Abstract
The Frenet frame is not suitable for describing the behavior of the curve in the Galilean space since it is not defined everywhere. In this study, an alternative frame, the so-called quasi-frame, is investigated in Galilean 4-space. Furthermore, the quasi-formulas in Galilean 4-space are deduced and quasi-curvatures are obtained in terms of the quasi-frame and its derivatives. Quasi-rectifying, quasi-normal, and quasi-osculating curves are studied in Galilean 4-space. We prove that there is no quasi-normal and accordingly normal curve in Galilean 4-space.
1 Introduction
The Galilean space is considered to be one of the Cayley–Klein spaces, and Roschel was the primary contributor to its development. A Galilean space is the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. In this situation, the only shape left is a plane. The limit transition is similar to that encountered when classical mechanics replaced special relativity.
The disadvantage of the Frenet frame is that it is not defined everywhere, namely, if the curve has points where they have zero curvature. At these points, normal and binormal vectors are not defined. Hence, many mathematicians investigated frames that are defined everywhere, even if the curve has zero curvature points. Many frames such as the modified frame, the Bishop frame, the Darboux frame, the equiform frame, and quasi-frame have been investigated and studied in Euclidean space [1–5], Minkowski space [6–11], and Galilean space [12–15].
In Euclidean three-space, the osculating curve is defined as the position vector of the curve residing in the plane consisting of its tangent vector and normal vector. The normal curve is defined as the position vector of the curve residing in the plane consisting of its normal vector and binormal vector. The rectifying curve is defined as the position vector of the curve residing in the plane consisting of its tangent vector and binormal vector. Some studies have been carried out on normal, osculating, and rectifying curves in Euclidean three and four spaces [16–20], Minkowski three and four spaces [21–24], Galilean three and four spaces [12,25–29] and in Sasakian space [30].
In 2015 [1], Dede et al. investigated an alternate adapted frame called the quasi-frame, which followed a space curve, rather than using the Frenet frame. This frame is easier and more accurate than the Frenet frame and the Bishop frame, and it is considered a generalization of the Frenet frame. Many studies have been carried out on the quasi-frame in Euclidean and Minkowski spaces [2,3,31,32]. Furthermore, more recent research studies on position vectors in Galilean three and four spaces were performed with the Frenet frame [33–36].
Rectifying curves, normal curves, and osculating curves are found in the Euclidean space . These curves meet the fixed point criterion proposed by Cesaro. It is well known that if all the normal planes or osculating planes of a curve in pass through a given point, then the curve either resides in a sphere or is a planar curve, depending on the two category it falls into. It is also well known that if all rectifying planes of a non-planar curve in run through a certain point, then the ratio of the curve’s torsion to its curvature is a non-constant linear function. For more details, see [16]. In addition, Ilarslan and Nesovic [17] provided some characterizations for osculating curves in . They also constructed osculating curves in as a curve whose position vector always lies in the orthogonal complement of its first binormal vector field. These characterizations were given for osculating curves in . As a consequence of their findings, they could classify osculating curves according to the curvature functions of those curves and provide both the necessary and sufficient conditions of osculating curves for arbitrary curves in .
The research is organized as follows: Section 3 introduces the quasi-frame, its relation with the Frenet frame, quasi-formulas, and the quasi-curvatures in Galilean 4-space. Section 4 describes the study of the position vectors in Galilean 4-space. Section 5 characterizes the quasi-rectifying curves. Section 6 introduces and describes the quasi-osculating curves. Section 7 finally proves that there is no normal curve in Galilean 4-space.
2 Preliminaries
In this section, we introduce some basic concepts of Galilean 4-space. The Galilean metric in Galilean 4-space is defined bywhere and . Based on this metric, the Galilean norm of the vector is given by
In addition, the Galilean cross-product of , and is defined aswhere (, , , and ) are the usual bases of [26,35].
The Galilean adds even more complexity by investigating all qualities that remain constant despite the spatial motions of objects. It was further clarified that this geometry may be defined as the investigation of properties of 4-dimensional space, the coordinates of which remain unchanged when subjected to a general Galilean transformation [27,29].
A curve in is a mapping from an open interval to defined aswhere and are differentiable functions. If the curve is parameterized by the arc length, then it takes the form
On the other hand, the Frenet frame in consists of four orthonormal vectors called the tangent, the principal normal, the first binormal, and the second binormal, and they are denoted, respectively, bywhere and are the first, second, and third Frenet curvatures, respectively. They can be given byIf the Frenet curvatures are constant, then we say the curve is a W-curve.The Frenet formulas of the curve are
Let be a unit speed curve in . If its position vector always lies in the orthogonal complement of or , then a curve is called an osculating curve in . If the position vector of always lies in the orthogonal complement of the normal vector . Let be an admissible curve in . We say that is a rectifying curve if the position vector of always lies in the orthogonal complement of N [26,35].
3 Quasi-frame and quasi-formulas in
In this section, we investigate the quasi-frame and its relation with the Frenet frame in . In addition, quasi-formulas in Galilean 4-space are investigated. Moreover, the quasi-curvatures are introduced. Let be a curve in .
The quasi-frame is an alternative to the Frenet frame and involves two fixed unit vectors. We define the quasi frame depending on four orthonormal vectors, called the unit tangent, called the unit quasi-normal vector, called the unit first quasi-binormal vector, and called the unit second quasi-binormal vector. The quasi-frame is defined asfor the projection vectors and and is , where the determinant of the matrix is equal to 1. Here, we choose for simple calculations and .
The transformation matrix keeps the tangent vector unchanged. Then, we consider three possible planes of rotations. The first rotation is in the plane spanned by and with an angle . The second rotation in the plane is spanned by and with an angle . The third rotation in the plane is spanned by and with an angle as in Figure 1. The quasi-frame can be written in terms of the Frenet frame asThe transformation matrix can be written as
FIGURE 1
Let the matrix of the quasi-frame be and the matrix of the Frenet frame be . In addition, let the curvature matrix of the quasi-frame be and the curvature matrix of the Frenet frame be . Then, we can write
Then, we can writeBy differentiating Eq. 1 with respect to , we haveBy substituting Eqs 2–4 into Eq. 5, we haveTherefore,
Corollary 3.1The quasi-frame is considered a generalization to the Frenet frame by putting . In addition, the quasi-formulas are considered generalizations to the Frenet formulas by putting .
Corollary 3.2The quasi-curvatures of the curve are given, respectively, by
4 Quasi-position vector curves in
In this section, we study the position vectors in .
We consider a curve in Galilean 4-space as a curve whose position vector satisfies the parametric equationfor some differentiable functions, and , where is the quasi-frame. By differentiating Eq. 7 with respect to arclength parameter s and using the quasi Eq. 6, we obtainHence,Let and are constants, so we can find asTherefore, we can write completely the curve
5 Quasi-rectifying curves
In this section, we define the quasi-rectifying curve in the Galilean 4-space and characterize quasi-rectifying curves .
Definition 1A curve in the Galilean 4-space is called a quasi-rectifying curve if it has no component in the quasi-normal direction, in other words if . In addition, the curve is called a quasi-rectifying curve if the position vector satisfies the parametric equationfor some differentiable functions, and , where is the quasi-frame.By differentiating Eq. 8 concerning arclength parameter s and using the quasi Eq. 6, we obtainHence,By solving Eqs 9–12 together, we get
6 Quasi-osculating curves
In this section, we define the quasi-osculating curve in the Galilean 4-space and characterize quasi-osculating curves .
Definition 2A curve in the Galilean 4-space is called a quasi-osculating curve if it has no component in the first quasi-binormal direction or the second quasi-binormal direction, in other words if or . In addition, the curve is called a quasi-osculating curve if the position vector satisfies the parametric equationorfor some differentiable functions, , , , and .
6.1 Quasi-osculating curve of type 1
We consider a curve in Galilean 4-space to be a quasi-osculating curve of type 1 if the position vector satisfies the parametric equationfor some differentiable functions, and , where is the quasi-frame. By differentiating Eq. 13 concerning arclength parameter s and using the quasi Eq. 6, we obtainHence,By solving Eqs 14–17 together, we get
6.2 Quasi-osculating curve of type 2
We consider a curve in Galilean 4-space to be a quasi-osculating curve of type 2 if the position vector satisfies the parametric equationfor some differentiable functions, and , where is the quasi-frame. By differentiating Eq. 18 with respect to arclength parameter s and using the quasi Eq. 6, we obtainHence,By solving Eqs 19–22 together, we get
7 Quasi-normal curves in
In this section, we prove that there is no quasi-normal curve in .
Definition 3A curve in the Galilean 4-space is called a quasi-normal curve if it has no component in the tangent direction, in other words if . In addition, the curve is called a quasi-normal curve if the position vector satisfies the parametric equation for some differentiable functions, and , where is the quasi-frame.
In the Galilean 4-space, there is no quasi-normal curve.
Suppose that is any curve in the Galilean 4-space. Then, the tangent is given byThus,Therefore, there is no quasi-normal curve in .
8 Conclusion
In this study, we investigate the definition of the quasi-frame in Galilean 4-space and obtain its relation with the Frenet frame in . In addition, the quasi-formulas and the quasi-curvatures are investigated. Furthermore, the quasi-rectifying curves and the quasi-osculating curves are studied according to the quasi-frame in . Finally, we proved that there is no quasi-normal curve and accordingly normal curve in .
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
AE: writing–review and editing, writing–original draft, visualization, supervision, software, methodology, and investigation. NE: writing–original draft, visualization, validation, software, resources, methodology, formal analysis, data curation, and conceptualization.
Funding
The authors declare that no financial support was received for the research, authorship, and/or publication of this article.
Acknowledgments
The authors would like to express their gratitude to the editor and reviewers for their thoughtful comments and suggestions, which greatly improved the quality and clarity of this paper.
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.
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Summary
Keywords
Galilean space, quasi-frame, quasi-formulas, quasi-curvatures, quasi-rectifying, quasi-osculating, quasi-normal
Citation
Elsharkawy A and Elsharkawy N (2024) Quasi-position vector curves in Galilean 4-space. Front. Phys. 12:1400730. doi: 10.3389/fphy.2024.1400730
Received
14 March 2024
Accepted
12 June 2024
Published
24 July 2024
Volume
12 - 2024
Edited by
William Cannon, Pacific Northwest National Laboratory (DOE), United States
Reviewed by
Özcan Bektaş, Samsun University, Türkiye
Sameh Shenawy, Modern Academy for Computer Science and Management Technology, Egypt
Updates
Copyright
© 2024 Elsharkawy and Elsharkawy.
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*Correspondence: Ayman Elsharkawy, ayman_ramadan@science.tanta.edu.eg
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.