This paper presents and proves an equation for the time horizon of symmetric trajectories with zero boundary conditions and bounded derivatives of arbitrary order. This equation holds regardless of the number of phases comprising the associated motion, which avoids case distinctions in calculations. Application examples of motions with minimum time, minimum velocity, and minimum acceleration are discussed. Furthermore, an algorithm is derived that reduces the time minimization problem as well as the problem of minimizing velocity, acceleration,... to solving a system of equations. This algorithm avoids nested case distinctions and complex optimizations.