On Generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space

Abstract

In this paper we define generalized Darboux frame of a a pseudo null curve $\alpha$ lying on a lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that $\alpha$ has two such frames and obtain generalized Darboux frame's equations. We obtain the relations between the curvature functions of $\alpha$ with respect to the Darboux frame and generalized Darboux frames. We also find parameter equations of the Darboux vectors of the Frenet, Darboux and generalized Darboux frames and give the necessary and the sufficient conditions for such vectors to have the same directions. Finally, we present related examples.

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

“…Generalized Darboux frames of the first and the second kind along a pseudo null curve and a spacelike curve with a non-null principal normal lying on a lightlike surface in Minkowski space E 3 1 are defined in [8] and [9] as the frames obtained by scaling the null normal vector field of the surface restricted to the curve. Such frames represent generalizations of the Darboux frame and have a property that the curve is geodesic, asymptotic and principal curvature line if and only if its generalized geodesic curvature, generalized normal curvature and generalized geodesic torsion is equal to zero, respectively at each point of the curve.…”

On null Cartan normal isophotic and normal silhouette curves on a timelike surface in Minkowski 3‐space

“…The introduced generalized Darboux frame of the first kind can be used for defining null Cartan rectifying isophotic and rectifying silhouette curves lying on the timelike surface. Additionally, spacelike osculating general helices and spacelike rectifying general helices lying on lightlike surface can be defined by means of generalized Darboux frame of the first kind obtained in the literature [25, 26]. These results have applications in numerical computation of the mentioned kinds of curves, in investigations of generalized Darboux frames along cuspidal edges [27] and in characterizations of pseudospherical rectifying, normal, and osculating Darboux images [28, 29] along null Cartan curves according to their generalized Darboux frames.…”