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

Ali Çakmak1,*, Sezai Kızıltuğ2, Gökhan Mumcu3
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 this 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

Edge of regression; Galilean space; curvature; helix; ruled surface

Çakmak, A., Kızıltuğ, S., Mumcu, G. (2023). Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space. CMES-Computer Modeling in Engineering & Sciences, 135(3), 2731–2742.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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