Bertrand lightcone framed curves in the Lorentz-Minkowski 3-space

TL;DR

This paper introduces Bertrand lightcone framed curves in Lorentz-Minkowski 3-space, analyzing their existence conditions and deriving pseudo-circular involutes and evolutes for mixed type curves with singular points.

Contribution

It extends the concept of Bertrand curves to lightcone framed curves with singularities in Lorentz-Minkowski space, providing existence conditions and geometric properties.

Findings

Existence conditions for Bertrand lightcone framed curves established.

Pseudo-circular involutes and evolutes derived for these curves.

Framework applicable to mixed type curves with singular points.

Abstract

We consider mixed types of not only regular curves but also curves with singular points in the Lorentz-Minkowski 3-space. In order to consider mixed type of curves with singular points, we consider the lightcone frame and lightcone framed curves. By using lightcone frame, we can consider Bertrand types for lightcone framed curves, so-called Bertrand lightcone framed curves. We clarify that the existence conditions of Bertrand lightcone framed curves in all cases. As a consequence, we find pseudo-circular involutes and evolutes of mixed type curves which appear as Bertrand lightcone framed curves.