Exploring quaternionic Bertrand curves: involutes and evolutes in $ \mathbb{E}^{4} $

Ayman Elsharkawy; Ahmer Ali; Muhammad Hanif; Fatimah Alghamdi; Ayman Elsharkawy; Ahmer Ali; Muhammad Hanif; Fatimah Alghamdi

doi:10.3934/math.2025213

AIMS Mathematics

2025, Volume 10, Issue 3: 4598-4619. doi: 10.3934/math.2025213

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Exploring quaternionic Bertrand curves: involutes and evolutes in $ \mathbb{E}^{4} $

1.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
2.
Department of Mathematics, University of Narowal, Pakistan
3.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23541, Saudi Arabia

Received: 26 December 2024 Revised: 14 February 2025 Accepted: 26 February 2025 Published: 03 March 2025
MSC : 11R52, 53A04

This study investigated the concepts of (0, 2)-involute and (1, 3)-evolute curves associated with quaternionic Bertrand curves within the context of four-dimensional Euclidean space. Using a type-2 quaternionic frame, we derived mathematical expressions that define these interacting and evolute curves. The (0, 2)-involute curve is characterized by tangents orthogonal to points on the original quaternionic Bertrand curve, while the (1, 3)-evolute curve is constructed using specific normal vectors related to curvature properties. We presented a comprehensive framework that clarifies the interrelationships between the curvature functions of involute and evolute pairs and their connections to the Frenet frame. This framework provides a geometric basis for analyzing curves in higher-dimensional spaces. The findings enhance the understanding of quaternionic curves and their geometric properties, contributing to the broader field of differential geometry.
- involute,
- evolute,
- quaternions,
- Euclidean space,
- Bertrand curves
Citation: Ayman Elsharkawy, Ahmer Ali, Muhammad Hanif, Fatimah Alghamdi. Exploring quaternionic Bertrand curves: involutes and evolutes in $ \mathbb{E}^{4} $[J]. AIMS Mathematics, 2025, 10(3): 4598-4619. doi: 10.3934/math.2025213

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Abstract

This study investigated the concepts of (0, 2)-involute and (1, 3)-evolute curves associated with quaternionic Bertrand curves within the context of four-dimensional Euclidean space. Using a type-2 quaternionic frame, we derived mathematical expressions that define these interacting and evolute curves. The (0, 2)-involute curve is characterized by tangents orthogonal to points on the original quaternionic Bertrand curve, while the (1, 3)-evolute curve is constructed using specific normal vectors related to curvature properties. We presented a comprehensive framework that clarifies the interrelationships between the curvature functions of involute and evolute pairs and their connections to the Frenet frame. This framework provides a geometric basis for analyzing curves in higher-dimensional spaces. The findings enhance the understanding of quaternionic curves and their geometric properties, contributing to the broader field of differential geometry.

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