The trajectory of a moving particle in space forms a curve. By moving a line along a curve, a surface called ruled surface is obtained. The striction point on a ruled surface is the foot of the common normal between two consecutive generators or ruling. The set of striction points defines the striction curve. In the present paper, we obtain surfaces passing through the spherical indicatrix curves formed on the unit sphere by the end points of the geodesic Frenet frame formed on this curve. We present conditions for these curves to be asymptotic curves or geodesic on the surface. We illustrate the method with several examples.