Many applications in MRI such as accelerated receive and transmit sequences require the synthesis of nonuniform 3-D gradient trajectories. Several methods have been proposed to design these gradient trajectories in a time-optimal manner, subject to hardware specific gradient magnitude and slew rate constraints. In this work a novel method is derived that designs time-optimal trajectories, solely based on a set of arbitrarily chosen control points in k-space. In particular, no path constraint is required for the k-space trajectory. It is shown that the above problem can be formulated as a constrained optimization problem. The fact that the objective function is derived in an analytic manner allows for designing time-optimal 3-D gradient trajectories within only few seconds without any significant numerical instabilities. The utilization of the shape of the trajectory--serving as a degree of freedom--results in significantly accelerated trajectories compared to current standard methods. This is proven in an extensive evaluation of the proposed method and in comparison with what can be considered the current Gold Standard method. The proposed Gradient Basis Function method provides significant benefits over current standard methods in terms of the duration of the trajectory (in average 9.2% acceleration), computation time (acceleration by at least 25% up to factors of 100), and robustness (no significant numerical instabilities).