@article{Elsharkawy2025, 
author = {Ayman Elsharkawy and Ahmer Ali and Muhammad Hanif and Fatimah Alghamdi},
title = {Exploring quaternionic Bertrand curves: involutes and evolutes in              E              4},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {3},
pages = {4598-4619},
keywords = {involute, evolute, quaternions, Euclidean space, Bertrand curves},
url = {https://www.sciopen.com/article/10.3934/math.2025213},
doi = {10.3934/math.2025213},
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.}
}