Malaysian Journal of Mathematical Sciences, December 2025, Vol. 19, No. 4
Integral Curves and Flc-Frames: A Comprehensive Study in Timelike Minkowski 3-Space
Elsharkawy, A., Elsayied, H. K., and Refaat, A.
Corresponding Email: ayman_ramadan@science.tanta.edu.eg
Received date: 7 February 2025
Accepted date: 13 May 2025
Abstract:
This work generalizes the Frenet-like frame to timelike curves in Minkowski space, providing a unified framework for cases where the Frenet frame fails, particularly when the second derivative of a curve vanishes. We derive the Flc-frame and Flc-equations for timelike curves, introducing the Flc-curvatures $K_1, K_2$ and $K_3$, and establish explicit relationships between them. A key contribution is the formulation of new integral curves generated from the Flc-frame, which allows for the explicit computation of Frenet vectors, curvature, and torsion in terms of the Flc-curvatures. We provide a detailed example to illustrate the application of the Flc-frame, demonstrating its effectiveness in analyzing the geometric properties of poly-timelike curves. Additionally, we explore the relationship between the Flc-curvatures and derive integral curves based on the Flc-normal and Flc-binormal vectors. These results extend the understanding of curve analysis in Minkowski 3-space and offer a robust framework for studying polynomial curves, particularly in scenarios where the Frenet frame fails.
Keywords: Minkowski 3-space; Flc-frame; Flc-equations; Flc-curvatures; polynomial curve; integral curves
