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

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.