Involutes of null Cartan curves and their representations in Minkowski 3-space

The involute–evolute pair is a classic theme, which has great research significance in mathematics and engineering. A Cartan null curve is a curve whose tangent vector is light-like on each point, and its module vanishes everywhere along the curve. Due to these special properties of null curves, some research on them shows more particular and interesting prospects. In this paper, the involutes of null Cartan curves are defined and investigated based on the classifications of curves in Minkowski space. The conclusion that the involutes of null Cartan curves only can be space-like is achieved, and the relationships on curvatures, torsions, frames, expressions between a null Cartan curve and its three kinds of space-like involutes are explored. Especially, the pseudo-null involute of a given null Cartan curve can be directly expressed via the structure function of the null curve; the first and second kind space-like involutes of a given null Cartan curve can also be expressed if the tangent distance function is determined. Meanwhile, the k-type null helices and their involutes are expressed by the structure functions of null curves. Last but not least, some typical examples are presented and visualized to characterize such curve pairs explicitly.