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cient conditions on them for which the given curve is a line of curvature on a surface. Finally, we give some nice examples, showing members of the surface pencil with their common line of curvatures to illustrate the method. 

## 2 Preliminaries 

Let us consider Minkowski 3-space R 3 1 = [ R $^{3}$, (+ , + , − ) ] and let the Lorentzian inner product of X = ( x$_{1}$, x$_{2}$, x$_{3}$ ) and Y = ( y$_{1}$, y$_{2}$, y$_{3}$ ) ∈ R 3 1 be 

A vector X ∈ R 3 1 is called a spacelike vector when 〈 X, X 〉 > 0 or X = 0 . It is called timelike and null (lightlike) vector in case of 〈 X, X 〉 < 0, 〈 X, X 〉 = 0 for X ̸ = 0 . respectively, [5]. The vector product of vectors X = ( x$_{1}$, x$_{2}$, x$_{3}$ ) and Y = ( y$_{1}$, y$_{2}$, y$_{3}$ ) ∈ R 3 1 is defined by [6] 

Let r = r ( s ) be a unit speed curve in R 3 $_{1}$. By κ ( s ) and τ ( s ) we denote the natural curvature and torsion of r ( s ) ,respectively. Consider the Frenet frame { T, N, B } associated with the curve r = r ( s ) such that T = T ( s ) , N = N ( s ) and B = B ( s ) are the unit tangent, the principal normal and the binormal vector fields, respectively. If r = r ( s ) is a spacelike curve, then the structural equations (or Frenet formulas) of this frame are given as 

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