In this article, a planar geodesic (2-dimensional minimum length curve between two points) on which the maximum curvature is constrained and with prescribed initial and terminal directions is studied. A generic problem is formulated by the minimum-time optimal control problem in free terminal time. It is shown that the optimal path ($G^2$) may contain a singular arc or not and that the general types of $G^2$ can he classified into the 3 classes of control sequences. Finally, the explicit form of $G^2$ is derived geometrically as well as algebraically form the main theorem of this article.