A study on a ruled surface with lightlike ruling for a null curve with Cartan frame (Q2890356)

The authors study the curvature functions of a ruled surface \(M\) with light-like ruling for a null curve \(\alpha\) with Cartan frame in 3-dimensional Minkowski space \(E_1^3\). They study the relations between Darboux frames \(\{T,V,U\}\) and Cartan frames \(\{T,N,W\}\) on such a curve, and prove that the location of the striction line relative to the directrix along the ruling \(V\) is NEWLINE\[NEWLINE\beta(s) = \alpha(s) - \left(-{1\over 2}\int{\kappa_g^2\over \tau_g^2} dt+c\right)V(s) NEWLINE\]NEWLINE where \(c\) is any constant, \(\kappa_g\) and \(\tau_g\) are called the geodesic curvature and torsion of the curve, respectively. They also discuss the relation of their calculations to the study of end-effector motion.