In this work, we presented a method of defining convex corridors that specify free space and provide linear constraints required for finding the optimal trajectory for polygonal robots in cluttered environments. By representing obstacles as point clouds, this method combines well with off the shelf range sensors. Obstacle points are represented in the robot configuration space and by using a pointwise Minkowski sum, the occupied regions are defined for every orientation of the polygonal robot. To perform covering of SE(2) in $\mathbb{R}3$, obstacles are then repeated between $\theta \u2208(\u22123\pi ,3\pi )$. Using a graph search or sample-based planner (in results we use a graph search), we are able to obtain an initial seed path. By using obstacle points to define hyperplanes, we define polytopes that enclose the chords of the seed path and inherently overlap, producing a chain of overlapping polytopes and hence convex regions. These chains of overlapping convex regions then allow us to perform a quadratic program and obtain a trajectory that minimizes a desired objective such as velocity, which would produce smooth, short trajectories, or any higher derivative desired the selection of which is driven by robot dynamics, kinematics, or power constraints. Finally, we compare our method to leading methods and show that while our method is competitive in free space computation, our method best serves the goal of quickly finding consecutive overlapping convex regions. Advantages of this methodology are that first, adaptable polytopes can be constructed that maximize the representation of free space within the configuration space of a robot and this does not require an optimization step to determine the polytopes as in Ref. [11]. Second, this overall methodology is scalable for many obstacles as opposed to MIQP methods. Disadvantages of the proposed method are that first, as there is no direct relation between the translational and rotational units, the selection of the scaling between the *x-*axis, *y-*axis, and *θ-*axis must be selected based on performance. Second, without prior knowledge of the arrangement of obstacles and the seed path, a naive approach is to rank the obstacle points for each chord; this can become computationally intensive depending on the number of obstacles points. In future work, we plan to extend this method to SE(3), by representing SE(3) in higher dimensions that allow for linear constraints. Also, in future work, we will investigate ways to improve computational performance of the method by considering autonomous ways to determine if re-ranking of points is required from chord to chord to increase computational efficiency.