New repulsive potential functions (RPFs) for point robot path planning are presented, where the RPF for path planning has a different magnitude at each direction of a RPF based on the angle between a goal and an obstacle, unlike a conventional RPF in which the same magnitude at each direction is obtained. In doing so the RPF attempts to overcome some of the typical problems that may pose to the conventional RPF. In particular, this paper presents a set of analysis for designing potential functions to avoid local minima for a number of representative scenarios. Specifically the following cases are addressed: a non-reachable goal problem (a case that the potential of the goal is over-whelmed by the potential of an obstacle), an obstacle collision problem (a case that the potential of the obstacle is overwhelmed by the potential of the goal), and a narrow passage problem (a case that the potential of the goal is overwhelmed by the potential of two obstacles). The proposed RPF scheme eliminates the non-feasible area for the three cases by the help of angle varying magnitude between a goal and an obstacle.