We consider planar navigation for a polygonal, holonomic robot in an obstacle-filled environment in SE(2). To determine the free space, we first represent obstacles as point clouds in the robot configuration space (C). A point-wise Minkowski sum of the robot and obstacle points is then calculated in C using obstacle points and robot convex hull points for varying robot configurations. Using graph search, we obtain a seed path, which is used in our novel method to compute overlapping convex regions for consecutive seed path chords. The resulting regions provide collision-free space useful for finding feasible trajectories that optimize a specified cost functional. The key contribution is the proposed method's ability to easily generate a set of convex, overlapping polytopes that effectively represent the traversable free space. This, in turn, lends itself to (a) efficient computation of optimal paths in and (b) extending these basic ideas to the special Euclidean space SE(2). We provide simulated examples and implement this algorithm on a KUKA youBot omnidirectional base.