Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 355 -- 373

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

GEOMETRIC FEATURES OF NORMAL CURVES IN EUCLIDEAN SPACES

Absos Ali Shaikh, Pushpinder Badyal, Sandeep Sharma and Kuljeet Singh

Abstract. A normal curve in the Euclidean space E3 is defined as a curve whose position vector lies entirely within the normal plane, ensuring orthogonality at each point to the tangent vector. In this article, we examine normal curves in E3 and E4. For E3, we derive a differential equation relating curvature and torsion to characterize normal curves. We also demonstrate that for unit-speed normal curves, the distance function remains constant, and there exist specific relationships between the curvatures and different components of the curve. In E4, we extend these results by characterizing normal curves in terms of the curvatures k1, k2, and k3, establishing conditions for constant curvatures, and analyzing their normal and binormal components.

2020 Mathematics Subject Classification: 53A04; 53A05; 53A15.
Keywords: Serret-Frenet frame; normal curve; curvature; torsion.

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Absos Ali Shaikh
Department of Mathematics, The University of Burdwan,
Burdwan-713104, West Bengal, India.
e-mail: aashaikh@math.buruniv.ac.in

Pushpinder Badyal
School of Mathematics, Shri Mata Vaishno Devi University,
Katra-182320, Jammu and Kashmir, India.
e-mail: pushpinder4970@gmail.com

Sandeep Sharma
School of Mathematics, Shri Mata Vaishno Devi University,
Katra-182320, Jammu and Kashmir, India.
e-mail: sandeep.greater123@gmail.com

Kuljeet Singh
School of Mathematics, Shri Mata Vaishno Devi University,
Katra-182320, Jammu and Kashmir, India.
e-mail: kulljeet83@gmail.com

https://www.utgjiu.ro/math/sma