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## 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 

<img src='content_image/299336.jpg'>

<img src='content_image/299339.jpg'>

<img src='content_image/299338.jpg'>

