Range-only localization has applications as diverse as underwater navigation, drone tracking and indoor localization. While the theoretical foundations of lateration---range-only localization for static points---are well understood, there is a lack of understanding when it comes to localizing a moving device. As most interesting applications in robotics involve moving objects, we study the theory of trajectory recovery. This problem has received a lot of attention; however, state-of-the-art methods are of a probabilistic or heuristic nature and are not well suited for guaranteeing trajectory recovery. In this paper, we pose trajectory recovery as a quadratic problem and show that we can relax it to a linear form, which admits a closed-form solution. We provide necessary and sufficient recovery conditions and in particular show that trajectory recovery can be guaranteed when the number of measurements is proportional to the trajectory complexity. Finally, we apply our reconstruction algorithm to simulated and real-world data.