Mixed-type curves and the lightcone frame in Minkowski 3-space

Abstract

In this paper, we study mixed-type curves in Minkowski 3-space. Mixed-type curves are regular curves, and there are both non-lightlike points and lightlike points in a mixed-type curve. For non-lightlike curves and null curves in Minkowski 3-space, we can study them by a Frenet frame or a Cartan frame, respectively. But for mixed-type curves, the two frames will not work. As far as we know, no one has yet given a frame to study them in Minkowski 3-space. So, we give the lightcone frame in order to provide a tool for studying this type curves in mathematical and physical research. As an application of the lightcone frame, we define an evolute of a mixed-type curve. We also give some examples to show the evolutes.