Higher-Order Kinematic Analysis of Particle Motion along Bishop-Framed Curves in Euclidean 3-Space

This paper investigates the higher-order kinematic properties of particle motion along curves equipped with Bishop frames in three-dimensional Euclidean space. We establish comprehensive expressions for jerk and snap vectors, representing the third and fourth derivatives of position with respect to time. Our analysis transcends traditional Frenet-Serret formulations by employing the Bishop frame, which offers enhanced stability and well-defined behavior at points where classical frames may fail. The study presents novel decompositions of these kinematic quantities along Bishop frame components and introduces alternative geometric representations using specialized radial-tangential coordinate systems. These results contribute to differential geometry applications in physics, engineering, and computer graphics where smooth trajectory analysis is essential.