Fig. 4.2. P$_{4}$ ( s, t ) as a member of the surface pencil and its line of curvature Now let P = P ( s, t ) be a parametric timelike surface and r = r ( s ) be a timelike curve. The surface is defined by the given curve as L$_{1}$ ≤ s ≤ L$_{2}$, T$_{1}$ ≤ t ≤ T$_{2}$ . Let n$_{1}$ ( n$_{1}$ = cos θN + sin θB ) be a vector orthogonal to the curve r ( s ), where θ = θ ( s ) is the Lorentzian spacelike angle between N and n$_{1}$. The curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if n$_{1}$ is parallel to the normal vector n ( s, t ) of the surface P ( s, t ) and S ( T ) = ωT, ω = 0, where S is the shape operator of the surface. Since the curve r ( s ) is a parametric curve on the surface P ( s, t ), there exists P ( s, t ) = r ( s ) + ( u ( s, t ) , v ( s, t ) , w ( s, t ))    T ( s ) N ( s ) B ( s )   $^{,}$ (4.7) 17