A rotary piston machine includes a casing and a rotor assembly, each with a hypotrochoidal. The rotor assembly includes apices that conform to the hypotrochoidal profile of the casing, and the relative motion of the casing and the rotor assembly create variable volumes. The variable volume allows the rotary piston machine to function as a compressor, an expander, an internal combustion engine, a positive displacement pump, or a fluid-driven motor.