International Journal of Analysis and Applications

Volume 18, Number 6 (2020), 900-919

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-900

RULED SURFACES WITH CONSTANT SLOPE RULING ACCORDING TO

DARBOUX FRAME IN MINKOWSKI SPACE

AYŞE YAVUZ1,∗, YUSUF YAYLI2

1Department of Mathematics and Science Education, Necmettin Erbakan University, Konya, Turkey

2Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey

∗Corresponding author: ayasar@erbakan.edu.tr

Abstract. In this study, three different types of ruled surfaces are defined. The generating lines of these

ruled surfaces are given by points on a curve X in Minkowski Space, while the position vector of X have

constant slope with respect to the planes (t, y) , (t, n) , (n, y). It is observed that the Lorentzian casual

characters of the ruled surfaces with constant slope can be timelike or spacelike. Furthermore, striction lines

of these surfaces are obtained and investigated under various special cases. Finally, new investigations are

obtained on the base curve of these types of ruled surfaces.

1. Introduction

A ruled surface is a special surface which is formed by moving a line along a given curve in 3-dimensional

Minkowski space. The line is called the generating line and the curve is called the direction curve of the

surface. Thus, a ruled surface has a parametrization

M(u,v) = α(u) + vX(u)

where α and X are curves. The curve α is called the directrix or base curve and X is called the director

curve of the ruled surface. Thus, the ruled surfaces in Minkowski space can be classified according to the

Lorentzian character of their ruling and surface normal. Developable ruled surfaces are surfaces which can

Received June 6th, 2020; accepted July 8th, 2020; published September 3rd, 2020.

2010 Mathematics Subject Classification. 53A04, 53A05.

Key words and phrases. ruled surface; surface with constant slope; Darboux frame; Minkowski space.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

900

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-900


Int. J. Anal. Appl. 18 (6) (2020) 901

be made isometric to part of the plane. The necessary and sufficient conditions for these surfaces to become

developable are characterized by vanishing Gaussian curvature. In this study “developable” and “torsal”

are used as synonyms, since a surface is developable if and only if it is a torsal ruled surface. Cylindrical,

conical, torse surfaces, a plane and surfaces of polyhedrons are examples of torsal surfaces. These surfaces

can be developed on a plane without any lap break. On the ground that their isometrics with planes are

becoming interesting to discover more ways to use these surfaces in different applications in [5] .

A regular curve in E31 , whose position vector is obtained by Frenet frame vectors on another regular curve,

is called Smarandache curve [1]. In this study, it is shown that if developable surface’s generating line is a

Smarandache curve and asymptotic or geodesic curve, then the basic curve is a general helix.

K. Malecek and others defined the surfaces with a constant slope with respect to the given surface

in Euclidean Space in [7]. At the same time in [10], Yavuz, Ateş and Yaylı investigated surface with a

constant slope ruling with respect to osculating plane by using Frenet Frame according to casual characters

in Minkowski space. By the definition of surfaces with a constant slope ruling with respect to the given

surface in study [7] , in this study new surface definitions are obtained. These surfaces in three types

were studied according to the Darboux frame in Minkowski Space. Furthermore, necessary and sufficient

conditions are given for these surfaces to become developable in Minkowski 3-space. Striction lines of the

surfaces are obtained and investigated under various special cases. Finally, the ruled surfaces with constant

slope ruling visualized of given curves as examples, separately.

2. Preliminaries

A tangent vector v on a semi-Riemanian manifold M is spacelike if g (v,v) > 0 or v = 0, is null if

g (v,v) = 0 and v 6= 0, timelike if g (v,v) < 0. The norm of a tangent vector v is given by |v| =
√
|g (v,v)|.

A curve in a manifold M is a smooth mapping α : I → M, where I is an open interval in the real line R.

A curve α in a semi-Riemannian manifold M is spacelike if all of its velocity vectors α
′
(s) are spacelike, is

null if all of its velocity vector α
′
(s) are null, timelike if all of its velocity vectors α

′
(s) are timelike [8].

In this study, the Darboux frames and formulas in the Minkowski space E31 are given with metric

g = −dx21 + dx
2
2 + dx

2
3.

Let S be an oriented surface in E31 and let consider a non-null curve α (s) lying fully on S . Since the curve

α (s) lies on the surface S there exists a frame along the curve α (s). This frame is called Darboux frame

and denoted by {t,y,n} which gives us an opportunity to investigate the properties of the curve according

to the surface. In this frame t is the unit tangent of the curve, n is the unit normal of the surface S along

curve α (s) and y is a unit vector given by y = ∓n × t. According to the Lorentzian casual characters of

the surface and the curve lying on surface, the derivative formulae of the Darboux frame can be changed as



Int. J. Anal. Appl. 18 (6) (2020) 902

follows:

i) If the surface is timelike, then the curve α (s) lying on surface can be spacelike or timelike. Thus, the

derivative formulae of the Darboux frame is given by

t′

y′

n′


 =




0 kg −εkn

kg 0 εtr

kn tr 0





t

y

n




〈t,t〉 = ε = ∓1,〈y,y〉 = −ε,〈n,n〉 = 1

ii) If the surface is spacelike, then the curve α (s) lying on surface is spacelike. Thus, the derivative formulae

of the Darboux frame is given by 

t′

y′

n′


 =




0 kg kn

-kg 0 tr

kn tr 0





t

y

n




where

〈t,t〉 = 1,〈y,y〉 = 1,〈n,n〉 = −1

Here,

kg (s) =
〈
α

′′
(s) ,y (s)

〉
is the geodesic curve, kn is the normal curvature defined by equality

kn (s) =
〈
α

′′
(s) ,n (s)

〉
and tr is the geodesic torsion of α(s) defined by

tr (s) = −
〈
n

′
(s) ,y (s)

〉
[2], [3], [4].

If u,v ∈ E31, Lorentzian vector product of u and v is to the unique vector by u×v that satisfies

〈u×v,w〉 = det (u,v,w)

where u×v vector product is defined as follows

u×v =

∣∣∣∣∣∣∣∣∣
−i j k

u1 u2 u3

v1 v2 v3

∣∣∣∣∣∣∣∣∣



Int. J. Anal. Appl. 18 (6) (2020) 903

[6].

The relations between geodesic curve, normal curvature, geodesic torsion and κ and τ are given, if both

surface and curve are timelike or spacelike, then

kg = κ cos θ

kn = κ sin θ

if surface is timelike and curve is spacelike, then

kg = κ cosh θ

kn = κ sinh θ

[2], [3], [4].

Let −→x and −→y be future pointing (or past pointing) timelike vectors in R31. Then there is a unique real

number θ > 0 such that

〈−→x ,−→y 〉 = −‖−→x‖‖−→y ‖cos hθ.

Let −→x and −→y be spacelike vectors in R31 that span a timelike vector subspace. Then there is a unique real

number θ > 0 such that

〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖cos hθ.

Let −→x and −→y be spacelike vectors in R31 that span a spacelike vector subspace. Then there is a unique real

number θ > 0 such that

〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖cos θ.

Let −→x be a spacelike vector and −→y be a timelike vector in R31. Then there is a unique real number θ > 0,

such that

〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖sinh θ.

[9].

3. Ruled Surfaces with Constant Slope Ruling According to Darboux Frame in Minkowski

3-Space

Let M be a ruled surface whose generating lines are given by points on the curve X in Minkowski Space

, while in all points they have the constant slope with respect to the tangent planes to the given surface

.These surfaces will be called ruled surfaces with constant slope ruling with respect to the given surface

in Minkowski Space. In this section, the developable conditions are investigated for the ruled surfaces with

constant slope ruling with respect to the given surface with Darboux Frame {t,n,y} and the striction lines of

the surface are obtained. At the same time, various relations and special cases about developable conditions



Int. J. Anal. Appl. 18 (6) (2020) 904

and striction line of the surfaces are given.

3.1. Ruled Surfaces with Constant Slope Ruling with Respect to the (t, y) Planes. Generating

lines of the surface M are given by points on the curve X(s) and they have the constant slope σ with respect

to the (t,y) planes to the curve at every point on the curve X(s). The surface will be called the ruled surfaces

with a constant slope ruling with respect to the (t,y) planes. We give the definition of a ruled surface with a

constant slope ruling with respect to the (t,y) planes according to casual characters in following three cases

in Minkowski space where

〈X(s),n(s)〉 = σ.

Case 3.1. If α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surface is

spacelike where 1 + σ2 > 0. Direction vector of generating line of the spacelike surface is given by

X(s) = sin w(s).t(s) + cos w(s).y(s) + σn(s)

and the surface is parametrized by

M1 (s,v) = α (s) + v (X(s)) .

Case 3.2. If α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is

spacelike where σ2 − 1 > 0 and the surface is timelike where σ2 − 1 < 0. Direction vector of generating line

of the surface is given by

u(s) = cosh w(s)t(s) + sinh w(s)y(s) + σn(s)

and the surface is parametrized by

M2 (s,v) = α (s) + v (u(s)) .

Case 3.3. If α(s) is a timelike curve with the principal spacelike normal vector field n(s), then surface is

timelike where σ2 + 1 > 0 and direction vector is given as follows

δ(s) = sinh w(s)t(s) + cosh w(s)y(s) + σn(s)

so the timelike surface is parametrized by

M3 (s,v) = α (s) + v (δ(s))

where the vector t is the direction vector of a tangent to the curve X, n is the direction vector of a normal

to the surface and y = n× t is the direction vector of intersection line of a tangent plane to the surface and

the normal plane curve X at the point.



Int. J. Anal. Appl. 18 (6) (2020) 905

Theorem 3.1. The spacelike surface M1(s,v) is developable if and only if

cos w(s) (sin w(s).kn + cos w(s).tr) = σ (sin w(s) (kg −w′(s)) + σtr) .

Proof. The surface M1 is developable if and only if det(t,X,X
′) = 0. Thus derivative of the direction vector

of generating lines of the surface is obtained as follows

X′(s) =
((
w

′
(s) −kg

)
cos w(s) + σkn

)
t(s) +

(
−
(
w

′
(s) + kg

)
sin w + σtr

)
y(s)

+ (kn sin w(s) + tr cos w(s)) .n(s)

det(t,X,X′) = 〈tΛX, X′〉 .

And

〈tΛX, X′〉

= cos w(s) (sin w(s).kn + cos w(s).tr) −σ (sin w(s) (kg −w′(s)) + σtr)

Thus developable condition for surface M1 (s,v) is given by

cos w(s) (sin w(s).kn + cos w(s).tr) = σ (sin w(s) (kg −w′(s)) + σtr) .

�

Corollary 3.1. Developable surface’s generating line X(s) is a asymptotic and Smarandache curve if and

only if α(s) is a general helix with

τ

κ
=

σx1
x22 −σ2

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Proof. If X(s) is asymptotic curve, then kn = 0

kn = 0 ⇒ cos2 w(s).tr = σ (sin w(s) (kg −w′(s)) + σtr)

and if surface and curve are the same character, then κ2 = k2n + k
2
g, so kg = κ, tr = τ. If we replace the

values in the last equation, we get the following equality

τ

κ
=

σx1
x22 −σ2

.

M1(s,v) developable surface’s generating line X(s) is a geodesic and Smarandache curve if and only if α(s)

is a general helix, so that

τ

κ
=

x1x2
σ2 −x22



Int. J. Anal. Appl. 18 (6) (2020) 906

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

�

Corollary 3.2. M1(s,v) spacelike developable surface’s generating line X(s) is a line curvature

kn
kg

=
σ

cos w(s)

where w(s) = const.

Theorem 3.2. The striction line on spacelike surface M1(s,v) is given by

β = α(s) −
−σkn − (w′ (s)−kg) cos w (s)

(w′ (s)−kg)


 (w′ (s)−kg) (sin2 w(s)−cos2 w(s))

−2σ (kn cos w (s) +tr sin w (s))




+ (kn sin w (s) +tr cos w (s))
2

+ σ2
(
−k2n+t

2
r

)
X (s) .

Remark 3.1. If X(s) is a geodesic curve and w (s) = −σ
∫
κ sec w (s) d (s) , then striction line of spacelike

surface is equal to base curve.

Remark 3.2. If X(s) is a asymptotic curve and w (s) =

∫
κd (s) where cos w (s) 6= 0, then striction line of

spacelike surface is equal to base curve.

Theorem 3.3. The spacelike surface M2(s,v) is developable if and only if

sinh w(s) (cosh w(s).kn + sinh w(s)tr) −σ (cosh w(s) (kg + w′(s)) + σtr) = 0

where σ2 − 1 > 0.

If σ2 − 1 < 0, then the surface is timelike. So the timelike surface is developable if and only if

ε sinh w(s) (cosh w(s).kn + sinh w(s)tr) −σ (cosh w(s) (kg + w′(s)) + σtr) = 0

where ε = 〈t,t〉 = ∓1.

Corollary 3.3. If sinhw(s) = x3 = const.and coshw(s) = x4 = const, then generating lines of the surface

u(s) is a Smarandache curve. Let M(s,v) be a spacelike developable surface and if u(s) be a Smarandache

and asymptotic curve, then α(s) is a general helix, so that

τ

κ
=

σx4
x23 −σ2



Int. J. Anal. Appl. 18 (6) (2020) 907

if also u(s) is a geodesic, then base curve is a general helix, so that

τ

κ
=

x3x4
σ2 −x23

Corollary 3.4. Let M2(s,v) be a spacelike developable surface and if u(s) is a line and Smarandache curve,

kn
kg

=
σ

x3

where sinhw(s) = x3 = const.and coshw(s) = x4 = const.

Corollary 3.5. If sinhw(s) = x3 = const.and coshw(s) = x4 = const , generating lines of the surface u(s)

be a Smarandache curve. Let M2(s,v) be a timelike developable surface and u(s) be a Smarandache curve ,

if at the same time u(s) be a asymptotic, then base curve is a general helix, so that

τ

κ
=

σx4
εx23 −σ2

if also u(s) is a geodesic, then base curve is a general helix, so that

τ

κ
=

εx3x4
εx23 −σ2

.

Proof. If M2(s,v) is a timelike developable surface and the base curve is spacelike, then κ
2 = k2g −k2n . So

we replace values kn = 0,kg = κ,tr = τ in condition of developable equation

ε sinh w(s) (sinh w(s).tr) = σ (cosh w(s) (kg) + σtr)

εx3 (x3.τ) = σ (x4.κ + στ)

τ

κ
=

σx4
εx23 −σ2

.

If u(s) is a geodesic, then we write values kg = 0,kn = −κ,tr = τ in condition of developable equation for

the timelike surface, we obtained as follows

εx3 (x4. (−κ) + x3.τ) = σ. (σ.τ)

τ

κ
=

εx3x4
εx23 −σ2

.

�

Remark 3.3. Let M2(s,v) be a timelike developable surface and if u(s) be a line and Smarandache curve,

kn
kg

=
σ

εx3

where sinhw(s) = x3 = const.and coshw(s) = x4 = const.



Int. J. Anal. Appl. 18 (6) (2020) 908

Theorem 3.4. The striction line on spacelike surface M2(s,v) is given by

β = α(s) −
−σkn − (w′ (s) −kg) sinh w (s)

(w′ (s)−kg)
2

+ σ2
(
−k2n+t

2
r

)
+ (2σ (w′ (s) −kg) (−kn sinh w (s) +tr cosh w (s)))

−(kn cosh w (s) +tr sinh w (s))
2

.X (s)

where σ2 − 1 > 0, and striction line on timelike surface is given by

β = α(s) −
−σkn − (w′ (s) −kg) sinh w (s)

(w′ (s)−kg)
2

+ σ2
(
−k2n+t

2
r

)
+ (2σ (w′ (s) −kg) (−kn sinh w (s) +tr cosh w (s)))

+ε2 (kn cosh w (s) +tr sinh w (s))
2

.X (s)

where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1.

Remark 3.4. If u(s) is a geodesic curve and

w (s) = −σ
∫

κ

sinh w (s)
d (s)

where σ2 − 1 > 0, then striction line of spacelike surface is equal to base curve.

Remark 3.5. If u(s) is a asymptotic curve and

w (s) =

∫
κd (s)

where σ2 − 1 > 0 and w (s) 6= 0, then striction line of spacelike surface is equal to base curve.

Theorem 3.5. The timelike surface M3 (s,v) is developable if and only if

ε cosh w(s) (−sinh w(s)kn + cosh w(s)tr) −σ (sinh w(s) (kg + w′(s)) + σtr) = 0.

Remark 3.6. If sinhw(s) = x3 = const.and coshw(s) = x4 = const ,generating lines of the surface δ(s) be

a Smarandache curve. Let M3(s,v) be a developable timelike surface and δ(s) be a Smarandache curve , if

at the same time δ(s) be a asymptotic, then the base curve is a general helix, so that

τ

κ
=

σx3
εx24 −σ2

,

if also δ(s) be a geodesic, then the base curve is a general helix, so that

τ

κ
=
−εx3x4
σ2 −εx24

.



Int. J. Anal. Appl. 18 (6) (2020) 909

Corollary 3.6. If u(s) be a line and Smarandache curve, then

kn
kg

=
σ

−εx4

where sinhw(s) = x3 = const.and coshw(s) = x4 = const.

Theorem 3.6. The striction line on timelike surface M3 (s,v) is obtained as follows

β = α−
−σkn − (w′ (s) −kg) . cosh w (s)

−(w′ (s)−kg)
2

+ σ2
(
−k2n+t

2
r

)
+ (2σ (w′ (s) −kg) (−kn cosh w (s) +tr sinh w (s)))

+ε2 (−kn sinh w (s) +tr cosh w (s))
2

X (s) .

Remark 3.7. If δ(s) is a geodesic curve and

w (s) = −σ
∫

κ

cosh w (s)
d (s) ,

then striction line of surface is equal to the base curve.

Remark 3.8. If δ(s) is a asymptotic curve and

w (s) =

∫
κd (s) ,

then striction line of timelike surface is equal to the base curve.

3.2. Ruled Surfaces with Constant Slope Ruling with Respect to the (t, n) Planes. Generating

lines of the surface M̃ are given by points on the curve X̃(s) and they have the constant slope σ with respect

to the (t,n) planes to the curve at every point on the curve X̃(s).The surface will be called the ruled surfaces

with constant slope ruling with respect to the (t,n) planes where

〈
X̃(s),y(s)

〉
= σ.

Case 3.4. If α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surfaces is

spacelike character where 1 + σ2 > 0. Direction vector of generating line of the spacelike surface is given by

X̃(s) = sin w(s).t(s) + cos w(s).n(s) + σ.y(s)

The surface with constant slope ruling M̃1 parametrization obtained by

M̃1 (s,v) = α (s) + v
(
X̃(s)

)
.



Int. J. Anal. Appl. 18 (6) (2020) 910

Case 3.5. If α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is

spacelike character where σ2−1 > 0 and the surfaces is timelike character where σ2−1 < 0. Direction vector

of generating line of the surface is given by

ũ(s) = cosh w(s).t(s) + sinh w(s).n(s) + σ.y(s)

and the surface is obtained by

M̃2 (s,v) = α (s) + v (ũ(s)) .

Case 3.6. If α(s) is a timelike curve with the principal spacelike normal vector field n(s), then the surface

is timelike where σ2 + 1 > 0 and direction vector is given as follows

δ̃(s) = sinh w(s).t(s) + cosh w(s).n(s) + σ.y(s).

The surface is parametrized by

M̃3 (s,v) = α (s) + v
(
δ̃(s)

)
.

Theorem 3.7. The spacelike surface M̃1 (s,v) is developable if and only if

cos w(s) (sin w(s)kg + cos w(s)tr) −σ ((kn −w′(s)) sin w(s) + σtr) = 0.

Corollary 3.7. Developable surface’s M̃1 (s,v) generating line X̃(s) is an asymptotic and Smarandache

curve if and only if α̃(s) is a general helix with

τ

κ
=

x1x2
σ2 −x22

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Corollary 3.8. M̃1(s,v) developable surface’s generating line X̃(s) is a geodesic and Smarandache curve

if and only if α̃(s) is a general helix, so that

τ

κ
=

σx1
x22 −σ2

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Corollary 3.9. X̃(s) is a line and Smarandache curve if and only if

kn
kg

=
σ

x2

where cosw(s) = x2 = const.



Int. J. Anal. Appl. 18 (6) (2020) 911

Theorem 3.8. The striction line on spacelike surface M̃1 (s,v) is given by

β = α−
σkg − (w′ (s) + kn) . cos w (s)

−w′ (s)2 −2w′ (s) .kn+k2n
(
sin2 w(s)−cos2 w(s)

)
+ σ2

(
−k2g+t

2
r

)
−2σ (kg. (w′ (s) + kn) . cos w(s) − tr. sin w (s) . (−w

′ (s) + kn))

+
(

sin w(s).kg+ cos w (s) .tr

)2
.X (s) .

Corollary 3.10. If X̃(s) is an asymptotic curve and

w (s) = σ

∫
κ

cos w (s)
d (s) ,

then striction line of surface is equal to the base curve.

Corollary 3.11. If X̃(s) is a geodesic curve and

w (s) = −
∫
κd (s)

where cos w (s) 6= 0, then striction line of the surface is equal to the base curve.

Theorem 3.9. The spacelike surface M̃2 (s,v) is developable if and only if

sinh w(s) (cosh w(s).kg + sinh w(s)tr) −σ (cosh w(s) (kn + w′(s)) + σtr) = 0

where σ2 − 1 > 0.

If σ2 − 1 < 0, then the surface is timelike ruled surface. So the timelike surface is developable if and only if

sinh w(s). (cosh w(s).kg + sinh w(s)tr) −σ (cosh w(s) (−εkn + w′(s)) + σεtr) = 0

where ε = 〈t,t〉 = ∓1.

Remark 3.9. If sinhw(s) = x3 = const.and coshw(s) = x4 = const , generating line of the surface ũ(s) be

a Smarandache curve. Let M̃2(s,v) be a developable spacelike surface and ũ(s) be a Smarandache curve , if

at the same time ũ(s) be an asymptotic, then the base curve is a general helix with

τ

κ
=

x3x4
σ2 −x23

,

if also ũ(s) be a geodesic, then the base curve is a general helix, so that

τ

κ
=

σ

x3
.



Int. J. Anal. Appl. 18 (6) (2020) 912

Remark 3.10. If sinhw(s) = x3 = const.and coshw(s) = x4 = const,generating lines of the surface ũ(s)

be a Smarandache curve. Let M̃2(s,v) be a developable timelike surface and ũ(s) be a Smarandache curve ,

if at the same time ũ(s) be an asymptotic, then the base curve is a general helix with

τ

κ
=

x3x4
x23 −εσ2

if also ũ(s) be a geodesic, then the base curve is a general helix with

τ

κ
=

εσx4
εσ2 −x23

if ũ(s) be a line curvature, then

kn
kg

=
−εσ
x3

.

Theorem 3.10. The striction line on spacelike surface M̃3(s,v) is given by

β = α−
σkg − (w′ (s) + kn) sinh w (s)

(w′ (s) +kn)
2

+ σ2
(
−k2g+t

2
r

)
+ ((2σ (w′ (s) + kn) kg sinh w (s) +tr cosh w (s)))

+ (kg cosh w (s) +tr sinh w (s))
2

X (s)

where σ2 − 1 > 0, and striction line on timelike surface is given by

β = α−
−σkg − (w′ (s) + kn) sinh w (s)

w′ (s)
2

+k2n−2.kn.w′ (s)
(
sinh2 w(s) −ε cosh2 w(s)

)
+σ2

(
k2g+t

2
r

)
+2σ (kg sinh w (s) . (w

′ (s) + kn) +ε.tr cosh w (s) . (w
′ (s) −εkn))

+ (kg cosh w (s) +tr sinh w (s))
2

X (s)

where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1.

Remark 3.11. If ũ(s) is an asymptotic curve and

w (s) = σ

∫
κ

sinh w (s)
d (s)

where σ2 − 1 > 0, then striction line of spacelike surface is equal to the base curve and

w (s) = −σ
∫

kg
sinh w (s)

d (s)

where σ2 − 1 < 0, then striction line of timelike surface is equal to the base curve.

Remark 3.12. If ũ(s) is a geodesic curve and

w (s) = −
∫
κd (s)



Int. J. Anal. Appl. 18 (6) (2020) 913

where σ2 − 1 > 0 then striction line of spacelike surface is equal to base curve and

w (s) =

∫
knd (s)

where σ2 − 1 < 0 and sinh w (s) 6= 0, then striction line of timelike surface is equal to base curve.

Corollary 3.12. The surface M̃3 (s,v) is torsal if and only if

cosh w(s) (sinh w(s)kg + cosh w(s)tr) −σ (sinh w(s) (−εkn + w′(s)) + εσtr) = 0

where ε = 〈t,t〉 = ∓1.

Corollary 3.13. If sinhw(s) = x3 = const.and coshw(s) = x4 = const, generating lines of the surface δ̃(s)

is a Smarandache curve. Let M̃3(s,v) be a developable timelike surface and δ̃(s) is a Smarandache curve ,

if at the same time δ̃(s) be a asymptotic, then basic curve is a general helix, so that

τ

κ
=

x3x4
σ2 −x24

if also δ̃(s) be a geodesic, then base curve is a general helix, so that

τ

κ
=

x3
σx4

if δ̃(s) be a line curvature, then

kn
kg

=
−εσ
x4

such that ε = 〈t,t〉 = ∓1.

Theorem 3.11. The striction line on timelike surface M̃3(s,v) is given by

β = α−
−σkg−(w′ (s) + kn) . cosh w (s)

−
(
w′ (s)

2
+k2n

)
−2w′ (s) .kn.

(
cosh2 w(s) + ε sinh2 w(s)

)
+σ2

(
k2g+t

2
r

)
+ (sinh w(s).kg + cosh w (s) .tr)

2

X (s) .

Remark 3.13. If generating line is a asymptotic curve on timelike surface M̃3(s,v) and

w (s) = −σ
∫

κ

cosh w (s)
d (s)

then striction line of surface is equal to base curve.



Int. J. Anal. Appl. 18 (6) (2020) 914

Remark 3.14. If generating line is a geodesic curve on timelike surface M̃3(s,v) and

w (s) = −
∫
κd (s)

then striction line of the surface is equal to the base curve.

3.3. Ruled Surfaces with Constant Slope Ruling with Respect to the (n,y) Planes. Surface M is

given by points on the curve X(s) and they have the constant slope σ with respect to the (n,y) planes.The

surface will be defined the ruled surfaces with constant slope ruling with respect to the (n,y) planes to the

curve where 〈
X(s), t(s)

〉
= σ.

Case 3.7. If α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surface is

spacelike where 1 + σ2 > 0. Direction vector of generating line of the spacelike surface is given by

X(s) = sin w(s).n(s) + cos w(s).y(s) + σt(s).

The surface with constant slope M1 is parametrized by

M1 (s,v) = α(s) + v
(
X(s)

)
.

Case 3.8. If α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is

spacelike where σ2 − 1 > 0 and the surface is timelike where σ2 − 1 < 0. Direction vector of generating line

of the surface is given by

u(s) = cosh w(s).n(s) + sinh w(s).y(s) + σt(s)

and the surface is obtained as follows

M2 (s,v) = α(s) + v (u(s)) .

Case 3.9. If α(s) is a timelike curve with the principal spacelike normal vector field n(s), then the surface

is timelike where σ2 + 1 > 0 and direction vector is defined as follows

δ(s) = sinh w(s).n(s) + cosh w(s).y(s) + σt(s).

The surface parametrization is given as follows

M3 (s,v) = α(s) + v
(
δ(s)

)
.

Theorem 3.12. The spacelike surface M1 (s,v) is developable if and only if

tr(sin
2 w(s) − cos2 w(s)) + σ (cos w(s)kn + sin w(s)kg) −w′(s) = 0.



Int. J. Anal. Appl. 18 (6) (2020) 915

Corollary 3.14. Developable surface’s generating line X(s) is a asymptotic and Smarandache curve if and

only if α(s) is a general helix, so that

τ

κ
=

σx1
x22 −x21

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Corollary 3.15. M1(s,v) developable surface’s generating line X(s) is a geodesic and Smarandache curve

if and only if α(s) is a general helix, so that

τ

κ
=

σx2
x22 −x21

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Corollary 3.16. X(s) is a line and Smarandache curve

kn
kg

= −
x2
x1

where sinw(s) = x1 = const.and cosw(s) = x2 = const.

Theorem 3.13. The striction line on the spacelike surface M1(s,v) is given by

β = α−
kn. sin w(s) −kg. cos w(s)(

w′(s) − t2r
)(

sin2 w(s) − cos2 w(s)
)
− 2tr.w′(s) + σ2.

(
k2n + k

2
g

)
+ (kn. sin w(s) −kg. cos w(s))

2

.X (s) .

Remark 3.15. If

kn
kg

= −cot w(s),

then the striction line on the surface M1(s,v) equal to the base curve.

Theorem 3.14. The spacelike ruled surface with constant slope M2 (s,v) is developable if and only if

sinh w(s) (cosh w(s).kn + ε sinh w(s)kg) −σ (cosh w(s) (tr + w′(s)) + σkg) = 0

ε = ±1. If σ2 −1 > 0, the surface is spacelike and ε = −1. If σ2 −1 < 0, the surface is timelike and ε = +1.



Int. J. Anal. Appl. 18 (6) (2020) 916

Corollary 3.17. If sinhw(s) = x3 = const.and coshw(s) = x4 = const,generating line of the surface u(s)

is a Smarandache curve. Let M2(s,v) be a developable surface and u(s) be a Smarandache curve , if at the

same time u(s) is an asymptotic, then the base curve is a general helix with

τ

κ
=
εx23 −σ2

σx4
,

if also u(s) is a geodesic , then the base curve is a general helix with

τ

κ
=
ε
(
σ2 −x3x4

)
σx4

,σ2 − 1 > 0

if u(s) is a line curvature, then

kn
kg

=
x3x4

σ2 − �x24
.

Remark 3.16. The striction line on the spacelike surface M2(s,v) is given by

β = α (s) +
kn. cosh w (s) −kg. sinh w (s)

(w′ (s) + tr)
2
.
(
sinh2 w (s) + cosh2 w (s)

)
+ σ2

(
k2g+k

2
n

)
+ (2σ (w′ (s) + tr) (kn. sinh w (s) +kg. cosh w (s)))

−(kn. cosh w (s)−kg. sinh w (s))
2

X (s)

where σ2 − 1 > 0, and striction line on the timelike surface is given by

β = α (s) −
kn. cosh w (s) + kg. sinh w (s)

(w′ (s) + tr)
2

+σ2
(
k2g+k

2
n

)
+ (kn. cosh w (s) + kg. sinh w (s))

2

+ (2σ (w′ (s) + tr) (ε.kn. sinh w (s) +kg. cosh w (s)))

.X (s)

where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1.

Theorem 3.15. The spacelike surface M3 (s,v) is developable if and only if

sinh w(s) (sinh w(s) (tr + w
′(s)) + σkg) − cosh w(s) (cosh w(s) (εtr + w′(s)) −εσkn)

where ε = 〈t,t〉 = ∓1.

Remark 3.17. If sinhw(s) = x3 = const.and coshw(s) = x4 = const, generating line of the surface δ(s) is

a Smarandache curve. Let M3(s,v) be a developable timelike surface and δ(s) be a Smarandache curve , if

at the same time δ(s) be an asymptotic, then the base curve is a general helix, so that

τ

κ
=

σx3
εx24 −x23



Int. J. Anal. Appl. 18 (6) (2020) 917

for ε = 1,
τ

κ
= σx3

if also δ(s) is a geodesic, then base curve is a general helix, so that

τ

κ
=

−εσ
x23 −εx24

for ε = 1,
τ

κ
= σ

if δ(s) is a line curvature, then
kn
kg

=
−εx4
x3

where ε = 〈t,t〉 = ∓1.

Theorem 3.16. The striction line on timelike surface M3(s,v) is given by

β = α (s) −
kn. sinh w(s) + kg. cosh w(s)

−(cosh w(s). (ε.tr + w′(s))−εσ.kn)
2

+ (sinh w(s). (tr + w
′(s)) +σ.kg)

2

+
(
kn. sinh w(s) + kg. cosh w(s)

)2
.X (s) .

4. Some Numerical Examples

In this section, we give examples of the surfaces with a constant slope ruling according to Darboux frame

in Minkowski Space with respect to the given planes.

Example 4.1. The curve α(s) given by

α(s) =
(
r sin

s

r
,r cos

s

r
,
s

r

)
and ruled surfaces with constant slope ruling is parametrized by

M1(s,v) = α(s) + v(sin w(s)t(s) + cos w(s)y(s) + σn(s)).

Surface is visualized in following figure for w(s) = π
2
,r = 10,σ = 2.



Int. J. Anal. Appl. 18 (6) (2020) 918

Figure 1.

Example 4.2. Ruled surfaces with constant slope ruling with respect to the {t,n} planes

M̃1(s,v) = α(s) + v(sin w(s)t(s) + cos w(s)n(s) + σy(s)).

is shown following figure for w(s) = s
π
,r = 2,σ =

√
3.

Figure 2.

Example 4.3. The curve β(s) is given by

β(s) =
(
r cosh

s

r
,r sinh

s

r
,
s

r

)
and the timelike surface with a constant slope ruling is shown following figure for w(s) = 3π

2
, r = 10,σ = 1

10
,

s�
(
−5, π

2

4

)
, u� (−1, 1).



Int. J. Anal. Appl. 18 (6) (2020) 919

Figure 3.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

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[2] H.H. Ugurlu, H. Kocayigit, The Frenet and Darboux Instantaneous Rotain Vectors of Curves on Timelike Surfaces, Math.

Comput. Appl. 1 (2) (1996), 133-141.

[3] S. Kızıltuğ, A. Çakmak, Developable Ruled Surfaces with Darboux Frame in Minkowski 3-Space. Life Science Journal

(2013), 10(4).

[4] S. Kızıltuğ, Y. Yaylı, Timelike Curves on Timelike Parallel Surfaces in Minkowski 3-Space E31, Math. Aeterna, 2 (2012),

689 - 700.

[5] S. N. Krivoshapko, S. Shambina, Design of Developable Surfaces and The Application of Thin-Walled Developable Struc-

tures, Serbian Architect. J. 4 (3) (2012), 298-317.

[6] R. López, Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. Int. Electron. J. Geom. 7 (1) (2014),

44-107.

[7] K. Malecek, J. Szarka, D. Szarkova, Surfaces with Constant Slope with Their Generalisation. J. Polish Soc. Geom. Eng.

Graph. 19 (2009), 67-77.

[8] B. O’Neill, Elementary Differential Geometry, Academic Press, New York, 1966.

[9] M. Önder, H.H. Uğurlu, Frenet Frames and Invariants of Timelike Ruled Surfaces, Ain Shams Eng. J. 4 (3) (2013), 507-513.

[10] A. Yavuz, F. Ateş, Y. Yaylı, Non-null Surfaces with Constant Slope Ruling with Respect to Osculating Plane. Adıyaman

Univ. J. Sci. 10 (2020), 240-255.


	1. Introduction
	2. Preliminaries
	3. Ruled Surfaces with Constant Slope Ruling According to Darboux Frame in Minkowski 3-Space
	3.1. Ruled Surfaces with Constant Slope Ruling with Respect to the ( t,y)  Planes 
	3.2. Ruled Surfaces with Constant Slope Ruling with Respect to the ( t,n)  Planes 
	3.3. Ruled Surfaces with Constant Slope Ruling with Respect to the ( n,y)  Planes 

	4. Some Numerical Examples 
	References