For lateral vehicle dynamics, planning trajectories for lane-keeping and lane-change can be generalized as a path planning task to stabilize a vehicle onto a target lane, which is a fundamental element in nowadays autonomous driving systems. On the other hand, two-point steering for lane-change and lane-keeping has been investigated by researchers from psychology as a sensorimotor mechanism of human drivers. In the first part of this paper, using knowledge of geometric control, we will first design a path planner which satisfies five design objectives: generalization for different vehicle models, convergence to the target lane, optimality, safety in lane-change maneuver and low computational complexity. Later, based on this path planner, a two-point steering path planner will be proposed and it will be proved rigorously that this two-point steering path planner possesses the advantage--steering radius of the planned trajectory is smaller than the intrinsic radius of reference line of the target lane. This advantage is also described as "corner-cutting" in driving. The smaller driving radius of the trajectory will result in higher vehicle speed along the winding roads and more comfortness for the passengers.