Null Hybrid Curves and Some Characterizations of Null Hybrid Bertrand Curves

Abstract

In this paper, we investigate null curves in R24, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in R24. We first derive Frenet formulas for a null curve in R23, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in R24. This approach is simpler and more efficient than the classical differential geometry methods and enables us to determine a null curve in R23 corresponding to the null curve in R24. Additionally, we provide an example of a null hybrid curve, demonstrate the construction of its Frenet frame, and calculate the curvatures of the curve. Finally, we introduce null hybrid Bertrand curves, and by using their symmetry properties, we provide some characterizations of these curves