1 Introduction On a Minkowski surface, tangent vectors are classified into timelike, spacelike or null and so a curve on the surface is said to be timelike, spacelike or null if its tangent vectors are always timelike, spacelike or null, respectively. In fact, a timelike curve corresponds to the path of an observer moving slower than the speed of light, spacelike curve corresponds to moving faster than the speed of light and null curve corresponds to moving at the speed of light. Surface with a common characteristic curve have been the subject of many recent stud- ies. Wang et.al.[1] studied the problem of constructing a surface pencil from a given spatial geodesic. They parametrized the surface by using the Frenet frame of the given curve and gave the necessary and sufficient condition to sat- ify the geodesic requirement. Kasap and Akyıldız [2] considered the surfaces with common geodesic in Minkowski 3-space. They studied spacelike surface with common spacelike geodesic and timelike surface with common spacelike or timelike geodesic. Li et.al.[3] derived the necessary and sufficient condition for a given curve to be a line of curvature on a surface. S ¸affak and Kasap [4] analyzed the problem of finding a surface family through a null geodesic with Cartan frame. Being inspired by the above studies, we extend the method of Lie et. al. [3] to derive the necessary and sufficient condition for the given curve to be a line of curvature for the parametric surface. By utilizing the Frenet frame, we derive necessary and sufficient condition for the correct parametric representation of the surface P ( s, t ) when the parameter s is the arc-length of the curve r ( s ) and find the necessary constraints on the coefficients of vectors of the frame ( which are called marching-scale functions) so that both the line of curvature and parametric requirement are met. Thus, we defined the spacelike and timelike surface pencil with common line of curvature. Also, we give two general forms of the marching-scale functions and obtain necessary and suffi- 2