Abstract
This study investigates fixed-axis spacelike ruled surfaces and their evolute offset counterparts within 
(Minkowski 3-space). The analysis utilizes the Blaschke frame associated with the striction curves of these surfaces. Spacelike ruled surfaces play a crucial role in various fields of both classical and modern physics. The research begins by introducing the fundamental concepts of fixed-axis spacelike ruled surfaces and defining a height function that establishes the necessary criteria for a ruled surface to be classified as a fixed-axis spacelike ruled surface. Subsequently, the study derives parameterization for both the fixed-axis spacelike ruled surfaces and their evolute offsets. Finally, several surface models are extended and visually represented through graphical illustrations.
Citation: Almoneef AA, Abdel-Baky RA (2025) Fixed-axis spacelike ruled surfaces and their evolute offsets. PLoS One 20(6):
e0325051.
https://doi.org/10.1371/journal.pone.0325051
Editor: Xuebo Zhang, Whale Wave Technology Inc., CHINA
Received: November 27, 2024; Accepted: May 5, 2025; Published: June 6, 2025
Copyright: © 2025 Almoneef, Abdel-Baky. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: This study was financially supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project in the form of a grant (PNURSP2025R337) received by AAA. No additional external funding was received for this study.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
In the context of movable frames, the most commonly used are the Frenet–Serret frame (
) for space curves and the Blaschke frame (
) for ruled surfaces. The Blaschke frame is determined by the velocity of the striction curve and the normal vector of the associated sphere, whereas the Frenet–Serret frame is defined by the velocity and acceleration of the curve. By differentiating these movable frames with respect to their own basis vectors, certain real-valued functions emerge. These functions are known as curvature and torsion in the case of the Frenet–Serret frame and as the Blaschke invariants for the Blaschke frame (see, for example, [1–3]).
In the realm of line geometry, the set of oriented lines embedded in a moving solid body primarily generates ruled surfaces. The geometry of ruled surfaces has been extensively applied in various fields, including Computer-Aided Manufacturing (
), Computer-Aided Geometric Design (
), geometric modeling, and motion analysis [3–7]. In recent years, the properties of ruled surfaces and their offset counterparts have been extensively studied in both Euclidean and non-Euclidean spaces. For example, Ravani and Ku [8] explored the theory of Bertrand curves for Bertrand ruled surface offsets using line geometry. They demonstrated that a ruled surface can possess an infinite number of Bertrand ruled surface offsets, similar to how a plane curve can have an infinite number of Bertrand mates. Building upon this work, Küçük and G ürsoy provided several examples of Bertrand offsets of trajectory ruled surfaces, analyzing their interrelations through projection domains and the corresponding spherical curve invariants [9]. In [10], Kasap and Kuruoğlu explored the relationships between the integral invariants of Bertrand ruled surfaces in Euclidean 3-space. In [11], they extended this research to the Bertrand offsets of ruled surfaces in 
. The involute-evolute offsets of the ruled surfaces were studied by Kasap et al. in [12]. Orbay et al. [13] introduced the study of Mannheim offsets for ruled surfaces, while Önder and Uğurlu investigated the invariants of Mannheim offsets for timelike (
) ruled surfaces and provided conditions for these surface offsets to be non-skew [14]. These offset surfaces are analyzed using the Blaschke frame, as defined in [8]. Based on the involute-evolute offsets of ruled surfaces in [12], Şentürk and Yüce computed the integral invariants of these offsets in relation to the geodesic 
[15]. Recently, Yoon examined evolute offsets of ruled surfaces in both Euclidean and Minkowski 3-spaces, considering stationary Gaussian and mean curvatures [16, 17]. There is a substantial body of literature on these topics, including various treatises such as [18–20].
Subsequently, to consolidate interdisciplinary papers, we wish to highlight some significant studies on ruled surfaces and surface families in various spaces [21–23].
To the best of our knowledge, there has been no prior work on the construction of evolute offsets for a fixed-axis skew 
-ruled surface in 
. This study aims to identify a set of invariants that describe the local shape of a fixed-axis 
-ruled surface and its evolute offsets. The conditions for two 
-ruled surfaces to be evolute offsets are developed, and the results are illustrated using computer-aided models. The findings presented in this paper offer valuable insight into surface theory, which could contribute to fields that require surface analysis.
2 Basic concepts
To meet the demands in the next sections, here, the crucial elements of the theory of curves in 
are briefly presented [1–4, 24, 25]. For vectors 
and 
, we know that
is named Lorentzian inner product. The cross product produces a vector given by
Since 
is an indefinite metric, recall that a vector 
can possess one of three causal natures; it can be 
if 
or 
, 
if 
and null or lightlike if 
and 
. The norm of 
is pointed by 
, then the hyperbolic and Lorentzian (de Sitter space) unit spheres are:
(1)and
(2)A 
line 
can be attended by a point 
and a 
unit vector 
on it, that is, 
. A parametric equation of 
is
(3)Thus, we set the moment 
with reference to a fixed origin point as
(4)where
(5)Wherefore, we can write that 
. Let 
be two 
lines assigned with 
and 
linearly independent. Then the distance 
among 
, 
is
(6)The Lorentzian distance in the customary sense will then be 
. The angle among 
, 
is specified as follows:
1) If they span a 
plane; there is a unique angle 
; 
such that
(7)2) If they span a 
plane, there is a unique angle 
; 
such that
(8)The spatial distance between 
, 
is located to be a relationship of real numbers.
(9)
2.1 Ruled surface
A ruled surface is a surface generated by a line 
that moves along a curve 
. The various positions of these lines are referred to as the generators of the surface. Such a surface has the parametric representation [1, 2 , 20, 21]:
(10)where 
; 
. In this setting, the curve 
is known as the striction curve, and v is the arc-length of 
(or 
). If 
is neither stationary nor null, and if 
is non-null, then the Blaschke frame for 
can be established as follows [20, 24]:
This provides a compact framework for analyzing the geometric properties of the ruled surface using the Blaschke frame. The Blaschke formula for the Blaschke frame is expressed as:
where 
represents the spherical curvature of 
. With respect to the Blaschke frame and the signs 
, 
, 
, the striction curve is defined as [20, 24]:
(11)Here, J(v), 
and 
are known as the curvature functions of 
. This formulation provides a systematic approach to describing the geometry of ruled surfaces in terms of their Blaschke frame components and curvature functions.
3 Main results
In this section, we explore and define the evolute offsets of a skew fixed-axis 
-ruled surface in 
, utilizing the symmetry of the evolute curves. We then provide the parameterization of the evolute offsets for both skew and non-skew fixed-axis 
-ruled surfaces. Furthermore, we examine the properties of these ruled surfaces and discuss a classification scheme for their various forms.
Based on the notations introduced in Section 2, we focus on a skew 
-ruled surface characterized by 
. From this, we derive the following results:
where
Then, the Blaschke formula is
(12)where 
is the Darboux vector, and
(13)The striction curve is defined by
(14)Therefore, a skew 
-ruled surface can be described as:
(15)J(v), 
and 
are the structure functions of 
; J(v) is the spherical curvature of 
, 
and 
is the distribution parameter of 
.
Definition 1. 
is a fixed-angle 
if its ruling has a fixed-angle with a definite line.
Definition 2. 
is a fixed-distance 
if its ruling has a fixed-distance with a definite line.
Definition 3. 
is a fixed-axis 
if its ruling has a fixed-spatial distance with a definite line.
Furthermore, the 
curvature-axis of 
is
(16)Let 
be the radius of curvature among 
and 
. Then
(17)Corollary 1. The curvature 
, the torsion 
, and J(v) of 
are
(18)Corollary 2. If 
, then 
is a Lorentzian small circle.
Proof. From Eq 18, we observe that 
, and 
, which indicates that 
is a Lorentzian small circle (assuming 
) ∎.
Definition 4. Let 
be a skew 
that satisfies Eq 15 in 
. A 
is an evolute offset of 
if there exists a bijection between their striction points such that the central normal of 
and the ruling of 
are colinear.
Let 
be an evolute offset of 
. Then,
(19)where
(20)Here f(v) is the distance function among the striction points of 
and 
[16, 17]. If 
, 
and 
are the Blaschke vectors of 
, and since 
at striction points, then,
(21)and
(22)In view of Eqs 17, (21), and (22) we reach
(23)where
If 
be the arc-length of 
, then 
. Therefore,
(24)where
(25)Corollary 3. 
, that is, 
is a Lorentzian small circle iff 
, that is, 
is a hyperbolic great circle.
3.1 Height functions
In matching with [26], a point 
will be 
curvature-axis of
; for all v such that 
, with 
. Here 
signalizes the p-th derivative of 
with reference to v. For the 1st curvature-axis 
of
, we locate 
, and 
. So, 
is at least a 
curvature-axis of 
. We now locate a high function 
, by 
. We let 
for any constant point 
. Then, we display the following:
Proposition 1. Via the last presuppositions, we occupancy:
i- w is fixed up to the 1st order iff 
,
, that is,
for 
and 
.
ii- w is fixed up to the 2nd order iff 
is 
curvature-axis of 
, that is,
iii- w is fixed up to the 3rd iff 
is 
curvature-axis of 
, that is,
iv- w is fixed up to the 4th order iff 
is 
curvature-axis of 
, that is,
Proof. i-Firstly, we derive
(26)Then,
(27)for real numbers c1, 
and 
, the consequence is apparent.
ii- The derivative of Eq 26 register that:
(28)By Eqs 26, and (28) we attain
iii- The derivative of Eq 28 is
Hence, we attain
iv- By the same pretexts, we can also control
The proof is complete ∎.
Via the Proposition 1, we conclude:
(a) The osculating circle (
) 
of 
is width by
which state that the 
must has link of at least 3rd order at 
iff J′′≠0.
(b) The curve 
and the 
has link at least 4-th order at 
iff J′=0, and J′′≠0.
In this track, by taking into examination the curvature-axes of 
, we can accomplish a sequence of curvature-axes 
, 
,..., 
. The proprietorships and the joint links among these curvature-axes are much amusing topics. For demand, it is uncomplicated to see that if 
, and J′=0, 
is locating at 
is fixed regarding to 
. In these circumstances, the curvature axis is fixed up to 2nd order, and 
is a constant angle 
.
Theorem 1. 
is a fixed-angle 
, that is, 
iff 
Since 
., from the Eqs 12, and (18), we reach to: 
. It is valuable to transfer the parameter v via 
, and let’s take 
. Then,
where a1, a2, a3 are constants fulfilling a2 = 0, and 
. It follows that
(29)for constants a2, a3 fulfilling 
, and 
. Let’s make
Therefore, we attain
Since 
, we realize that 
. By setting the upper sign, we receive
(30)Then,
(31)Let
(32)where 
and 
are differentiable functions of 
. Differentiating the last equation via v and appointing the Blaschke formulae, we possess
(33)From Eqs 14, and (33), one finds that:
(34)which signify we can manifest that
(35)
4 Evolute offsets of a fixed-axis 
SL-ruled surface
In this section, we conclude and inspect the evolute offsets of a constant-axis skew and non-skew 
-ruled surfaces. We then develop a theory analogous to the theory of evolute curves for these surfaces. To achieve this, we present the following theorem.
Theorem 2. Let 
be a skew 
-ruled surface as defined in Eq 15. Then 
is a fixed-axis 
-ruled surface iff (i) J = const., and (ii) 
.
Proof. The necessity of the conditions follows directly from Theorem 1. For the sufficiency, we proceed as follows: Without loss of generality, a constant Lorentzian frame 
can be used with the 
). The striction curve can be determined by
or in view of Eqs 31, we attain
(36)where 
is the distance along the 
axis (Figure 1). The stations of both points 
and 
determined on that of 
; the minimal distance 
is based on 
.
From Eqs 30-(32) and Eq 36 we acquire
(37)If 
, then Eq 27 leads to
(38)The proof is done ∎.
Since 
and 
are all constants, this exhibits that 
is a cylindrical helix with the 
-axis. Furthermore, for 
, from Eqs 18, and (38), we accomplish
(39)In view of Eqs 34, and (39), we fulfill
(40)Hence, from Eqs 15, (30), (39), and (40), we attain
(41)where
(42)Via Eqs 19, and (41) the surface 
is
(43)
4.1 Classification of 
M and 
M
In the following, we set 
. For specific values of 
, and 
, we consider the following:
(1) Let 
, 
, 
, and 
. The fixed-axis 
, and its evolute offset appear in Fig 2.
(2) Let 
, 
, 
, and 
. The fixed-axis 
, and its evolute offset appear in Fig 3.
(3) Since 
, then 
is an 
non-skew or tangential developable (
). For 
, 
, 
, and 
, the fixed-axis 
, and its evolute offset 
are located in Fig 4.
(4) Since 
, then 
is a 
binormal (
). For 
, 
, 
, and 
, the fixed-axis 
, and its evolute offset 
are located in Fig 5.
(5) If 
is a 
cone, then 
. From Eq 36, we conclude
(44)Then,
(45)which show that 
, and 
. Further, employing 
into the Eq 39 we deduce 
. Consequently, we acquire
(46)and
(47)The fixed-axis 
, and its evolute offset 
are arranged in Fig 6; where 
and 
.
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