Thus, we get where This follows that n$_{1}$ ( s ) //n ( s, t$_{0}$ ), L$_{1}$ ≤ s ≤ L$_{2}$ , if and only if there exits a function λ ( s ) = 0 such that Secondly, since S ( T ) = ωT, ω = 0 , we obtain where s$_{0}$ is the starting value of the arc-length and θ = θ ( s ) . In this paper, we assume that s$_{0}$ = 0 . Combining (3.2), (3.3) and (3.4), we have the following theorem. Theorem 3 A spacelike curve r ( s ) with spacelike binormal is a line of curva- ture on the surface P ( s, t ) if and only if the followings are satisfied: θ ( s ) = − s ∫ s 0 τds + θ (0) , 8 θ ( s ) = − ∫ s 0 τ ds + θ$_{0,}$ (3.4) φ$_{1}$ ( s, t$_{0}$ ) = 0 , φ$_{2}$ ( s, t$_{0}$ ) = λ ( s ) cosh θ, φ$_{3}$ ( s, t$_{0}$ ) = λ ( s ) sinh θ. (3.3) φ$_{1}$ ( s, t$_{0}$ ) = ∂v ( s,t 0 ) ∂s ∂w ( s,t 0 ) ∂t − ∂w ( s,t 0 ) ∂s ∂v ( s,t 0 ) ∂t , φ$_{2}$ ( s, t$_{0}$ ) = ( 1 + ∂u ( s,t 0 ) ∂s ) ∂w ( s,t 0 ) ∂t − ∂w ( s,t 0 ) ∂s ∂u ( s,t 0 ) ∂t , φ$_{3}$ ( s, t$_{0}$ ) = 1 + ∂u ( s,t 0 ) ∂s ∂v ( s,t 0 ) ∂t − ∂v ( s,t 0 ) ∂s ∂u ( s,t 0 ) ∂t . n ( s, t$_{0}$ ) = φ$_{1}$ ( s, t$_{0}$ ) T ( s ) + φ$_{2}$ ( s, t$_{0}$ ) N ( s ) + φ$_{3}$ ( s, t$_{0}$ ) B ( s ) , ((1 + ∂u ( s,t ) ∂s + v ( s, t ) κ ( s )) ∂w ( s,t ) ∂t − ( ∂w ( s,t ) ∂s + v ( s, t ) τ ( s )) ∂u ( s,t ) ∂t ) N ( s ) + ((1 + ∂u ( s,t ) ∂s + v ( s, t ) κ ( s )) ∂v ( s,t ) ∂t − ( ∂v ( s,t ) ∂s + u ( s, t ) κ ( s ) + w ( s, t ) τ ( s )) ∂u ( s,t ) ∂t ) B ( s )