This paper addresses the problem of computing a time-optimal trajectory between two configurations of a car-like robot. Since 1990, two remarkable results of this problem have been achieved: the sufficient family containing all types of optimal trajectories between any two configurations, and then the optimal synthesis determining a type of optimal trajectories for two specified configurations. Our contribution is to propose an algorithm which can determine a precise trajectory to link two specified configurations. Moreover, this algorithm has three features: first it can achieve an optimal trajectory ending at the exact final configuration; second it is complete, i.e. it allows to determine an optimal trajectory to link any two configurations; and third it has very low computational cost. The rules for designing this algorithm result from a new geometric reasoning in the switching space, which is another contribution of this paper.