In the collision avoidance problem, it is a common practice to approximate the shapes of the robots and obstacles by circles or spheres. However, such approximations can be overly conservative when the objects are more elongated in one direction when compared to another, and/or non-convex. This paper develops two analytical collision avoidance laws, which do not require any approximations to the object shapes. These laws are based on the collision cone approach which has previously been used to develop exact collision conditions for objects with arbitrary shapes, moving on a plane. The first collision avoidance law governs the magnitude of the acceleration (applied at an arbitrary angle), while the second collision avoidance law governs the direction of the acceleration (applied with a magnitude of 0 or 1). The effect of time delays on the performance of the laws is analyzed. Simulation results are presented to demonstrate the working of these collision avoidance laws.