12), we can calculate ( )v and ( )v as follows: 2 2cosh (1 tanh ( )) cosh ( ( ) tanh )

( ) and ( ) ,g gv vv v

a a

(4.4)

where 1. It follows from these formulae that ( ( ) tanh ( )) 1,a v v so ( )v is a

spacelike Bertrand curve.

Conversely, let be a spacelike Bertrand curve. There exist real constants ,A B different from

0 such that ( ) ( ) 1.A s B s Here we put ,a A tanh / .B a We assume that 0a and

choose 1 with cosh / 0.a We consider the Frenet frame ( ), ( ), ( )s s sT N B for the

spacelike curve ( ).s In this trihedron ( ), ( )s sT N are spacelike vectors and ( )sB is timelike

vector. For these vectors, we can write ( ) ( ) ( )s s s T N B and ( ) ( ) ( ).s s s B N T Now we

define a spacelike curve on 21S as

( ) cosh ( ) sinh ( ) .f s s s T B (4.5)

Thus we have

( ) cosh ( ( ) tanh ( )) ( ) cosh ( ).f s s s s sa N N (4.6)

Let v be the arc-length parameter of ,f then we have / cosh / .dv ds a Moreover, we have

( ) cosh cosh ( ) sinh ( )dvaf s s sds

T B (4.7)

and

tanh ( ) tanh (cosh ( ) sinh ( )) cosh ( )

sinh ( cosh ( ) sinh ( )).

df dva f s a s s sdv ds a

s s

T B N

B T (4.8)

By using Eqs. (4.7) and (4.8), we obtain

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0

0

0

0 0( ) tanh ( ) ( ) cosh (cosh ( ) sinh ( ))

sinh ( cosh ( ) sinh ( ))

( ) ( ).

v v s

s

s

s

s

s

a f t dt a f t f t dt t t dt

t t dt

t dt s

T B

B T

T

(4.9)

This completes the proof.

As a consequence of this lemma, we can make a connection between spacelike Bertrand

curves and helices.

Corollary 4.3. The unit speed spacelike curve f on 21S is a part of a pseudo-circle if and only if

the corresponding spacelike Bertrand curve is a helix.

Proof. By using Eq. (4.4), we have 2sinh 2 ( ) cosh ( )

( ) and ( ) .2

g gv vv v

a a

(4.10)

The unit speed spacelike curve f on 21S is a part of a pseudo-circle if and only if

( )g v constant. This condition is equivalent to the condition that both ( )v and ( )v are non-

zero constants. The proof is completed.

Under the assumption that 2 ( ) 1,g v we define a curve in 31 :E

2

( ) ( ) ( )( ) .

( ) 1g

f

g

v f v vd v

v

s (4.11)

We remark that ( )fd v is located in 21S if and only if 2 ( ) 1,g v otherwise it is in 2. We call fd

the de Sitter evolute of f or the pseudo-spherical evolute of .f

Then we have the following proposition:

Proposition 4.4. Let 21:f I S be a unit speed spacelike curve and 3

1: I E be a spacelike

Bertrand curve corresponding to .f Then the Darboux indicatrix of is equal to the de Sitter

evolute of .f

16

Proof. By Eq. (4.4), we have 2 2cosh (1 tanh ( )) cosh ( ( ) tanh )

( ) and ( ) .g gv vv v

a a

For the spacelike curve , we have

( ) ( ) tanh ( ) dvv a f v vds

T s and ( ) ( ).v vN t (4.12)

Then we get

( ) ( ) ( ) ( ) tanh ( ) .dvv v v a v f vds

B N T s (4.13)

We can easily show that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .gdvv v v v v v f v vds

D T B s (4.14)

Therefore we have ( ) = ( ) / ( ) ( ).fv v v d vC D D This completes the proof.

We have the following theorem:

Theorem 4.5. Let 21:f I S be a unit speed spacelike curve and 3

1: I E be a spacelike

Bertrand curve corresponding to .f Then ( )v lies on the spacelike constant slope surface

( , )x u v lying in the spacelike cone.

Proof. By Lemma 4.2, taking the derivative of Eq. (4.2) with respect to v we obtain

( ) ( ) tanh ( ) ( ).v af v a f v f v (4.15)

We can take a as cosh cosha u and so tanh cosh sinh ,a u where ,u are

constants. Thus by the Statement (5) in Theorem 4.1, ( )v is v parameter curve of spacelike

constant slope surface ( , )x u v lying in the spacelike cone and ( )v lies on it. This completes the

proof.

We now state the relation between spacelike Bertrand curves and spacelike constant slope

spacelike surfaces lying in the spacelike cone.


space 31E and x be on the spacelike cone. If ( )x v is v parameter curve of spacelike constant

17

slope surface ( , )x u v lying in the spacelike cone, then 0

( )vx v dv is a spacelike Bertrand curve.

Proof. From the Statement (5) in Theorem 4.1, we get

( ) cosh cosh ( ) cosh sinh ( ) ( )x v u f v u f v f v (4.16)

for u constant, where ( ) tanh ln .u u By integrating ( ),x v we have the equation as

0 0 0

( ) cosh cosh ( ) cosh sinh ( ) ( ) .v v vx v dv u f v dv u f v f v dv (4.17)

Since coefficients of ( )f v and ( ) ( )f v f v are constants, we can take cosh coshu as

cosh coshu a and so cosh sinh tanh .u a Therefore we obtain

0 0 0

( ) ( ) tanh ( ) ( ) .v v vx v dv a f v dv a f v f v dv (4.18)

By Lemma 4.2, 0

( )vx v dv is a spacelike Bertrand curve. This completes the proof.

5. Examples and their pictures

We now give some examples of spacelike constant slope surfaces and Bertrand curves

and draw their pictures by using Mathematica.

Example 5.1. Thanks to the Statement (5) of Theorem 3.1, we may choose the unit speed

spacelike curve as ( ) (sinh ,0, cosh )f v v v on 2. Then we have ( ) ( ) (0,1,0).f v f v Thus,

the spacelike constant slope surface lying in the timelike cone is given by

( , ) sinh cosh(coth ln )sinh ,sinh(coth ln ),cosh(coth ln )cosh .x u v u u v u u v

For 1.5, the picture of this surface is given by the Fig. 1. Thus, for ,u e the timelike

Bertrand curve is given by

0

( ) sinh cosh(coth )(cosh 1),sinh(coth ) ,cosh(coth )sinh .v

x v dv e v v v

So, the picture of this curve is given by the Fig. 2.

18

Fig. 1. 1.5, ( ) (sinh ,0,cosh ),f v v v the spacelike constant slope surface lying in the timelike

cone.

Fig. 2. 1.5, ,u e the timelike Bertrand curve.

Example 5.2. In the Statement (5) of Theorem 4.1, we may choose the unit speed spacelike curve

as ( ) (sin ,cos ,0)f v v v on 21 .S Then we have ( ) ( ) (0,0,1).f v f v Thus, the spacelike

constant slope surface lying in the spacelike cone is given by

( , ) cosh cosh(tanh ln ) sin , cosh(tanh ln )cos ,sinh(tanh ln ) .x u v u u v u v u

For 1.5, the picture of this surface is given by the Fig. 3. Thus, for ,u e the spacelike

Bertrand curve is given by

19

0

( ) cosh cosh(tanh )(cos 1),cosh(tanh )sin ,sinh(tanh ) .v

x v dv e v v v

Since the spacelike curve ( )f v is a part of a pseudo-circle, from Corollary 4.3, this spacelike

Bertrand curve is a helix. Thus, the picture of this curve is given by the Fig. 4.

Fig. 3. 1.5, ( ) (sin , cos ,0),f v v v the spacelike constant slope surface lying in the spacelike

cone.

Fig. 4. 1.5, ,u e the spacelike Bertrand curve.

20

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On Spacelike Constant Slope Surfaces and Bertrand Curves in Minkowski 3-Space