On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$

Abstract

In this paper, we introduce generalized Darboux frames of the first and the second kind along a null Cartan curve lying on a timelike surface in Minkowski space ${E}^{3}_{1}$ and define null Cartan rectifying isophotic and rectifying silhouette curves in terms of the vector field that belongs to generalized Darboux frame of the first kind. We investigate null Cartan rectifying isophotic and rectifying silhouette curves with constant geodesic curvature $k_g$ and geodesic torsion $\tau_g$ and obtain the parameter equations of their axes. We prove that such curves are the null Cartan helices and the null Cartan cubics. We show that the introduced curves with a non-zero constant curvatures $k_g$ and $\tau_g$ are general helices, relatively normal-slant helices and isophotic curves with respect to the same axis. In particular, we find that null Cartan cubic lying on a timelike surface is rectifying isophotic and rectifying silhouette curve having a spacelike and a lightlike axis. Finally, we give some examples.