3 Spacelike Surface Pencil with a Common Line of Curvature Let P = P ( s, t ) be a parametric spacelike surface and r = r ( s ) be a spacelike curve with spacelike binormal. The surface is defined by the given curve as L$_{1}$ ≤ s ≤ L$_{2}$, T$_{1}$ ≤ t ≤ T$_{2}$, where u ( s, t ) , v ( s, t ) and w ( s, t ) are called the marching-scale functions and { T ( s ) , N ( s ) , B ( s ) } is the Frenet frame associated with the curve r ( s ) . Since the curve r ( s ) is a parametric curve on the surface P ( s, t ), there exists a parameter t$_{0}$ ∈ [ T$_{1}$, T$_{2}$ ] such that P ( s, t$_{0}$ ) = r ( s ), L$_{1}$ ≤ s ≤ L$_{2}$, that is, Let n$_{1}$ ( n$_{1}$ = cosh θN + sinh θB ) be a vector orthogonal to the curve r ( s ), where θ = θ ( s ) is the Lorentzian timelike 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. Firstly, we derive the condition for n$_{1}$ to be parallel to the normal vector n ( s, t ) of the surface P ( s, t ) : The normal vector can be expressed as n ( s, t ) = ∂P ( s,t ) ∂s × ∂P ( s,t ) ∂t = ( − ( ∂w ( s,t ) ∂s + v ( s, t ) τ ( s )) ∂v ( s,t ) ∂t + ( ∂v ( s,t ) ∂s + u ( s, t ) κ ( s ) + w ( s, t ) τ ( s )) ∂w ( s,t ) ∂t ) T ( s ) + 7 u ( s, t$_{0}$ ) = v ( s, t$_{0}$ ) = w ( s, t$_{0}$ ) ≡ 0 , L$_{1}$ ≤ s ≤ L$_{2}$. (3.2) P ( s, t ) = r ( s ) + ( u ( s, t ) , v ( s, t ) , w ( s, t ))    T ( s ) N ( s ) B ( s )   $^{,}$ (3.1)