This article addresses the optimal (minimal input energy) design of scan trajectories, which is important in applications such as the imaging and manipulation of nano-scale surface phenomena using scanning tunneling microscopes (STM), MEMS-based micro-scanners, quick-return mechanisms and cams used in manufacturing, and general repeating processes. The contribution of this article is the systematic solution of the optimal scan-trajectory design problem. As opposed to existing techniques that require pre-specification of the desired output trajectory (such prespecifications can be ad hoc), the optimal output trajectory is found as a result of the proposed input-energy minimization approach. In this sense, the proposed approach leads to a systematic solution of the optimal output-trajectory-design problem. The proposed optimal scanning method is applied to an experimental scanning tunneling microscope; simulation and experimental results are presented to illustrate the efficacy of the proposed approach to design optimal scan-trajectories.