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# Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

1 Department of Mathematics, Faculty of Arts and Sciences, Bitlis Eren University, Bitlis, 13000, Turkey

2 Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan, 24002, Turkey

3 Department of Mathematics, Faculty of Arts and Science, Erzincan Binali Yıldırım University, Erzincan, 24002, Turkey

* Corresponding Author: Ali Çakmak. Email:

(This article belongs to the Special Issue: On Innovative Ideas in Pure and Applied Mathematics with Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(3), 2731-2742. https://doi.org/10.32604/cmes.2023.024517

**Received** 31 May 2022; **Accepted** 09 August 2022; **Issue published** 23 November 2022

## Abstract

In this paper, we define the curve at a constant distance from the edge of regression on a curve*r*(

*s*) with arc length parameter

*s*in Galilean 3-space. Here,

*d*is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve

*r*(

*s*) in 3-dimensional Galilean space. We build the Frenet frame of the constructed curve with respect to two types of the vector

*d*and we indicate the properties related to the curvatures of the curve . Also, for the curve , we give the conditions to be a circular helix. Furthermore, we discuss ruled surfaces of type

*A*generated via the curve and the vector

*D*which is defined as tangent of the curve in 3-dimensional Galilean space. The constructed ruled surfaces also appear in two ways. The first is constructed with the curve and the non-isotropic vector

*D*. The second is formed by the curve and the non-isotropic vector

*D*. We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable. Finally, we provide examples and visuals to back up our research.

## Keywords

Klein pronounced a different definition of geometry in his introductory speech at the University of Erlangen in 1872. He explained that geometry, given by a subgroup

One of the important research areas in differential geometry is the theory of curves examined in various spaces. In particular, it has been examined in a lot of papers and remarkable results have been obtained in the 3-dimensional Galilean space [4–10].

The notion of the curves at a constant distance from the edge of regression has been introduced by Vogler. He has studied the curves traced on a torse at a constant distance from its edge of regression. The torse of a space curve in

This subject has been studied in Euclidean 3-space since the 1970s, and it is a method that generates a new curve from the curve through the Frenet frame of the curve. For the first time, we will discuss this issue in 3-dimensional Galilean space. While the curve is produced by using the unit vector which is defined by the Frenet frame apparatus of a curve in Euclidean 3-space, we will have produced the curve by considering two situations in the Galilean 3-space. This is because, in Galilean space, vectors are treated in two ways, isotropic and non-isotropic.

In this paper, we first recall the essential preliminaries on the Galilean 3-space. Then, we define curves in the Galilean 3-space and give the curvature properties of these curves. In the main part of our study, we define a curve noted by

Let us consider a curve

The Galilean space

Now, let us consider the basic definitions and notions.

Let

If

Let

The Galilean vector product of two vectors in

where

A vector

Definition 2.1. An angle

If the vectors

If the vectors

2.1 Curves in Galilean 3-Space

Let

Let

In this case, the functions

Then,

The using Eq. (4), then the tangent vector of

If we take the derivation of Eq. (5), we get

And from Eqs. (5) and (7), we write

Using Eqs. (7) and (8), we write

As a consequence, the unit binormal vector B(x) of

and then the frame

Proposition 2.1. The Frenet formulae of a unit speed curve

where

is the curvature of

is the torsion of

3 Curve at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space

Definition 3.1. Suppose that r is a curve in Galilean 3-space and {T, N, B} is the Frenet frame at the point P = r(s) of r. Let

such that

Now, let us construct the Frenet frame of

Case 3.1. d is non-isotropic. In this case,

where d(s) = T(s).

Theorem 3.1. If r(s) is a curve with arc length parameter s, then the arc length parameter of the curve

Proof. By differentiating Eq. (14)

If we take the norm of two sides of Eq. (15), we have

which completes the proof.

Theorem 3.2. Let

and

Proof. By differentiating Eq. (14) and using Eq. (11), we have

which gives us Eq. (16). If we take derivation of Eq. (21) according to s, we get

Using Eq. (22) in Eq. (12), then we get Eq. (19). From Eq. (9), we easily obtain the Eq. (17) and from Eq. (10), we get Eq. (18). Considering Eq. (13), we obtain Eq. (20).

In these calculations we used

Corollary 3.1. Let

Proof. We know that if the curvatures

In this case, we have

Case 3.2. d is isotropic. In this case,

where

Theorem 3.3. If r(s) is a curve with arc length parameter s, then the arc length parameter of the curve

Proof. By differentiating Eq. (23), we have

If we take the norm of two sides of Eq. (24), we have

Theorem 3.4. Let

and

Proof. By differentiating Eq. (23) and using Eq. (11), we obtain

In these calculations, we use

Corollary 3.2. Let

Proof. If the curvatures of r are constants,

In this case, we have

4 Ruled Surfaces Generated by the Curve

The ruled surfaces in

4.1 Ruled Surface of Type A Generated by

A ruled surface of type A in

where

where Frenet trihedron

Additionally, the parameter of distribution

We know that if

Theorem 4.1. Suppose that (

Proof. By taking derivative of

If we take

Now, we consider that v is a constant in the ruled surface

where

Finally considering Eq. (1), the angle

4.2 Ruled Surface of Type A Generated by

Similarly to Section 4.1, a ruled surface of type A in

where the curve

Thus, the following theorem can be written:

Theorem 4.2. Suppose that (

Proof. By taking derivative of

We take as

Hence,

Example 5.1. Consider the curve given by the parametrization

The Frenet frame fields of the curve of

Considering Eq. (14), the curve

for

for

Example 5.2. Let us consider the curve given by Eq. (38) in Example 5.1. The ruled surface

In Eq. (39), the curve

is generator. For

Example 5.3. Let us consider the curve given by Eq. (38) in Example 5.1. The ruled surface

In Eq. (40), the curve

is directrix and the non-isotropic vector

is generator. For

In this study, we present a method that generates a new curve from the curve using the Frenet frame of a curve that is parameterized by arc length in

Acknowledgement: The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*135*

*(3)*, 2731-2742. https://doi.org/10.32604/cmes.2023.024517

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 135, no. 3, pp. 2731-2742. 2023. https://doi.org/10.32604/cmes.2023.024517

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