In this paper, we study inextensible flows of spacelike curves on oriented time-like surface in M₁³. We research inextensible flows of spacelike curves according to Darboux frame in M₁³.

In this paper, a new modified roller coaster surface according to a modified orthogonal frame is investigated in Euclidean 3-space. In this method, a new modified roller coaster surface is modeled. Both the Gaussian curvature and mean curvature of roller coaster surfaces are investigated. Subsequently, we obtain several characterizations in Euclidean 3-space.