Abstract: This invention presents a four-region catadioptric tile. The four-region catadioptric tile includes a first surface, a second surface, and a filler material layer between the first and second surfaces. The four regions includes a first region wherein the first surface is convex and the second surface is convex, a second region wherein the first surface is convex and the second surface is concave, a third region wherein the first surface is concave and the second surface is convex, and a third region wherein the first surface is concave and the second surface is concave. The surface morphology of each of the four-region catadioptric tiles can be repeated across an entire tessellated projection screen or an angularly dependent retroreflector.