International Journal of Analysis and Applications

Volume 18, Number 1 (2020), 1-15

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

DOI: 10.28924/2291-8639-18-2020-1

ON THE EQUIFORM DIFFERENTIAL GEOMETRY OF AW(k)-TYPE CURVES

IN PSEUDO-GALILEAN 3-SPACE

M. KHALIFA SAAD1,2,∗ AND H. S. ABDEL-AZIZ2

1Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Madinah, KSA

2Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt

∗Corresponding author: mohammed.khalifa@iu.edu.sa

Abstract. The aim of this paper is to study AW(k)-type (1 ≤ k ≤ 3) curves according to the equiform

differential geometry of the pseudo-Galilean space G13. We give some geometric properties of AW(k) and

weak AW(k)-type curves. Moreover, we give some relations between the equiform curvatures of these curves.

Finally, examples of some special curves are given and plotted to support our main results.

1. Introduction

The geometry of space is associated with mathematical group. The idea of invariance of geometry under

transformation group may imply that, on some spacetimes of maximum symmetry there should be a principle

of relativity which requires the invariance of physical laws without gravity under transformations among

inertial systems [1]. The theory of curves and the curves of constant curvature in the equiform differential

geometry of the isotropic spaces I13 , I
2
3 and the Galilean space G3 are described in [2] and [3], respectively.

The pseudo-Galilean space is one of the real Cayley-Klein spaces. It has projective signature (0, 0, +,−)

according to [2]. The absolute of the pseudo-Galilean space is an ordered triple {w,f,I} where w is the ideal

plane, f a line in w and I is the fixed hyperbolic involution of the points of f. In [4], from the differential

Received 2019-10-24; accepted 2019-11-20; published 2020-01-02.

2010 Mathematics Subject Classification. 53A04, 53A35, 53C40.

Key words and phrases. AW(k)-type curves; spacelike and timelike curves; general helix; equiform geometry; pseudo-Galilean

space.

c©2020 Authors retain the copyrights

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

1

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-1


Int. J. Anal. Appl. 18 (1) (2020) 2

geometric point of view, K. Arslan and A. West defined the notion of AW(k)-type submanifolds. Since then,

many works have been done related to AW(k)-type submanifolds (see, for example, [5–10]). In [9], Özgür and

Gezgin studied a Bertrand curve of AW(k)-type and furthermore, they showed that there is no such Bertrand

curve of AW(1) and AW(3)-types if and only if it is a right circular helix. In addition, they studied weak

AW(2)-type and AW(3)-type conical geodesic curves in Euclidean 3-space E3. Besides, In 3-dimensional

Galilean space and Lorentz space, the curves of AW(k)-type were investigated in [6, 8]. In [7], the authors

gave curvature conditions and characterizations related to AW(k)-type curves in En and in [10], the authors

investigated curves of AW(k)-type in the 3-dimensional null cone.

This paper is organized as follows. In section 2, the basic notions and properties of a pseudo-Galilean

geometry are reviewed. In section 3, properties of the equiform geometry of the pseudo-Galilean space G13

are given. Section 4 contains a study of AW(k)-type equiform Frenet curves. Finally, some examples of

special curves in G13 are included in section 5.

2. Basic concepts

In this section, we recall some basic notions from pseudo-Galilean geometry [11,12]. In the inhomogeneous

affine coordinates for points and vectors (point pairs) the similarity group H8 of G
1
3 has the following form

x̄ = a + b.x,

ȳ = c + d.x + r. cosh θ.y + r. sinh θ.z,

z̄ = e + f.x + r. sinh θ.y + r. cosh θ.z, (2.1)

where a,b,c,d,e,f,r and θ are real numbers. Particularly, for b = r = 1, the group (2.1) becomes the group

B6 ⊂ H8 of isometries (proper motions) of the pseudo-Galilean space G13. The motion group leaves invariant

the absolute figure and defines the other invariants of this geometry. It has the following form

x̄ = a + x,

ȳ = c + d.x + cosh θ.y + sinh θ.z,

z̄ = e + f.x + sinh θ.y + cosh θ.z. (2.2)

According to the motion group in the pseudo-Galilean space, there are non-isotropic vectors A(A1,A2,A3)

(for which holds A1 6= 0) and four types of isotropic vectors: spacelike (A1 = 0, A22 − A23 > 0), timelike

(A1 = 0, A
2
2 −A23 < 0) and two types of lightlike vectors (A1 = 0,A2 = ±A3). The scalar product of two

vectors u = (u1,u2,u3) and v = (v1,v2,v3) in G
1
3 is defined by

〈u,v〉 =


 u1v1, if u1 6= 0 or v1 6= 0,u2v2 −u3v3 if u1 = 0 and v1 = 0.



Int. J. Anal. Appl. 18 (1) (2020) 3

We introduce a pseudo-Galilean cross product in the following way

u×G13 v =

∣∣∣∣∣∣∣∣∣
0 −j k

u1 u2 u3

v1 v2 v3

∣∣∣∣∣∣∣∣∣
,

where j = (0, 1, 0) and k = (0, 0, 1) are unit spacelike and timelike vectors, respectively. Let us recall basic

facts about curves in G13, that were introduced in [13–15].

A curve γ(s) = (x(s),y(s),z(s)) is called an admissible curve if it has no inflection points (γ̇× γ̈ 6= 0) and

no isotropic tangents (ẋ 6= 0) or normals whose projections on the absolute plane would be lightlike vectors

(ẏ 6= ±ż). An admissible curve in G13 is an analogue of a regular curve in Euclidean space [12].

For an admissible curve γ(s) : I ⊆ R → G13, the curvature κ(s) and torsion τ(s) are defined by

κ(s) =

√
|ÿ(s)2 − z̈(s)2|

(ẋ(s))2
, τ(s) =

ÿ(s)
...
z (s) −

...
y (s)z̈(s)

|ẋ(s)|5 ·κ2(s)
, (2.3)

expressed in components. Hence, for an admissible curve γ : I ⊆ R → G13 parameterized by the arc length s

with differential form ds = dx is given by

γ(x) = (x,y(x),z(x)). (2.4)

The formulas (2.3) have the following form

κ(x) =
√
|y′′ (x)2 −z′′ (x)2|, τ(x) =

y
′′
(x)z

′′′
(x) −y

′′′
(x)z

′′
(x)

κ2(x)
. (2.5)

The associated trihedron is given by

e1 = γ
′(x) = (1,y

′
(x),z

′
(x)),

e2 =
1

κ(x)
γ

′′
(x) =

1

κ(x)
(0,y

′′
(x),z

′′
(x)),

e3 =
1

κ(x)
(0,�z

′′
(x),�y

′′
(x)), (2.6)

where � = +1 or � = −1, chosen by criterion det(e1,e2,e3) = 1, that means∣∣∣y′′ (x)2 −z′′ (x)2∣∣∣ = �(y′′ (x)2 −z′′ (x)2).
The curve γ given by (2.4) is timelike (resp. spacelike) if e2(s) is a spacelike (resp. timelike) vector. The

principal normal vector or simply normal is spacelike if � = +1 and timelike if � = −1. For derivatives of the

tangent e1, normal e2 and binormal e3 vector fields, the following Frenet formulas in G
1
3 hold:

e′1(x) = κ(x)e2(x),

e′2(x) = τ(x)e3(x),

e′3(x) = τ(x)e2(x). (2.7)



Int. J. Anal. Appl. 18 (1) (2020) 4

3. Frenet equations according to the equiform geometry of G13

This section contains some important facts about equiform geometry. The equiform differential geometry

of curves in the pseudo-Galilean space G13 has been described in [11]. In the equiform geometry a few specific

terms will be introduced. So, let γ(s) : I → G13 be an admissible curve in the pseudo-Galilean space G13, the

equiform parameter of γ is defined by

σ :=

∫
1

ρ
ds =

∫
κds,

where ρ = 1
κ

is the radius of curvature of the curve γ. Then, we have

ds

dσ
= ρ. (3.1)

Let h be a homothety with center at origin and the coefficient µ. If we put γ̄ = h(γ), then it follows

s̄ = µs and ρ̄ = µρ,

where s̄ is the arc-length parameter of γ̄ and ρ̄ is the radius of curvature of this curve. Therefore, σ is an

equiform invariant parameter of γ (see [11]).

Notation 3.1. The functions κ and τ are not invariants of the homothety group, then from (2.3) it follows

that κ̄ = 1
µ
κ and τ̄ = 1

µ
τ.

Now we define the Frenet formulas of the curve γ with respect to its equiform invariant parameter σ in

G13. The vector

T =
dγ

dσ
,

is called a tangent vector of the curve γ. From (2.6) and (3.1), we get

T =
dγ

ds

ds

dσ
= ρ ·

dγ

ds
= ρ · e1. (3.2)

Also, the principal normal and the binormal vectors are respectively, given by

N = ρ · e2, B = ρ · e3. (3.3)

It is easy to show that {T, N, B} is an equiform invariant frame of γ. On the other hand, the derivatives of

these vectors with respect to σ are given by


T

N

B



′

=



ρ̇ 1 0

0 ρ̇ ρτ

0 ρτ ρ̇






T

N

B


 . (3.4)

The functions K : I → R defined by K = ρ̇ is called the equiform curvature of the curve γ and T : I → R

defined by T = ρτ = τ
κ

is called the equiform torsion of this curve. In the light of this, the formulas (3.4)



Int. J. Anal. Appl. 18 (1) (2020) 5

analogous to the Frenet formulas in the equiform geometry of the pseudo-Galilean space G13 can be written

as 


T

N

B



′

=



K 1 0

0 K T

0 T K






T

N

B


 . (3.5)

The equiform parameter σ =
∫
κ(s)ds for closed curves is called the total curvature, and it plays an important

role in global differential geometry of Euclidean space. Also, the function τ
κ

has been already known as a

conical curvature and it also has interesting geometric interpretation.

Notation 3.2. Let γ : I → G13 be a Frenet curve in the equiform geometry of G13, the following statements

are true ( for more details, see [11, 13] ):

(1) If γ(s) is an isotropic logarithmic spiral in G13. Then, K =const. 6= 0 and T = 0,

(2) If γ(s) is a circular helix in G13. Then, K =0 and T =const. 6= 0,

(3) If γ(s) is an isotropic circle in G13. Then, K =0 and T = 0.

4. AW(k)-type curves in the equiform geometry of G13

Let γ(s) : I → G13 be a curve in the equiform geometry of the pseudo-Galilean space G13. The curve γ is

called a Frenet curve of osculating order l if its derivatives:

γ′(s),γ′′(s),γ′′′(s), ...,γ(l)(s),

are linearly dependent and

γ′(s),γ′′(s),γ′′′(s), ...,γ(l+1)(s),

are no longer linearly independent for all s ∈ I.

To each Frenet curve of order 3, one can associate an orthonormal 3-frame {T, N, B} along γ, such that

γ′(s) = 1
ρ
T, called the equiform Frenet frame (Eqs. (3.5)).

Now, we consider equiform Frenet curves of osculating order 3 in G13 and discuss some important results.

Let γ(s) : I → G13 be a Frenet curve in the equiform geometry of the pseudo-Galilean space. By the use

of Frenet formulas (3.5), we obtain the higher order derivatives of γ as follows

γ′(s) =
dγ

dσ

dσ

ds
=

1

ρ
T,

γ′′(s) =
1

ρ2
N,

γ′′′(s) =
1

ρ3
(−KN+T B) ,

γ′′′′(s) =
1

ρ4
[(2K2+T 2 −K′)N + (T ′ − 3KT )B].



Int. J. Anal. Appl. 18 (1) (2020) 6

Notation 4.1. Let us write

Q1 =
1

ρ2
N, (4.1)

Q2 =
1

ρ3
(−KN+T B) , (4.2)

Q3 =
1

ρ4
[(2K2+T 2 −K′)N + (T ′ − 3KT )B]. (4.3)

Notation 4.2. γ′(s),γ′′(s),γ′′′(s) and γ′′′′(s) are linearly dependent if and only if Q1,Q2 and Q3 are linearly

dependent.

Definition 4.1. [5] Frenet curves (of osculating order 3) in the equiform geometry of the pseudo-Galilean

space G13 are called curves of type:

(1) equiform AW(1) if they satisfy Q3 = 0,

(2) equiform AW(2) if they satisfy ‖Q2‖
2
Q3 = 〈Q3,Q2〉Q2,

(3) equiform AW(3) if they satisfy ‖Q1‖
2
Q3 = 〈Q3,Q1〉Q1,

(4) weak equiform AW(2) if they satisfy

Q3 = 〈Q3,Q∗2〉Q
∗
2, (4.4)

(5) weak equiform AW(3) if they satisfy

Q3 = 〈Q3,Q∗1〉Q
∗
1, (4.5)

where

Q∗1 =
Q1
‖Q1‖

,

Q∗2 =
Q2 −〈Q2,Q∗1〉Q∗1
‖Q2 −〈Q2,Q∗1〉Q∗1‖

. (4.6)

Proposition 4.1. Let γ : I → G13 be a Frenet curve (of osculating order 3) in the equiform geometry of the

pseudo-Galilean space G13, therefore

(i) γ is of type weak equiform AW(2) if and only if

2K2 + T 2 −K′ = 0, (4.7)

(ii) γ is of type weak equiform AW(3) if and only if

T ′ − 3KT (s) = 0. (4.8)

Proof. Using Definition 4.1 and Notation 4.1, the proof will be obvious. �



Int. J. Anal. Appl. 18 (1) (2020) 7

Theorem 4.1. Let γ : I → G13 be a Frenet curve (of osculating order 3) in the equiform geometry of the

pseudo-Galilean space G13. Then γ is of type equiform AW(1) if and only if

−K′ + 2K2 + T 2 = 0,

3KT −T ′ = 0. (4.9)

Proof. Since γ is of type equiform AW(1), then from (4.3), we obtain

1

ρ4
[(2K2+T 2(s) −K′)N + (T ′ − 3KT )B] = 0.

As we know, the vectors N and B are linearly independent, so we can write

2K2+T 2 −K′ = 0 and T ′ − 3KT = 0.

The converse statement is straightforward and therefore, the proof is completed. �

Theorem 4.2. Let γ : I → G13 be a Frenet curve (of osculating order 3) in the equiform geometry of the

pseudo-Galilean space G13. Then γ is of type equiform AW(2) if

K2T −KT ′ + TK′ −T 3 = 0. (4.10)

Proof. Assuming that γ is a Frenet curve in the equiform geometry of G13 , then from (4.2) and (4.3), one

can write

Q2 = a11N + a12B,

Q3 = a21N + a22B,

where a11,a12, a21 and a22 are differentiable functions. Since Q2 and Q3 are linearly dependent, hence

coefficients determinant equals zero, that is ∣∣∣∣∣∣
a11 a12

a21 a22

∣∣∣∣∣∣ = 0, (4.11)
where

a11 =
−1
ρ3
K, a12 =

1

ρ3
T ,

a21 =
1

ρ4
[−K′ + 2K2 + T 2],

a22 =
1

ρ4
[−3KT + T ′]. (4.12)

From (4.11) and (4.12), we obtain (4.10). �



Int. J. Anal. Appl. 18 (1) (2020) 8

Theorem 4.3. Let γ : I → G13 be a Frenet curve (of osculating order 3) in the equiform geometry of G13.

Then γ is of equiform AW(3)-type if

T ′ − 3KT = 0. (4.13)

Proof. Using Definition 4.1 and Eqs. (4.1) and (4.3), we obtain (4.13). �

5. Computational examples

We consider some examples (timelike and spacelike curves [11, 12]) which characterize equiform gen-

eral (circular) helices with respect to the Frenet frame {T, N, B} in the equiform geometry of G13 which

satisfy some conditions of equiform curvatures (i)K = K(s),T = T (s) (ii)K =const. 6= 0,T =const. 6= 0

(iii)K =const. 6= 0,T =0.

Example 5.1. Consider the equiform timelike general helix r : I −→ G13,I ⊆ R which parameterized by

the arc length s with differential form ds = dx is given by

r(x) = (x,y(x),z(x)),

where

x(s) = s,

y(s) =
e−as

(a2 − b2)2
((
a2 + b2

)
cosh (bs) + 2ab sinh (bs)

)
,

z(s) =
e−as

(a2 − b2)2
(
2ab cosh (bs) +

(
a2 + b2

)
sinh (bs)

)
;

a,b ∈ R−{0} .

The corresponding derivatives of r are as follows

r′ =

(
1,
−e−as

(a2 − b2)
(a cosh (bs) + b sinh (bs)) ,

e−as

(b2 −a2)
(b cosh (bs) + a sinh (bs))

)
,

r′′ =
(
0,e−as cosh (bs) ,e−as sinh (bs)

)
,

r′′′ =
(
0,e−as (−a cosh (bs) + b sinh (bs)) ,e−as (b cosh (bs) −a sinh (bs))

)
.

The tangent vector of r has the form

e1 = (x
′,y′,z′)

=

(
1,
−e−as

(a2 − b2)
(a cosh (bs) + b sinh (bs)) ,

e−as

(b2 −a2)
(b cosh (bs) + a sinh (bs))

)
,

and the two normals (normal and binormal) of the curve are, respectively

e2 = (0, cosh (bs) , sinh (bs)) ,

e3 = (0, sinh (bs) , cosh (bs)) ; det[e1, e2, e3] = 1.



Int. J. Anal. Appl. 18 (1) (2020) 9

Therefore, the curvature and torsion of r are respectively, given by

κ = e−as, τ = b .

From the equiform Frenet formulas, we can express the vector fields T, N, B as follows

T =

(
eas,

−1
(a2 − b2)

(a cosh (bs) + b sinh (bs)) ,
1

(b2 −a2)
(b cosh (bs) + a sinh (bs))

)
,

N = (0,eas cosh (bs) ,eas sinh (bs)) ,

B = (0,eas sinh (bs) ,eas cosh (bs)) ,

respectively. In the light of this, the equiform curvatures are given by

K = aeas,T = −beas.

Figure 1. Equiform timelike general helix with K = 5e5s,T = −2e5s.

Example 5.2. Let r : I −→ G13,I ⊆ R be the equiform spacelike general helix, and it is given by

r(x) = (x,y(x),z(x)),



Int. J. Anal. Appl. 18 (1) (2020) 10

where

x(s) = s,

y(s) =
e−as

(a2 − b2)2
(
2ab cosh (bs) +

(
a2 + b2

)
sinh (bs)

)
,

z(s) =
e−as

(a2 − b2)2
((
a2 + b2

)
cosh (bs) + 2ab sinh (bs)

)
;

a,b ∈ R−{0} .

For the coordinate functions of r, we have

r′ =

(
1,

e−as

(b2 −a2)
(b cosh (bs) + a sinh (bs)) ,

−e−as

(a2 − b2)
(a cosh (bs) + b sinh (bs))

)
,

r′′ =
(
0,e−as sinh (bs) ,e−as cosh (bs)

)
,

r′′′ =
(
0,e−as (b cosh (bs) −a sinh (bs)) ,e−as (b sinh (bs) −a cosh (bs))

)
.

Also, the associated trihedron is given by

e1 =

(
1,

e−as

(b2 −a2)
(b cosh (bs) + a sinh (bs)) ,

−e−as

(a2 − b2)
(a cosh (bs) + b sinh (bs))

)
,

e2 = (0, sinh (bs) , cosh (bs)) ,

e3 = (0,−cosh (bs) ,−sinh (bs)) .

The curvature and torsion of this curve are

κ = e−as, τ = −b .

Furthermore, the tangent, normal and binormal vector fields in the equiform geometry of G13 are obtained as

follows

T =

(
eas,

1

(b2 −a2)
(b cosh (bs) + a sinh (bs)) ,

−1
(a2 − b2)

(a cosh (bs) + b sinh (bs))

)
,

N = (0,eas sinh (bs) ,eas cosh (bs)) ,

B = (0,−eas cosh (bs) ,−eas sinh (bs)) ,

respectively.

The equiform curvatures of r are

K = aeas,T = −beas.

Example 5.3. Consider the equiform timelike circular helix r : I −→ G13,I ⊆ R is given by

r(x) = (x,y(x),z(x)),



Int. J. Anal. Appl. 18 (1) (2020) 11

Figure 2. Equiform spacelike general helix with K = 5e5s,T = −2e5s.

where

x(s) = s,

y(s) =
a3s

b (b2 −a2)

(
b sinh

(
b

a
ln(as)

)
−a cosh

(
b

a
ln(as)

))
,

z(s) =
a3s

b (b2 −a2)

(
b cosh

(
b

a
ln(as)

)
−a sinh

(
b

a
ln(as)

))
;

a,b ∈ R−{0} .

For this curve, the equiform vector fields are obtained as follows

T =

(
s

a
,
as

b
cosh

(
b

a
ln(as)

)
,
as

b
sinh

(
b

a
ln(as)

))
,

N =

(
0,
s

a
sinh

(
b

a
ln(as)

)
,
s

a
cosh

(
b

a
ln(as)

))
,

B =

(
0,
s

a
cosh

(
b

a
ln(as)

)
,
s

a
sinh

(
b

a
ln(as)

))
,

respectively.

It follows that

K =
1

a
,T =

−b
a2
.



Int. J. Anal. Appl. 18 (1) (2020) 12

Figure 3. Equiform timelike circular helix with K = 1
2
,T = −5

4
.

Example 5.4. Let the equiform spacelike circular helix r : I −→ G13,I ⊆ R be

r(x) = (x,y(x),z(x)),

where

x(s) = s,

y(s) =
a3s

b (b2 −a2)

(
b cosh

(
b

a
ln(as)

)
−a sinh

(
b

a
ln(as)

))
,

z(s) =
a3s

b (b2 −a2)

(
b sinh

(
b

a
ln(as)

)
−a cosh

(
b

a
ln(as)

))
;

a,b ∈ R−{0} .

Here, the equiform differential vectors respectively, are as follows

T =

(
s

a
,
as

b
sinh

(
b

a
ln(as)

)
,
as

b
cosh

(
b

a
ln(as)

))
,

N =

(
0,
s

a
cosh

(
b

a
ln(as)

)
,
s

a
sinh

(
b

a
ln(as)

))
,

B =

(
0,−

s

a
sinh

(
b

a
ln(as)

)
,−

s

a
cosh

(
b

a
ln(as)

))
.

Equiform curvature and equiform torsion are calculated as follows

K =
1

a
,T =

b

a2
.



Int. J. Anal. Appl. 18 (1) (2020) 13

Figure 4. Equiform spacelike circular helix with K = 1
3
,T = 4

9
.

Example 5.5. Let r : I −→ G13,I ⊆ R be a equiform timelike isotropic logarithmic spiral which parameterized

by the arc length s with differential form ds = dx, and is given by

r(x) = (x,y(x), 0),

where

x(s) = s,

y(s) =
as + b

a2
(ln(as + b) − 1) ,

z(s) = 0;

a,b ∈ R−{0} .

For this curve, we get

r′ =

(
1,

ln(as + b)

a
, 0

)
,

r′′ =

(
0,

1

as + b
, 0

)
,

r′′′ =

(
0,

−a
(as + b)

2
, 0

)
,



Int. J. Anal. Appl. 18 (1) (2020) 14

and

e1 =

(
1,

ln(as + b)

a
, 0

)
,

e2 = (0, 1, 0) ,

e3 = (0, 0, 1) ; κ =
1

as + b
, τ = 0.

In this case, equiform Frenet vectors and equiform curvatures are as follows

T =

(
as + b,

(as + b) ln(as + b)

a
, 0

)
,

N = (0,as + b, 0) ,

B = (0, 0,as + b) , K = a,T = 0.

respectively.

Figure 5. Equiform timelike isotropic logarithmic spiral with K = 2,T = 0.

From aforementioned calculations, according to (Proposition 4.2 and Theorems 4.1 − 4.3), the first four

examples are not characterize curves of equiform AW(1), weak equiform AW(2) or weak equiform AW(3)-

types. On the other hand, the last example shows that the curve is of equiform AW(2) and AW(3)-types

and it is not of equiform AW(1)-type. Also, this curve is of weak equiform AW(2) and not of weak equiform

AW(3)-types.



Int. J. Anal. Appl. 18 (1) (2020) 15

6. Conclusion

In this paper, we have considered some special curves of equiform AW(k)-type of the pseudo-Galilean 3-

space. Also, using the equiform curvature conditions of these curves, the necessary and sufficient conditions

for them to be equiform AW(k) and weak equiform AW(k)-types are obtained. Furthermore, some examples

to support our main results are given and plotted.

Acknowledgment

This research was supported by Islamic University of Madinah. We would like to thank our colleagues

from Deanship of Scientific Research who provided insight and expertise that greatly assisted the research.

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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3 , Rad JAZU, 1986, 39-44.

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[11] Z. Erjavec and B. Divjak, The equiform differential geometry of curves in the pseudo-Galilean space, Math. Commun. 13

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	1. Introduction
	2. Basic concepts
	3. Frenet equations according to the equiform geometry of G31
	4. AW(k)-type curves in the equiform geometry of G31
	5. Computational examples
	6. Conclusion
	Acknowledgment
	References