Fixed-axis spacelike ruled surfaces and their evolute offsets

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.

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 .

[Figure omitted. See PDF.]

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.

[Figure omitted. See PDF.]

(2) Let , , , and . The fixed-axis , and its evolute offset appear in Fig 3.

[Figure omitted. See PDF.]

(3) Since , then is an non-skew or tangential developable (). For , , , and , the fixed-axis , and its evolute offset are located in Fig 4.

[Figure omitted. See PDF.]

(4) Since , then is a binormal (). For , , , and , the fixed-axis , and its evolute offset are located in Fig 5.

[Figure omitted. See PDF.]

(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 .

[Figure omitted. See PDF.]

5 Conclusion

In this study, we examined the evolute offsets of fixed-axis skew -ruled surface in , by leveraging the symmetry of evolute curves. Through our analysis, we formulated the parameterization of evolute offsets for both skew and non-skew ruled surfaces while also characterizing their geometric properties. We established key structural relationships between the curvature-axis functions and the associated height functions, leading to a deeper understanding of the curvature behavior of skew -ruled surface. In particular, we derived conditions under which a skew -ruled surface becomes a fixed-axis -ruled surface and formulated a theorem analogous to the classical theory of evolute curves, providing a framework for further studies on these ruled surfaces.

Our results highlight that a surface is a fixed-axis -ruled surface if and only if the spherical curvature function J remains constant and the function is also constant. These findings contribute to the broader study of ruled surfaces in Lorentzian geometry and offer a foundation for potential applications in kinematics and differential geometry. Future research may explore generalizations of these results in higher-dimensional spaces or in relation to other classes of ruled surfaces.

These results are expected to be useful in the field of . In future work, we plan to further investigate the classification of singularities as outlined in [27, 28].

References

1. 1. Gugenheimer HW. Differential geometry. New York: McGraw-Hill; 1956, pp. 162–9.

2. 2. Bottema O, Roth B. Theoretical kinematics. New York: North-Holland Press; 1979.

3. 3. Karger A, Novak J. Space kinematics and Lie groups. New York: Gordon and Breach Science Publishers; 1985.

4. 4. Papaioannou SG, Kiritsis D. An application of Bertrand curves and surfaces to CAD/CAM. Comput-Aided Design. 1985;17(8):348–52.

* View Article

* Google Scholar

5. 5. Schaaf JA, Ravani B. Geometric continuity of ruled surfaces. Comput Aided Geom Des. 1998;15(3):289–310.

* View Article

* Google Scholar

6. 6. Peternell M, Pottmann H, Ravani B. On the computational geometry of ruled surfaces. Comput-Aided Des. 1999;31(1):17–32.

* View Article

* Google Scholar

7. 7. Pottman H, Wallner J. Computational line geometry. Heidelberg: Springer; 2001.

8. 8. Ravani B, Ku TS. Bertrand offsets of ruled and developable surfaces. Comput-Aided Des. 1991;23(2):145–52.

* View Article

* Google Scholar

9. 9. Küçük A, Gürsoy O. On the invariants of Bertrand trajectory surface offsets. Appl Math Comput. 2004;151(3):763–73.

* View Article

* Google Scholar

10. 10. Kasap E, Kuruoğlu N. Integral invariants of the pairs of the Bertrand ruled surface. Bull Pure Appl Sci Sect E Math. 2002;21;37–44.

* View Article

* Google Scholar

11. 11. Kasap E, Kuruoğlu N. The Bertrand offsets of ruled surfaces. Acta Math Vietnam. 2006;31:39–48.

* View Article

* Google Scholar

12. 12. Kasap E, Yüce S, Kuruoğlu N. The involute-evolute offsets of ruled surfaces. Iranian J Sci Tech Trans. 2009;33:195–201.

* View Article

* Google Scholar

13. 13. Orbay K, Kasap E, Aydemir İ. Mannheim offsets of ruled surfaces. Math Probl Eng. 2009;2009(1).

* View Article

* Google Scholar

14. 14. Önder M, Uğurlu HH. Frenet frames and invariants of timelike ruled surfaces. Ain Shams Eng J. 2013;4(3):507–13.

* View Article

* Google Scholar

15. 15. Senturk GY, Yuce S. Properties of integral invariants of the involute-evolute offsets of ruled surfaces. Int J Pure Appl Math. 2015;102(4).

* View Article

* Google Scholar

16. 16. Yoon DW. On the evolute offsets of ruled surfaces in Euclidean 3-space. Inter J Pure App Maths. 2016;108(4):985–97.

* View Article

* Google Scholar

17. 17. Yoon DW. On the evolute offsets of ruled surfaces in Minkowski 3-space. Turk J Math. 2016;40:594–604.

* View Article

* Google Scholar

18. 18. Aldossary MT, Abdel-Baky RA. On the Blaschke approach of Bertrand offsets of spacelike ruled surfaces. AIMS Math. 2022;7(10):17843–58.

* View Article

* Google Scholar

19. 19. Alluhaibi N, Abdel-Baky RA, Naghi M. On the Bertrand offsets of timelike ruled surfaces in Minkowski 3-space. Symmetry. 2022;14(4):673.

* View Article

* Google Scholar

20. 20. Nazra SH, Abdel-Baky RA. Bertrand offsets of ruled surfaces with Blaschke frame in Euclidean 3-space. Axioms. 2023;12(7):649.

* View Article

* Google Scholar

21. 21. Li Y, Eren K, Ersoy S. On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space. AIMS Math. 2023;8(9):22256–73.

* View Article

* Google Scholar

22. 22. Baltić V, Eren K, Savić A, Ersoy S. Constant angle ruled surfaces with a pointwise 1-type Gauss map. Mathematics. 2024;12(12):1861.

* View Article

* Google Scholar

23. 23. Savić A, Eren K, Ersoy S, Baltić V. Alternative view of inextensible flows of curves and ruled surfaces via alternative frame. Mathematics. 2024;12(13):2015.

* View Article

* Google Scholar

24. 24. López R. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int Electron J Geom. 2014;7:44–107.

* View Article

* Google Scholar

25. 25. O’Neil B. Semi-Riemannian geometry, with applications to relativity. New York: Academic Press; 1983.

26. 26. Bruce JW, Giblin PJ. Curves and singularities. 2nd edn. Cambridge: Cambridge University Press; 1992.

27. 27. Almoneef AA, Abdel-Baky RA. Singularity properties of Spacelike circular surfaces. Symmetry. 2023;15(4):842.

* View Article

* Google Scholar

28. 28. Nazra SH, Abdel-Baky RA. Singularities of non-lightlike developable surfaces in Minkowski 3-space. Mediterr J Math. 2022;20(1).

* View Article

* Google Scholar

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

About the Authors:

Areej A. Almoneef

Contributed equally to this work with: Areej A. Almoneef, Rashad A. Abdel-Baky

Roles: Data curation, Formal analysis, Funding acquisition, Methodology, Resources, Writing – original draft, Writing – review & editing

E-mail: [email protected]

Affiliation: Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Saudi Arabia

ORICD: https://orcid.org/0000-0001-7041-3730

Rashad A. Abdel-Baky

Contributed equally to this work with: Areej A. Almoneef, Rashad A. Abdel-Baky

Roles: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – original draft

Affiliation: Department of Mathematics, Faculty of Science, University of Assiut, Assiut, Egypt

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References

1. Gugenheimer HW. Differential geometry. New York: McGraw-Hill; 1956, pp. 162–9.

2. Bottema O, Roth B. Theoretical kinematics. New York: North-Holland Press; 1979.

3. Karger A, Novak J. Space kinematics and Lie groups. New York: Gordon and Breach Science Publishers; 1985.

4. Papaioannou SG, Kiritsis D. An application of Bertrand curves and surfaces to CAD/CAM. Comput-Aided Design. 1985;17(8):348–52.

5. Schaaf JA, Ravani B. Geometric continuity of ruled surfaces. Comput Aided Geom Des. 1998;15(3):289–310.

6. Peternell M, Pottmann H, Ravani B. On the computational geometry of ruled surfaces. Comput-Aided Des. 1999;31(1):17–32.

7. Pottman H, Wallner J. Computational line geometry. Heidelberg: Springer; 2001.

8. Ravani B, Ku TS. Bertrand offsets of ruled and developable surfaces. Comput-Aided Des. 1991;23(2):145–52.

9. Küçük A, Gürsoy O. On the invariants of Bertrand trajectory surface offsets. Appl Math Comput. 2004;151(3):763–73.

10. Kasap E, Kuruoğlu N. Integral invariants of the pairs of the Bertrand ruled surface. Bull Pure Appl Sci Sect E Math. 2002;21;37–44.

11. Kasap E, Kuruoğlu N. The Bertrand offsets of ruled surfaces. Acta Math Vietnam. 2006;31:39–48.

12. Kasap E, Yüce S, Kuruoğlu N. The involute-evolute offsets of ruled surfaces. Iranian J Sci Tech Trans. 2009;33:195–201.

13. Orbay K, Kasap E, Aydemir İ. Mannheim offsets of ruled surfaces. Math Probl Eng. 2009;2009(1).

14. Önder M, Uğurlu HH. Frenet frames and invariants of timelike ruled surfaces. Ain Shams Eng J. 2013;4(3):507–13.

15. Senturk GY, Yuce S. Properties of integral invariants of the involute-evolute offsets of ruled surfaces. Int J Pure Appl Math. 2015;102(4).

16. Yoon DW. On the evolute offsets of ruled surfaces in Euclidean 3-space. Inter J Pure App Maths. 2016;108(4):985–97.

17. Yoon DW. On the evolute offsets of ruled surfaces in Minkowski 3-space. Turk J Math. 2016;40:594–604.

18. Aldossary MT, Abdel-Baky RA. On the Blaschke approach of Bertrand offsets of spacelike ruled surfaces. AIMS Math. 2022;7(10):17843–58.

19. Alluhaibi N, Abdel-Baky RA, Naghi M. On the Bertrand offsets of timelike ruled surfaces in Minkowski 3-space. Symmetry. 2022;14(4):673.

20. Nazra SH, Abdel-Baky RA. Bertrand offsets of ruled surfaces with Blaschke frame in Euclidean 3-space. Axioms. 2023;12(7):649.

21. Li Y, Eren K, Ersoy S. On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space. AIMS Math. 2023;8(9):22256–73.

22. Baltić V, Eren K, Savić A, Ersoy S. Constant angle ruled surfaces with a pointwise 1-type Gauss map. Mathematics. 2024;12(12):1861.

23. Savić A, Eren K, Ersoy S, Baltić V. Alternative view of inextensible flows of curves and ruled surfaces via alternative frame. Mathematics. 2024;12(13):2015.

24. López R. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int Electron J Geom. 2014;7:44–107.

25. O’Neil B. Semi-Riemannian geometry, with applications to relativity. New York: Academic Press; 1983.

26. Bruce JW, Giblin PJ. Curves and singularities. 2nd edn. Cambridge: Cambridge University Press; 1992.

27. Almoneef AA, Abdel-Baky RA. Singularity properties of Spacelike circular surfaces. Symmetry. 2023;15(4):842.

28. Nazra SH, Abdel-Baky RA. Singularities of non-lightlike developable surfaces in Minkowski 3-space. Mediterr J Math. 2022;20(1).