Figure 2: (Top) A simple coordinate transformation that compresses a section of the $x$ axis. As a result, rays follow a distorted trajectory, shown on the top right, but emerge from the compressed region travelling in exactly the same direction with the same phase as before. (Bottom) Requiring that a ray pass through the compressed region with the same phase change as through uncompressed space enables us to predict the metamaterial properties that would realize this trajectory for a ray.