As mobile robots operate with limited resources in large obstructed environments, their success is dependent on how efficiently they move while they avoid collision with obstacles and other robots. Therefore, planning optimal motions and devising optimal coordination strategies are two important and challenging fundamental problems in mobile robotics, which have received significant attention in the last couple of decades. Both of those problems can be reduced to shortest path, or equivalently geodesic, problems in appropriate geometric settings. Geodesic problems have been studied in two disciplines: (1) optimal control theory, and (2) computational geometry. Optimal control theory has focused on the differential constraints of robotic systems, while computational geometry has focused on shortest path problems in an environment with obstacles. We introduce a unified approach that is inspired by main results in both disciplines. In this book, we demonstrate our technique, which combines the celebrated Pontryagin's Maximum Principle from optimal control theory with visibility graph methods from computational geometry.