Collision checking and avoidance is an import part of the perception and planning system for autonomous driving. We present a new analytic approach to calculate the probability of a future collision and extend another already known solution to be suitable for ground vehicle navigation. Our new concept of the collision octagon facilitates in both cases the derivation of an analytic solution. Both approaches are compared to each other using simulated and real world scenarios. By comparing the results of the analytic solutions to the corresponding Monte Carlo simulations, their accuracy and real-time capability is demonstrated. The suitability of the analytic solutions for real world autonomous systems is further proven by integrating them into the trajectory prediction and planning system of the self-driving car of the Freie Universität Berlin.