Rectifying curves under conformal transformation

Shaikh, Absos Ali; Lone, Mohamd Saleem; Ghosh, Pinaki Ranjan

doi:10.1016/j.geomphys.2021.104117

Cited by 7 publications

(6 citation statements)

References 7 publications

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“…and discovered that, under isometry of surfaces, the component of the position vector of a space curve along the surface normal remains invariant. This paper's primary objective is to expand on the work of Lone [5,12,13], where he studied the geometric invariants of normal curves under conformal transformation in E 3 . In [5], the author investigated the invariant properties of normal curves under conformal transformation and also studied the normal and tangential components of the normal curves under the same motion.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

WITHDRAWN: Geometric invariance for the conformal image of rectifying curve in Euclidean space R3

Sharma

Singh

Lone

2022

Preprint

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A rectifying curve is a space curve that plays an important role in the field of differential geometry. This article is a follow-up of the work done in two recent papers [4] and [5], where along with various results on normal, rectifying, and osculating curves, several other properties of these curves are investigated. In this paper, we investigate some geometric invariance for the conformal image of a rectifying curves on regular surfaces under conformal transformation in the Euclidean space R3. The main objective of this paper is to discuss the invariant sufficient condition for the conformal image of a rectifying curve under the conformal, homothetic, and isometric transformations. The normal components of the rectifying curves are also computed, and it is demonstrated that these components remained invariant under the isometry of the surfaces in R3. We also investigated that, for a rectifying curve, the Christoffel symbols are invariant under isometry of surfaces. Mathematics Subject Classification (2010). Secondary 53A15, 53A05, 53A04, 51M05.

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Section: Definitions and Preliminariesmentioning

confidence: 99%

WITHDRAWN: Geometric invariance for the conformal image of rectifying curve in Euclidean space R3

Sharma

Singh

Lone

2022

Preprint

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“…In the same way the plane spanned by { n, b} is called the normal plane. Whenever we talk about the position vector of the curve, which defines the different kinds of curves [12,16,17] :…”

Section: Preliminariesmentioning

confidence: 99%

“…It was found that, under the assumption that surfaces are isometric, the component of the position vector of a space curve along the surface normal remains constant. This paper's primary objective is to expand on the work of Lone et al [5,12,13], they studied the geometric invariants of normal curves under conformal transformation in the Euclidean space E 3 . In [5],the author explored the behaviour of the normal and tangential components of the normal curves under the same motion as well as the invariant characteristics of normal curves under conformal transformation.…”

Section: Introductionmentioning

confidence: 99%

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

Sharma

Singh²

2023

Int. J. Anal. Appl.

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An essential space curve in the study of differential geometry is the rectifying curve. In this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. The normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the Christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. Moreover, we also studied some properties for the first fundamental form of the surfaces.

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“…Both of these problems were discussed in [7,9,10]. Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion.…”

Section: Introductionmentioning

confidence: 99%

“…Later in [11], we generalize the notion of study by conformal transformation, wherein we study the conformal transformation of rectifying curves lying on smooth surfaces. In this paper( [11]), in addition to various geometric invariants, we obtain a sufficient condition with respect to which a rectifying curve retains its nature under conformal motion. Now, the first hand possible studies depending upon the position vector field and the conformality will be the investigation of osculating and normal curves.…”

Section: Introductionmentioning

confidence: 99%

Conformal image of an osculating curve on a smooth immersed surface

Shaikh

Lone

Ghosh

2020

Journal of Geometry and Physics

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The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.2010 Mathematics Subject Classification. 53C22, 53A04, 53A05.

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Rectifying curves under conformal transformation

Cited by 7 publications

References 7 publications

WITHDRAWN: Geometric invariance for the conformal image of rectifying curve in Euclidean space R3

WITHDRAWN: Geometric invariance for the conformal image of rectifying curve in Euclidean space R3

Some Aspects of Rectifying Curves on Regular Surfaces Under Different Transformations

Conformal image of an osculating curve on a smooth immersed surface

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