Computing globally optimal motion plans requires exploring the configuration space to identify reachable free space regions as well as refining understanding of already explored regions to find better paths. We present the rapidly-exploring roadmap (RRM), a new method for single-query optimal motion planning that allows the user to explicitly consider the trade-off between exploration and refinement. RRM initially explores the configuration space like a rapidly exploring random tree (RRT). Once a path is found, RRM uses a user-specified parameter to weigh whether to explore further or to refine the explored space by adding edges to the current roadmap to find higher quality paths in the explored space. Unlike prior methods, RRM does not focus solely on exploration or refine prematurely. We demonstrate the performance of RRM and the trade-off between exploration and refinement using two examples, a point robot moving in a plane and a concentric tube robot capable of following curved trajectories inside patient anatomy for minimally invasive medical procedures.