Boyuan Xu

In this paper, we provide an effective tool for studying spacelike curves on null planes in mathematical and physical research. As an application, we propose a unified method for constructing generalized null Cartan curves.

In this paper, we define generalized null Cartan rectifying curves which may have singular points. We propose a construction method for generalized null Cartan rectifying curves, which is more general than the one for regular null rectifying curves, as it also applies to non-regular cases. Moreover, we explore the relationships between generalized null Cartan rectifying curves, generalized null Cartan helices and centrodes.

We use the Bishop frame to investigate magnetic curves and magnetic vector fields in Euclidean 3-space, and address the case of magnetic curves with inflection points. Moreover, we study the differential geometric properties of magnetic flux surfaces, especially a special kind of magnetic flux surfaces (normal magnetic flux surfaces). We provide a physical explanation for the inflection points of the magnetic curves, which refer to the points where the charged particle experiences zero Lorentz force. These points also correspond to the singularities of the normal magnetic flux surfaces.

In this paper, we define Legendre curves and introduce frontals with their associated moving frames on the lightcone in Minkowski 3-space. By treating an optical fiber as a curve on the lightcone, we analyze the geometric phase of linearly polarized light wave propagating along the optical fiber. We examine the transformation of polarization vectors along optical fibers and generate Rytov curves. The singularities of Rytov curves are investigated. Furthermore, we define electromagnetic curves on the lightcone and analyze the motion trajectories of charged particles under the influence of electromagnetic fields. Finally, we demonstrate our results through concrete examples.

We give a method for constructing generalized null Cartan helices, which may have singularities, by using regular space-like plane curves and smooth functions in Minkowski 3-space. We also study the principle normal worldsheet, which is deeply related to generalized null Cartan helices from the viewpoint of singularity theory.