function λ ( s ) = 0 such that 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 s$_{0}$ = 0 . Combining (4.2), (4.3) and (4.4), we have the following theorem. Theorem 6 A spacelike curve r ( s ) with timelike binormal is a line of curvature on the surface P ( s, t ) if and only if the followings are satisfied 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). then, we can simply express the sufficient condition for which the curve r ( s ) is a line of curvature on the surface P ( s, t ) as (i) If we choose u ( s, t ) = p ∑ k =1 a$_{1}$$_{k}$l ( s ) k U ( t ) k , v ( s, t ) = p ∑ k =1 a$_{2}$$_{k}$m ( s ) k V ( t ) k and w ( s, t ) = p ∑ k =1 a$_{3}$$_{k}$n ( s ) k W ( t ) k θ ( s ) = − s ∫ s 0 τds + θ (0) , u ( s, t$_{0}$ ) = v ( s, t$_{0}$ ) = w ( s, t$_{0}$ ) ≡ 0 , φ$_{1}$ ( s, t$_{0}$ ) ≡ 0 , φ$_{2}$ ( s, t$_{0}$ ) = λ ( s ) cosh θ, φ$_{3}$ ( s, t$_{0}$ ) = λ ( s ) sinh θ. θ ( s ) = − s ∫ s 0 τ ds + θ$_{0,}$ (4.4) φ$_{1}$ ( s, t$_{0}$ ) = 0 , φ$_{2}$ ( s, t$_{0}$ ) = λ ( s ) cosh θ, φ$_{3}$ ( s, t$_{0}$ ) = λ ( s ) sinh θ. (4.3) 14