content_image/325608.jpg content_image/325604.jpg content_image/325605.jpg content_image/325606.jpg content_image/325610.jpg content_image/325607.jpg
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/325608.jpg'>

Thus, we get 

where 

<img src='content_image/325604.jpg'>

<img src='content_image/325605.jpg'>

<img src='content_image/325606.jpg'>

This follows that n$_{1}$ ( s ) //n ( s, t$_{0}$ ), L$_{1}$ ≤ s ≤ L$_{2}$ , if and only if there exits a 

<img src='content_image/325610.jpg'>

<img src='content_image/325607.jpg'>

