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

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space E-1(3) and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature kg and constant geodesic torsion tau(g). We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non-zero constant curvatures k(g) and tau(g) have a remarkable property that they are general helices, relatively normal-slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions.