In this paper we present an algorithm to compute the time-optimal trajectory from one point in multi-dimensional Cartesian space to another with arbitrary start and target velocity vectors and with limited acceleration norm. Based on Pontryagin's minimum principle, we devise the conditions that have to be fulfilled and formulate a cost function for the time-optimal solution. The resulting boundary value problem is an optimization problem, which is exemplarily solved using the Nelder-Mead method. Exemplary curves are compared against cubic polynomials and trajectories generated by the Reflexxes Motion Libraries.