content_image/314211.jpg content_image/314212.jpg content_image/314213.jpg content_image/314210.jpg content_image/314214.jpg
Secondly, since S ( T ) = ωT, ω ̸ = 0 , 

where s$_{0}$ is the starting value of arc length and θ = θ ( s ) . In this paper, we assume that s$_{0}$ = 0 . 

Combining (4.8), (4.9) and (4.10), we have the following theorem. 

Theorem 8 A timelike curve r ( s ) is a line of curvature on the surface P ( s, t ) if and only if the followings are satisfied 

<img src='content_image/314211.jpg'>

<img src='content_image/314212.jpg'>

<img src='content_image/314213.jpg'>

We call the set of surfaces defined by (4.1) - (4.4) or (4-7) - (4-10) timelike surface pencil with a common line of curvature . Any surface P ( s, t ) satisfying these conditions is a member of this family. 

Now, we analyse two different types of the marching-scale functions 

u ( s, t ) , v ( s, t ) and w ( s, t ) in the Eq. (4.1). 

(i) If we choose marching-scale functions as 

<img src='content_image/314210.jpg'>

is a line of curvature of the surface P ( s, t ) as 

<img src='content_image/314214.jpg'>

