EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5710
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quasi Ruled Surfaces in Euclidean 3-space

Ayman Elsharkawy1,∗, Hoda K. Elsayied1, Aya Refaat1
1 Department of Mathematics, Faculty of Science, University of Tanta, Tanta, Egypt

Abstract. This paper introduces three distinct types of ruled surfaces, namely, the quasi-
tangent surfaces , the quasi-normal surfaces, and the quasi-binormal surfaces. These types are
determined by the orientation of their direction curves tangent, normal, and binormal to the base
curve, respectively. This paper does not only introduce these surfaces but also determines their
fundamental properties, including the first, the second, and the third fundamental forms, as well
as the Gaussian and the Mean curvatures. Also, the geodesic curvature, the normal curvature,
and the geodesic torsion associated with the base curve for each type of surface are investigated.
Furthermore, the conditions for the base curve to be as a geodesic, an asymptotic line, and a
principal line for each type of surface are provided. Also, the conditions for these curves to be
considered developable and minimal surfaces are introduced. Moreover, two illustrative examples
are introduced to obtain our results.
2020 Mathematics Subject Classifications: 53A05
Key Words and Phrases: Ruled surface, Euclidean space, Quasi frame, developable surface,
minimal surface

1. Introduction

The study of ruled surfaces has garnered significant attention from researchers in
recent decades due to their broad applications in various fields, including spatial mecha-
nism design, computer-aided geometric design, architecture, civil engineering, and solid
modeling [5–7, 23, 24, 31]. In differential geometry, ruled surfaces are generated by the
motion of a straight line, called a ruling, along a base curve in space. These surfaces
are central to many theoretical and practical advancements, including the study of de-
velopable ruled surfaces, which are characterized by zero Gaussian curvature and can
be unfolded onto a plane without distortion [8, 26], and minimal ruled surfaces, which
minimize surface area and are characterized by vanishing Mean curvature [29].

A significant body of work has explored the relationship between ruled surfaces and
helical curves within the framework of the Frenet frame in three-dimensional Euclidean
space [2, 3, 26, 32]. Historical contributions include the foundational work of Karger

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5710

Email addresses: ayman_ramadan@science.tanta.edu.eg (A. Elsharkawy) hkelsayied1989@yahoo.com
(H. K. Elsayied), aya30899252@science.tanta.edu.eg (A. Refaat)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 2 of 18

and Novak in 1978, introducing Frenet frames and invariants for ruled surfaces [27], and
Pottmann et. al. exploration of rational ruled surfaces and their offsets in 1996 [34].
Later, Peternel et al. addressed computational aspects of ruled surfaces in 1999 [33],
while recent research has extended these investigations to the differential geometry of
ruled surfaces in Minkowski space [1, 4, 9, 28, 30, 35, 36, 39].

The quasi-frame has emerged as an alternative to the Frenet frame for studying
curves and surfaces. Defined by a fixed projection vector and the angle between the
principal normal and the quasi-normal vector field, the quasi-frame simplifies computa-
tions compared to the Frenet and Bishop frames. This simplicity has made the quasi-
frame a valuable tool in exploring geometrical properties and applications in Euclidean,
Minkowski, and Galilean spaces [13, 14, 18, 22, 25]. Furthermore, variants of quasi-
frames, such as equiform and modified frames, have been utilized in diverse contexts
[10–12, 15, 17, 19–21, 24, 37, 38].

This paper is organized as follows. In Section 2, we provide fundamental definitions
and concepts used throughout the paper. Section 3 introduces three new types of ruled
surfaces based on the quasi-frame: QRT-surfaces, QRN-surfaces, and QRB-surfaces. For
each surface type, we discuss their fundamental properties and provide a detailed analysis.
Finally, we present two illustrative examples in Section 4 to validate the theoretical results
and demonstrate their practical relevance.

2. Preliminaries

Let E3 be an Euclidean 3-space equipped with the metric <,> given by

<,>= du2 + dv2 + dw2,

where (u, v, w) is a coordinate system of E3. For a space curve α(s) : (a, b) ∈ I → R3

represented by its arc-length s let {tq(s), nq(s), bq(s)} be the Quasi frame along α(s) in
which tq(s), nq(s) and bq(s) are the Quasi-tangent, Quasi-normal and Quasi-binormal
vectors, respectively, given in [22] by

tq(s) =
α′(s)

∥ α′(s) ∥
, nq(s) =

tq ×M

∥ tq ×M ∥
, bq(s) = tq × nq, (2.1)

where ′ is the derivative with respect to s and M is the projection vector which we could
choose M = (0, 1, 0), (1, 0, 0) or (0, 0, 1). The quasi-frame becomes singular in all cases
where T and M are parallel and in these cases we change the projection. The Quasi
formulae are given in [25] by

d

ds

 tq(s)
nq(s)
bq(s)

 =

 0 κ1 κ2
−κ1 0 κ3
−κ2 −κ3 0

 tq(s)
nq(s)
bq(s)

 , (2.2)

where the functions κ1, κ2 and κ3 are the first, second, and third Quasi-curvatures of
the curve, respectively, given by

κ1 = κ(s) cos(ϕ), (2.3)



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 3 of 18

κ2 = − κ(s) sin(ϕ), (2.4)

κ3 =
dϕ

ds
+ τ(s), (2.5)

where κ(s) and τ(s) are the Frenet curvature and Frenet torsion, respectively [13, 14, 22,
25].
If ϕ is the angle between the Frenet normal n and the Quasi normal nq and the relation
between the Quasi frame and usual orthonormal Frenet frame {t, n, b} given by

tq(s) = t(s), (2.6)
nq(s) = cos(ϕ)n(s) + sin(ϕ)b(s), (2.7)
bq(s) =− sin(ϕ)n(s) + cos(ϕ)b(s). (2.8)

A ruled surface W can be defined as a surface formed by the movement of a line L in
space. Suppose α(s) represents a regular curve in Euclidean 3−space and Y (s) represents
the direction vector of the line L. The parametric representation of the ruled surface W
can be given by

W (s, u) = α(s) + uY (s),

where α(s) denotes the base curve [2]. The striction line and the distribution parameter
of the ruled surface W can be given respectively as

α∗(s) = α(s) +
< tq(s), Y

′(s) >

∥ Y ′(s) ∥2
Y (s), (2.9)

and
λ(s) =

det [ tq(s), Y (s), Y ′(s)]

∥ Y ′(s) ∥2
, (2.10)

where tq(s) is unit tangent vector field of the base curve α(s). The ruled surface W is
developable if and only if λ(s) = 0. If ∥ Y ′(s) ∥= 0, then the ruled surface does not have
any striction curve. In this case, the ruled surface is cylindrical. Thus, the base curve
can be taken as a striction curve [3].

The standard unit normal vector field N on a surface W can be defined by

N =
Ws ×Wu

∥ Ws ×Wu ∥
, (2.11)

where Ws and Wu are partial derivatives of W (s, u) with respect to s and u. The 1st

F.F, the 2nd F.F and 3rd F.F of the surface W (s, u) are given, respectively, by

I =E(ds)2 + 2F ds du+G(du)2, (2.12)

II =L(ds)2 + 2M ds du+N(du)2, (2.13)

III =e(ds)2 + 2f ds du+ g(du)2, (2.14)



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 4 of 18

where
E =< Ws,Ws >,F =< Ws,Wu >,G =< Wu,Wu >,

L =< Wss, N >,M =< Wsu, N >,N =< Wuu, N >,

e =< Ns, Ns >, f =< Ns, Nu > and G =< Nu, Nu > .

Also, the Gaussian curvature K and the Mean curvature H are given, respectively,
by

K =
LN −M2

EG− F 2
, H =

EN − 2MF +GL

2(EG− F 2)
. (2.15)

The G− curvature κg, the N− curvature κn and the G− torsion τg which associate
the curve α(s) on the surface W can be computed as follows

κg =< N(s)× tq(s), t
′
q(s) >, κn =< N(s), α′′(s) >, τg =< N ×N ′, t′q(s) > (2.16)

where N represents the unit normal vector of the surface along the curve α(s) and tq
denotes the unit tangent vector of α(s).

Definition 2.1. [16] Let α(s) be a regular curve lying on a surface W (s, u).
(a) The curve α(s) is said to be geodesic curve if only if G− curvature vanishes.
(b) The curve α(s) is said to be an asymptotic line if and only if N− curvature vanishes.
(c) The curve α(s) is said to be a principal line if and only if G− torsion vanishes.

Definition 2.2. (a) A regular surface is flat (developable) if and only if its Gaussian
curvature vanishes identically.
(b) A regular surface is a minimal surface if and only if the Mean curvature vanishes
identically.

3. Main Result

This section has three subsections that introduce three different types of ruled
surfaces according to the Quasi frame called QRT-surfaces, QRN-surfaces, and QRB-
surfaces, respectively. Also, discuss their fundamental properties.

3.1. Quasi-Tangent Ruled Surfaces According to the Quasi frame

In this section, we establish the definition of ruled surfaces that arise from a regular
curve (referred to as the base curve) and are generated by the Quasi tangent vector tq,
which represents the direction vector. Additionally, we explore the fundamental proper-
ties associated with this particular type of ruled surface.

Definition 3.1. Let α(s) be a regular curve with Quasi frame {tq, nq, bq}, then the para-
metric representations of the QTR-Surface W T (s, u) with the ruling u given by

W T (s, u) = α(s) + u tq(s). (3.1)



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 5 of 18

Theorem 3.1. The striction curve of the Quasi-Tangent Ruled Surface is also a base
function of the surface.

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s) from Equation (2.9) the
striction curve defined by

α∗(s) = α(s) +
< tq(s), t

′
q(s) >

∥ t′q(s) ∥2
tq.

By the Equations in (2.2)

α∗(s) = α(s) +
< tq(s), (κ1 nq(s) + κ2 bq(s)) >

∥ t′q(s) ∥2
tq,

or
α∗(s) = α(s) +

κ1 < tq(s), nq(s) > + κ2 < tq(s), bq(s) >

∥ t′q(s) ∥2
tq,

Therefore,
α∗(s) = α(s).

Theorem 3.2. The first fundamental form of the Quasi-Tangent Ruled Surface W T (s, u)
is given by

I = 1 + u2 (κ1
2 + κ2

2)(ds)2 + 2 dsdu+ (du)2.

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s). Considering Equations
(2.2) and (2.6), the first partial derivatives of the QTR-surface with respect to s and u
are given by

W T
s = tq(s) + u ( κ1 nq(s) + κ2 bq(s)), (3.2)

W T
u = tq(s). (3.3)

From Equations (2.12),(3.2) and (3.3) the coefficients of the 1st F.F are given by

E = < W T
s,W

T
s > = 1 + u2 (κ1

2 + κ2
2),

F = < W T
s,W

T
u > = 1, (3.4)

G = < W T
u,W

T
u > = 1.

Theorem 3.3. The second fundamental form of the Quasi-Tangent Ruled Surface W T (s, u)
is given by

II = −
u
(
κ2 (κ2 κ3 − κ1

′) + κ1 κ2
′ + κ1

2 κ3
)

√
κ12 + κ22

(ds)2.



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 6 of 18

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s). The cross product of
Equations (3.2) and (3.3) given by

W T
s ×W T

u = u (κ2 nq(s)− κ1 bq(s)), (3.5)

taking the norm we get

∥ W T
s ×W T

u ∥= u2(κ1
2 + κ2

2). (3.6)

From Equations (2.11), (3.5) and (3.6), the unit normal vector can be defined by

NT =
W T

s ×W T
u

∥ W T
s ×W T

u ∥
=

u ( κ2 nq(s) − κ1 bq(s) )√
E − 1

, (3.7)

where E = 1+ u2 (κ1
2 + κ2

2). Considering Equations (2.2),(3.2) and (3.3), the second
partial derivatives of the QTR-surface with respect to s and u are given by

W T
ss = −u (κ1

2 + κ2
2)tq(s) + (κ1 + u(−κ2κ3 + κ1

′)nq(s)

+(κ2 + u(−κ1κ3 + κ2
′))bq(s), (3.8)

W T
su = κ1 nq(s) + κ2 bq(s), W T

uu = 0.

From Equations (2.13), and (3.8) the coefficients of the 2nd F.F are given by

L =< W T
ss, N >=

−u2
(
κ2 (κ2 κ3 − κ1

′) + κ1 κ2
′ + κ1

2 κ3
)

√
E − 1

,

M =< W T
su, N >= 0, (3.9)

N =< W T
uu, N >= 0.

Theorem 3.4. The third fundamental form of the Quasi-Tangent Ruled Surface W T (s, u)
is given by

III =

(
κ2 (κ2κ3 − κ1

′) + κ1κ2
′ + κ1

2κ3
)2

(κ12 + κ22)
2 (ds)2.

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s). The first partial
derivatives of Equation (3.7) with respect to s and u given by

(NT )s =
κ1
(
κ2 (κ2κ3 − κ1

′) + κ1κ2
′ + κ1

2κ3
)

(κ12 + κ22)
3/2

nq(s)

+
κ2
(
κ2 (κ2κ3 − κ1

′) + κ1κ2
′ + κ1

2κ3
)

(κ12 + κ22)
3/2

bq(s), (3.10)



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 7 of 18

and
(NT )u = 0. (3.11)

From Equations (2.14), (3.10) and (3.11) the coefficients of 3rd F.F are given by

e = < NT
s, N

T
s > =

(
κ2 (κ2κ3 − κ1

′) + κ1κ2
′ + κ1

2κ3
)2

(κ12 + κ22)
2 =

L√
E − 1

,

f = < NT
s, N

T
u > = 0,

g = < NT
u, N

T
u > = 0,

where E = 1 + u2 (κ1
2 + κ2

2).

Theorem 3.5. The Gaussian curvature K and the Mean curvature H of the Quasi-
Tangent Ruled Surface W T (s, u) are given, respectively, by

K = 0, H =
κ2 (κ1

′ − κ2κ3) + κ1 (−κ2
′)− κ1

2κ3

2u (κ12 + κ22)
3/2

.

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s). From Equations (2.15),
(3.4) and (3.9) we deduce the result.

Theorem 3.6. The geodesic curvature κg, the normal curvature κn and the geodesic
torsion τg which associate the base curve on the Quasi-Tangent Ruled Surface W T (s, u)
are given, respectively, by

κg = −
√
κ12 + κ22, κn = 0, τg = 0.

Proof. Let W T (s, u) be a QTR-Surface with base curve α(s). From Equations (2.2)
and (2.16) we deduce the result.

Corollary 3.1. Let α(s) be a regular curve lying on a surface W T (s, u).
(a) The curve α(s) is said to be a geodesic curve if only if α(s) is a straight line.
(b) The curve α(s) is always asymptotic line.
(c) The curve α(s) is a always principal line.

Corollary 3.2. (a) The QTR-surface is always flat (developable) surface.
(b) For non-straight lines the QTR-surface is a minimal surface if and only if

κ3 =
−κ21

d
du(

κ2
κ1
)

κ21 + κ22
.



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 8 of 18

3.2. Quasi-Normal Ruled Surfaces According to the Quasi frame

In this section, we outline the characteristics of QNR-Surfaces. These surfaces are
formed by a regular curve (known as the base curve) and are generated using the quasi-
normal vector Nq, which represents the direction vector. Furthermore, we will examine
the fundamental properties that are inherent to this type of ruled surface.

Definition 3.2. Let η(s) be a regular curve with quasi frame {tq, nq, bq}, then the para-
metric representations of the QNR-Surface WN (s, u) with the ruling u given by

WN (s, u) = η(s) + u nq(s). (3.12)

The theories presented in this section are built upon the same foundational evidence
as those discussed in the first section.

Theorem 3.7. The striction curve of the Quasi-Normal Ruled Surface is given by

η∗(s) = η(s)− κ1
κ12 + κ32

nq

Theorem 3.8. The first fundamental form of the Quasi-Normal Ruled Surface W T (s, u)
is given by

I = (u2κ3
2 + (1− u κ1

2))(ds)2 + (du)2,

where

E = < WN
s,W

N
s > =u2κ3

2 + (1− u κ1
2),

F = < WN
s,W

N
u > =0,

G = < WN
u,W

N
u > =1.

Theorem 3.9. The second fundamental form of the Quasi-Normal Ruled Surface WN (s, u)
is given by

II = (
κ2
(
u2κ1

2 + u2κ3
2 − 2uκ1 + 1

)
+ u (uκ3κ1

′ + κ3
′(1− uκ1))√

u2κ12 + u2κ32 − 2uκ1 + 1
)(ds)2

+
2κ3√

u2κ12 + u2κ32 − 2uκ1 + 1
ds du,

where

L = < WN
ss, N >=

κ2
(
u2κ1

2 + u2κ3
2 − 2uκ1 + 1

)
+ u (uκ3κ1

′ + κ3
′(1− uκ1))√

u2κ12 + u2κ32 − 2uκ1 + 1
,

M = < WN
su, N >=

κ3√
u2κ12 + u2κ32 − 2uκ1 + 1

,

N = < WN
uu, N >= 0.



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 9 of 18

Theorem 3.10. The third fundamental form of the Quasi-Normal Ruled Surface WN (s, u)
is given by

III = e(ds)2 + 2f dsdu+ g(du)2,

where

e =< NN
s, N

N
s >=

κ23
(
u4κ1

′2 + (uκ1 − 1)2
)
− 2u3κ3κ1

′κ3
′(uκ1 − 1)

(u2κ32 + (1− uκ1)2)
2

+
2uκ2

(
u2κ23 + (uκ1 − 1)2

)
(uκ3κ1

′ + κ3
′(1− uκ1))

(u2κ32 + (1− uκ1)2)
2

+
κ22
(
u2κ23 + (uκ1 − 1)2

)2
+ u2κ3

′2(uκ1 − 1)2 + u2κ43

(u2κ32 + (1− uκ1)2)
2 =

L2

E
+ M2,

f =< NN
s, N

N
u >=

κ3
(
κ2
(
u2κ21 + u2κ23 − 2uκ1 + 1

)
+ u (uκ3κ1

′ + κ3
′(1− uκ1))

)
(u2κ32 + (1− uκ1)2)

2 =
LM

E
,

g =< NN
u, N

N
u >=

κ23
(u2κ32 + (1− uκ1)2)

2 =
M2

E
.

Theorem 3.11. The Gaussian curvature K and the Mean curvature H of the Quasi-
Normal Ruled Surface WN (s, u) are given, respectively, by

K =
−κ3

2

(u2κ32 + (1− uκ1)2)
2 = −M2

E
,

H =
κ2
(
u2κ23 + (uκ1 − 1)2

)
+ u (uκ3κ1

′ + κ3
′(1− uκ1))

2
(
u2κ23 + (uκ1 − 1)2

)3/2 =
L

2E
.

Theorem 3.12. the geodesic curvature κg, the normal curvature κn and the geodesic
torsion τg which associate the base curve on the Quasi-Normal Ruled Surface WN (s, u)
are given, respectively, by

κg =
κ1 − uκ21√

u2κ23 + (uκ1 − 1)2
,

κn =
κ2(1− uκ1)√

u2κ23 + (1− uκ1)2
,

τg =
−κ1

(
κ2
(
u2κ23 + 1

)
+ u (κ3

′ + uκ3κ1
′)
)
− u2κ31κ2 + uκ21 (2κ2 + uκ3

′) + uκ2κ
2
3

u2κ23 + (uκ1 − 1)2
.

Corollary 3.3. Let η(s) be a regular curve lying on a surface WN (s, u).
(a) The curve η(s) is a geodesic curve if only if κ1 = 0 or κ1 =

1
u .



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 10 of 18

(b) The curve η(s) is an asymptotic line if only if κ2 = 0 or κ1 =
1
u .

(c) The curve η(s) is a principal line if and only if

κ2(s) =
uκ1(s)

(
dκ3(s)

dt + uκ3(s)
dκ1(s)

dt

)
− u2κ1(s)

2 dκ3(s)
dt

−κ1(s)
(
u2κ23(s) + 1

)
− u2κ1(s)3 + 2uκ21(s) + uκ23(s)

.

Corollary 3.4. (a) The QNR-surface is a flat (developable)surface if and only if κ3 = 0.
(b) A QNR-surface is a minimal surface if and only if

κ2(s) = −
u
(
uκ3(s)

dκ1(s)
dt + dκ3(s)

dt (1− uκ1(s))
)

u2κ1(s)2 + u2κ3(s)2 − 2uκ1(s) + 1
.

3.3. Quasi-Binormal Ruled Surfaces According to the Quasi frame

In this section, we define the QBR-Surfaces that are created by combining a regular
curve (referred to as the base curve) with the Quasi binormal vector bq, which determines
the direction. Additionally, we explore and discuss the fundamental properties associated
with this particular type of ruled surface.

Definition 3.3. Let γ(s) be a regular curve with Quasi frame {tq, nq, bq}. Then the
parametric representations of the QBR-Surface WB(s, u) with the ruling u is given by

WB(s, u) = γ(s) + u bq(s). (3.13)

The theories presented in this section are built upon the same foundational evidence
as those discussed in the first section.

Theorem 3.13. The striction curve of the Quasi-Binormal Ruled Surface is given by

γ∗(s) = γ(s)− κ2
κ22 + κ32

bq.

Theorem 3.14. The first fundamental form of the Quasi-Binormal Ruled Surface WB(s, u)
is given by

I = (u2κ3
2 + (1− u κ2

2))(ds)2 + (du)2,

Theorem 3.15. The second fundamental form of the Quasi-Binormal Ruled Surface
WB(s, u) is given by

II = (
−κ1

(
u2κ3

2 + (1− u κ2
2
)
+ u (uκ3κ2

′ + κ3
′(1− uκ2))√

u2κ32 + (1− u κ2)2
)(ds)2

+
2κ3√

u2κ32 + (1− u κ2)2
ds du,



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 11 of 18

Theorem 3.16. The third fundamental form of the Quasi-Binormal Ruled Surface WB(s, u)
is given by

III = e(ds)2 + 2f dsdu+ g(du)2.

where

e =< NN
s, N

N
s >=

κ23
(
u4κ2

′2 + (1− uκ2)
2
)
+ 2u3κ3κ2

′κ3
′(1− uκ2)

(u2κ32 + (1− uκ2)2)
2

−
2uκ1

(
u2κ23 + (1− uκ2)

2
)
(uκ3κ2

′ + κ3
′(1− uκ2))

(u2κ32 + (1− uκ2)2)
2

+
κ21
(
u2κ23 + (1− uκ2)

2
)2

+ u2κ3
′2(1− uκ2)

2 + u2κ43

(u2κ32 + (1− uκ2)2)
2 =

L2

E
+ M2,

f =< NN
s, N

N
u >=

κ3
(
κ1
(
u2κ23 + (1− uκ2)

2
)
+ u (uκ3κ2

′ + κ3
′(1− uκ2))

)
(u2κ32 + (1− uκ2)2)

2 =
LM

E
,

g =< NN
u, N

N
u >=

κ23
(u2κ32 + (1− uκ2)2)

2 =
M2

E
.

Theorem 3.17. The Gaussian curvature K and the Mean curvature H of the Quasi-
Binormal Ruled Surface WB(s, u) are given, respectively, by

K = −M2

E
=

−κ3
2

(u2κ32 + (1− uκ2)2)
2 ,

H =
L

2E
=

−κ1
(
u2κ23 + (1− uκ2)

2
)
+ u (uκ3κ2

′ + κ3
′(1− uκ2))

2
(
u2κ23 + (1− uκ2)2

)3/2 .

Theorem 3.18. the geodesic curvature κg, the normal curvature κn and the geodesic
torsion τg which associate the base curve on the Quasi-Binormal Ruled Surface WB(s, u)
are given, respectively, by

κg =
κ2(1− uκ2)√

u2κ23 + (1− uκ2)2
,

κn =
−κ1(1− uκ2)√
u2κ23 + (1− uκ2)2

,

τg =
−κ2

(
κ1
(
u2κ23 + 1

)
+ u (κ3

′ + uκ3κ2
′)
)
− u2κ32κ1 + uκ22 (2κ1 + uκ3

′) + uκ1κ
2
3

u2κ23 + (1− uκ2)2
.



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 12 of 18

Corollary 3.5. Let γ(s) be a regular curve lying on a surface WN (s, u).
(a) The curve γ(s) is a geodesic curve if only if κ2 = 0 or κ2 =

1
u .

(b) The curve γ(s) is an asymptotic line if only if κ1 = 0 or κ2 =
1
u .

(c) The curve γ(s) is a principal line if and only if

κ1(s) = −
uκ2(s)

(
−κ2(s) + uκ2(s)

2 + uκ3(s)
2
)
(κ′3(s)(uκ2(s)− 1)− uκ3(s)κ

′
2(s))

uκ2(s)− 1

.

Corollary 3.6. (a) The QBR-surface is a flat (developable)surface if and only if κ3 = 0.
(b) A QBR-surface is a minimal surface if and only if

κ1(s) = −
u
(
uκ3(s)

dκ2(s)
ds + dκ3(s)

ds (1− uκ2(s))
)

−u2κ3(s)2 − (uκ2(s)− 1)2
.

Example 3.1. Let ζ(s) be a general helix curve given by the parametrization

ζ(s) =

(
4 cos

(s
5

)
, 4 sin

(s
5

)
,
3s

5

)
.

By Equation (2.1) where choose M = (0, 1, 0) the Quasi-frame obtained by

t =

(
−4

5
sin
(s
5

)
,
4

5
cos
(s
5

)
,
3

5

)
,

nq =

(
0,

3√
8 cos

(
2s
5

)
+ 17

,−
4 cos

(
s
5

)√
8 cos

(
2s
5

)
+ 17

)
,

bq =

( −8 cos
(
2s
5

)
− 17

5
√
8 cos

(
2s
5

)
+ 17

,−
8 sin

(
2s
5

)
5
√
8 cos

(
2s
5

)
+ 17

,−
12 sin

(
s
5

)
5
√
8 cos

(
2s
5

)
+ 17

)
.

Also by Equation (2.3) the Quasi-Curvatures along ζ(s) are obtained by

κ1 =−
12 sin

(
s
5

)
25
√
8 cos

(
2s
5

)
+ 17

,

κ2 =
4 cos

(
s
5

)
5
√
8 cos

(
2s
5

)
+ 17

,

κ3 =−
48 sin2

(
s
5

)
25
(
8 cos

(
2s
5

)
+ 17

) .



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 13 of 18

The QTR-Surface, the QNR-Surface, and the QBR-Surface are, respectively, given in
Figure 3.1 by

W T (s, u) =

(
4 cos

(s
5

)
− 4

5
u sin

(s
5

)
,
4

5
u cos

(s
5

)
+ 4 sin

(s
5

)
,

3s

5
+

3u

5

)
,

WN (s, u) =

(
4 cos

(s
5

)
,

3u√
8 cos

(
2s
5

)
+ 17

+ 4 sin
(s
5

)
,
3s

5
−

4u cos
(
s
5

)√
8 cos

(
2s
5

)
+ 17

)
,

WB(s, u) =

(
4 cos

s

5
− 1

5
u

√
8 cos

2s

5
+ 17, 4 sin

(s
5

)
−

8u sin
(
2s
5

)
5
√

8 cos
(
2s
5

)
+ 17

,
3

5
(s−

4u sin
(
s
5

)√
8 cos

(
2s
5

)
+ 17

)

)
.

Figure 1: Ruled Surfaces generated by a general helix

Example 3.2. Let ξ(s) be a regular curve parameterized by

ξ =

(
3

2
cos
(s
2

)
+

1

6
cos

(
3s

2

)
,
3

2
sin
(s
2

)
+

1

6
sin

(
3s

2

)
,
√
3 cos

(s
2

))
.



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 14 of 18

By Equation (2.1) where choose M = (0, 0, 1) the Quasi-frame obtained by

t =

(
1

4

(
−3 sin

(s
2

)
− sin

(
3s

2

))
, cos3

(s
2

)
,−1

2

√
3 sin

(s
2

))
,

nq =

(
2
√
2 cos3

(
s
2

)√
3 cos(s) + 5

,
3 sin

(
s
2

)
+ sin

(
3s
2

)√
6 cos(s) + 10

, 0

)
,

bq =

(
sin2

(
s
2

)
(cos(s) + 2)√

2 cos(s) + 10
3

,−sin(s)(cos(s) + 1)

2
√
2 cos(s) + 10

3

,−
√
3 cos(s) + 5

2
√
2

)
.

Also by Equation (2.3) the Quasi-Curvatures along ξ(s) are obtained by

κ1 =−
3 cos2

(
s
2

)√
6 cos(s) + 10

,

κ2 =
cos
(
s
2

)√
2 cos(s) + 10

3

,

κ3 =−
3
√
3 sin2(s) csc

(
s
2

)
12 cos(s) + 20

.

The QTR-Surface, the QNR-Surface, and the QBR-Surface are, respectively, given in
Figure 3.2 by

W T (s, u) =

(
1

12

(
−3u

(
3 sin

(s
2

)
+ sin

(
3s

2

))
+ 18 cos

(s
2

)
+ 2 cos

(
3s

2

))
,

1

12

(
9u cos

(s
2

)
+ 3u cos

(
3s

2

)
+ 2

(
9 sin

(s
2

)
+ sin

(
3s

2

)))
,

− 1

2

√
3
(
u sin

(s
2

)
− 2 cos

(s
2

)))
,

WN (s, u) =

(
1

6

(
12
√
2u cos3

(
s
2

)√
3 cos(s) + 5

+ 9 cos
(s
2

)
+ cos

(
3s

2

))
,

u
(
3 sin

(
s
2

)
+ sin

(
3s
2

))√
6 cos(s) + 10

+
3

2
sin
(s
2

)
+

1

6
sin

(
3s

2

)
,
√
3 cos

(s
2

))
,

WB(s, u) =

(
u sin2

(
s
2

)
(cos(s) + 2)√

2 cos(s) + 10
3

+
3

2
cos
(s
2

)
+

1

6
cos

(
3s

2

)
,

− u sin(s)(cos(s) + 1)

2
√

2 cos(s) + 10
3

+
3

2
sin
(s
2

)
+

1

6
sin

(
3s

2

)
,



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 15 of 18

√
3 cos

(s
2

)
−

u
√
3 cos(s) + 5

2
√
2

)
.

Figure 2: Ruled Surfaces generated by ξ(s)

Abbreviation Full Form
QRT Quasi-Tangent Ruled Surface
QRN Quasi-Normal Ruled Surface
QRB Quasi-Binormal Ruled Surface
F.F Fundamental Form

G- curvature geodesic curvature
N- curvature normal curvature
G- torsion geodesic torsion

Table 1: List of Abbreviations

4. Conclusion

This paper introduced three distinct types of ruled surfaces based on the quasi-
frame: quasi-tangent (QRT), quasi-normal (QRN), and quasi-binormal (QRB) surfaces.
These surfaces are generated by the motion of a straight line (ruling) along a base curve,
with the direction of the ruling determined by the quasi-tangent, quasi-normal, and quasi-
binormal vectors, respectively. We have thoroughly investigated their first, second, and
third fundamental forms, as well as their Gaussian and mean curvatures. In addition,
we have also explored the geodesic curvature, normal curvature, and geodesic torsion
associated with the base curve for each type of surface. Furthermore, we have established



A. Elsharkawy, H. K. Elsayied, A. Refaat / Eur. J. Pure Appl. Math, 18 (1) (2025), 5710 16 of 18

the conditions under which the base curve can be classified as a geodesic, an asymptotic
line, or a principal line on each type of surface. We have also derived the conditions
for these surfaces to be developable or minimal. To illustrate the theoretical results, we
provided two detailed examples. All the results derived in this paper can be specialized
to the Frenet frame by setting κ2 = 0. This connection between the quasi-frame and
the Frenet frame allows for a broader interpretation of the results and provides a bridge
between different geometric frameworks.

Future research could explore the application of quasi-frames in higher-dimensional
spaces, such as Minkowski and Galilean spaces. Additionally, the study of quasi-ruled
surfaces in the context of computer-aided design and architecture could lead to new
practical applications. Further investigations into the singularities of the quasi-frame
and their implications for geometric modeling are also warranted.

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	Introduction
	Preliminaries
	Main Result
	Quasi-Tangent Ruled Surfaces According to the Quasi frame 
	 Quasi-Normal Ruled Surfaces According to the Quasi frame 
	 Quasi-Binormal Ruled Surfaces According to the Quasi frame 

	Conclusion