Abstract
In this work, we investigate the synthesis problem of slant spacelike ruled surfaces and associated Bertrand offsets in (Minkowsk 3-space). We provide the parametric equation for a non-developable spacelike ruled surface by using the Blaschke frame . This results in the amplitude to control a family of curvature functions defining the domestic form of this . Therefore, we found the appropriate criteria to be slant . Thus, several new Bertrand offsets for slant are investigated and constructed.
1 Introduction
The fundamental principle of a directed line’s motion in connection with a solid body is referred to as the concept in spatial kinematics. This notion holds great importance in conventional differential geometry and has been the subject of extensive research by numerous scholars, as demonstrated by [1–7]. From a geometric perspective, the properties of and their offset surfaces have been analyzed in both Euclidean and non-Euclidean spaces. Bertrand curves were examined in the field of line-geometry by Ravani and Ku, revealing that can possess an infinite number of , similar to how a plane curve can possess an infinite number of mates [8]. Küçük and Gürsoy provided certain characterizations of related to the trajectory of by studying the relationships between the projection areas for the spherical curves of and their integral invariants [9]. Kasap and Kuruoğlu conducted an analysis of the integral invariants of the couple of in the Euclidean 3-space , as documented in [10]. By considering the orthonormal frame along striction curve of a ruled surface, Önder has defined slant ruled surfaces in the Euclidean 3-space [11]. Moreover, Kaya and Önder have studied the position vectors and some differential equation characterizations for slant ruled surfaces in the Euclidean 3-space [12–14]. They have also defined a new type of slant ruled surface as the Darboux slant ruled surface and characterized for this type of slant surfaces [15]. In [16], Önder introduced some characterizations for a non-null ruled surface to be a slant ruled surface in Minkowski 3-space . In their study, Kasap and Kuruoğlu investigated of in Minkowski 3-space , as documented in [17]. [18] demonstrated the involute–evolute offsets of . Orbay et al. began studying the Mannheim offsets of in [19]. Önder and Uğurlu conducted a study on the relationships between invariants of Mannheim offsets of . They also formulated many considerations for the development of these surface offsets [20, 21]. In view of the involute–evolute offsets of the ruled surface in [7], Şentürk and Yüce described the integral invariants of the involute–evolute offsets of s using the geodesic Frenet frame [22].
In recent times, Yoon has investigated the evolute offsets of in with a stationary Gaussian curvature and mean curvature [23]. A plethora of comprehensive treatises has been published on this subject, as demonstrated by the numerous written works, such as [24–27]. However, to the best of our knowledge, no prior work has focused on constructing of slant , utilizing the geometric attributes of the striction curve . Here, we intend to fill the gap in the existing literature.
In this paper, with the identification of slant curves, we treat the structure issue of the of a slant family in Minkowski 3-space . Therefore, we extend the parametrization of for any slant non-developable . Furthermore, we inquire into the ownerships of these surfaces and grant their distribution. Meanwhile, we extend some interpretative paradigms to display surfaces with their along mutual geodesic, line of curvature, and asymptotic curve. Our ramifications in this paper may be beneficial in any area that demands documentation around surfaces due to the descriptions supplying insights into surfaces theory.
2 Basic concepts
Let indicate the Minkowski 3-space [28, 29]. For vectors and in is named the Lorentzian inner product. We also explain a vectorSince is an indefinite metric, recall that a vector can have one of three causal natures; it can be if or , timelike if , and null or lightlike if and . The norm of is explained by ; then, the hyperbolic and Lorentzian (de Sitter space) unit spheres areand
2.1 Ruled surface
is a surface produced by a line mobile on a curve . The several locations of the line coined the producers or rulings of the surface. Such a surface, thus, has the ruled form [1–6]such that ; . In this circumstance, the curve is the striction curve and is the arc length of the spherical non-null curve . If is not stationary or not null or null, then the Blaschke Frame for will be registered aswhere are named the ruling, the central normal, and the central tangent, respectively. The Blaschke formula is from Equation 4where is the spherical curvature of . In view of with signs , , and , is, , and are titled the curvature parameters of . The geometrical view of these parameters is proved as follows: is the spherical curvature of the spherical image curve ; depicts the angle through the tangent of and the ruling of ; and is the distribution parameter of , from Equation 3 at the ruling .
In this study, we will meditate a non-developable nominated by . Then,where is the Darboux vector from Equation 6, andTherefore, a non-developable can be perceived as follows:The unit normal vector isNote that is identical with , which is the central normal at the striction point. The curvature axis of is from Equations 1, 2Let be the radii of curvature through and . Then, from Equation 11
Definition 1
[
16]
In, a surface can be determined by the induced metric on it. Hence, a surface is calledsurface iff the metric is Lorentzian metric.
surface iff the metric is a positive definite Riemannian metric.
Null surface iff the metric is null.
Corollary 1The curvature, the torsion, and the geodesic curvatureoffulfill that
Corollary 2Ifis a specified, thenis a Lorentzian circle.
Proof. Through Equation 13, we can see that , which is stationary, yields , and is stationary, which reveals is a Lorentzian circle (If ) or a Lorentzian great circle (when ).
Let’s state the Darboux frame ; let be the tangent unit to , is the surface unit normal along , and be the tangent unit to . Therefore, we can writeandLet be the arc length of , that is, . Then, from Equations 14, 15 the Darboux formula is expressed aswhere, , and are the geodesic curvature, the normal curvature, and the geodesic torsion of , respectively. Therefore, using Equations 16, 17
1) is a geodesic curve iff ;
2) is a asymptotic curve iff ;
3) is a curvature line iff .
Remark 1From Equation 8and the above notations, we state that(a) if, thenis atangential developable, and(b) if, thenis abinormal surface, and(c) if, thenis acone, and
Definition 2[14] A ruled surface is named a slant ruled surface if all its rulings have a stationary angle with a definite line.
3 Bertrand offsets for slant surfaces
In this section, we contemplate and analyze the for slant . Then, a theory hassling to the theory of the Bertrand curves can be broadened for such surfaces.
In comparable with [30], a point will be heading an curvature axis of the curve; for all such that , but . Here, signalizes the tth derivative of with regard to . For the first curvature axis of, we find , and . So, is at least an curvature axis of . We now sign a height function , by . We set the notation for any specified point .
Proposition 1
i)will be specified in the first evaluation iff, that is,for real numbers, and.
ii)will be specified in the second evaluation iffis thecurvature axis of, that is,
iii)will be specified in the third evaluation iffis thecurvature axis of, that is,
iv)will be specified in the fourth evaluation iffis thecurvature axis of, that is,
Proof. For , we determineSo, we realizefor real numbers , , and , the consequence is evident.2- Derivation of Equation 18 displays thatBy Equations 18–20, we determine3- Differentiation of Equation 20 displays thatThus, we gain4- By corresponding debates, we can also determineThe proof is finished.
In view of
Proposition 1, we determine
(a) The osculating circle of is displayed bywhich are pointed via the situation that the osculating circle must have touch of at least third order at iff .
(b) The curve and the osculating circle have touched at least fourth order at iff and .
Through this method, by catching into meditation the curvature axes of , we can attain a concatenation of curvature axes , ,…, . The ownerships and the joint links via these curvature axes are much pleasant troubles. For example, it is facile to catch that if and , located at is specified relative to . At this position, the curvature axis is fixed up to second order and is a slant .
A non-developableis a slantiff its geodesic curvatureis fixed.
Definition 3Letandbe two non-developable ruledsurfaces in. is entitled aofif there exists a bijection via their rulings such thatandpossess a reciprocal central normal at the conformable striction points.
Let be a of and is the of , as shown in Equations 7–9. Then, the surface can be allocated bywhereHere, is the distance through the proportional striction points of and . Through the differentiation of Equation 21 via and considering Equation 22, we assignSince at the congruent striction points of and , we gain is fixed. Furthermore, given that is the angle among the rulings of and , that is,By differentiation of Equation 23, we gainSince , then we realize is fixed. Moreover, at the congruent striction points of and , we observe that . Then, by Equation 24If (resp. ), then and are parallel (resp. oriented) offsets.
The coupleis fixed at the corresponding striction points ofand.
It is apparent from Theorem 2 that a non-developable , frequently, has a binary infinity of . Every can be displayed by a fixed linear offset and a fixed-angle offset . Therefore, if is a of , then is also a of .
Let be the unit normal of . Then, as shown in Equation 10, we locatewhere is the distribution parameter of .
The dissimilarity between the normal to a and its is apparent from Equations 10, 26. This demonstrates that the of a is often not a parallel offset. Therefore, the parallel circumstances through in view of can be exhibited by the following:
andare parallel offsets iff, with , their Blaschke frames, being conformable.
Proof. Let , that is, and are parallel offsets. Then, by Equations 10, 26, we acquirewhich is assumed true for any value , that is, , and .
Let the two events hold true, that is, and . Then, substituting them into using Equation 27, we acquirewhich indicates that and are parallel offsets since the previous is a zero vector.
Using the same approach, but with a developable surface , we encounter the following:
Corollary 3A developableand its developableare parallel offsets iff their Blaschke frames are identical.
Corollary 4A developableand its non-developablecannot be parallel offsets.
Furthermore, we also detectwhereBy takeoff , we locate using Equations 28, 29This presents a new perspective of of surfaces, specifically focusing on their geodesic curvatures.
andaresurfaces iffEquation 30is fulfilled.
Corollary 5andare parallel offsets iff.
Corollary 6andare oriented offsets iff.
For being fixed, from Equations 7, 12, we have the ODE, . In accordance with several algebraic manipulations, the solution iswhere is fixed and . Then,Therefore, from Equations 8, 30, 31, is expressed as Hence, from Equations 9, 30–33, the slant is expressed asFurthermore, by Equations 20, 25, 33, is expressed aswhere and can control the shape of ; here, we will set , , , and .
3.1 Classifications of the slant and its
From Equations 34, 35, the slant and its can be distributed as follows:1) Let be a asymptotic curve, i.e., . The slant and its parallel (oriented) are shown in Figure 1 ;Figure 2; .
FIGURE 1
FIGURE 2
2) Let be a geodesic curve, i.e.,where is a real constant. The slant and its parallel (oriented) are shown in Figure 3 (Figure 4); and .
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
FIGURE 12
4 Conclusion
This work explores the features of slant curves and develops and classifies slant surfaces and their in Minkowski 3-space using the Blaschke domain. Next, we construct contemporary surfaces in Lorentzian line space and determine their . In addition, we also obtain various groupings by a slant and its striction curve. These advancements are expected to enhance the usefulness of model-based manufacturing in mechanical outputs and geometric patterning. The authors intend to correlate this study across several domains and examine the classification of singularities, as identified in [31, 32].
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material; further inquiries can be directed to the corresponding author.
Author contributions
AA: conceptualization, data curation, formal analysis, funding acquisition, methodology, resources, writing–original draft, and writing–review and editing. RA-B: conceptualization, data curation, formal analysis, investigation, methodology, resources, supervision, validation, visualization, and writing–original draft.
Funding
The author(s) declare that financial support was received for the research, authorship, and/or publication of this article: Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R337).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
Darboux vector, height functions, developable surface
Citation
Almoneef AA and Abdel-Baky RA (2024) Slant spacelike ruled surfaces and their Bertrand offsets. Front. Phys. 12:1484936. doi: 10.3389/fphy.2024.1484936
Received
22 August 2024
Accepted
31 October 2024
Published
05 December 2024
Volume
12 - 2024
Edited by
Riccardo Meucci, National Research Council (CNR), Italy
Reviewed by
Jean-Marc Ginoux, Université de Toulon, France
Mehmet Önder, Manisa Celal Bayar University, Türkiye
Updates
Copyright
© 2024 Almoneef and Abdel-Baky.
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*Correspondence: Areej A. Almoneef, aaalmoneef@pnu.edu.sa
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.