This paper presents a path planning algorithm that can efficiently check for interference with potential obstacles while piecewise continuously computing the required space of moving car-like vehicles using cubic Bezier curves. Our collision-checking algorithm uses trajectories generated from a vehicle's front outer corner and rear inner axle, as well as partially overlapped rectangles. These outer and inner trajectories are computed from the trajectory generated by the center of the rear axle of the vehicle, which considers the dimensions of the vehicle, and the tangential and normal vectors of the trajectory. To validate the continuity and efficacy of our collision-checking algorithm, the collision-checking algorithm is applied to a spline-based RRTau, where the kinematics (or minimum turning radius) of car-like vehicles is satisfied using cubic Bezier curves. We show the benefits of our method through simulations and experimental results by using an autonomous ground vehicle.