Abstract: In a rotary-piston engine, the contour of a normal section of the piston is either a composite trochoid or a translated composite trochoid. Each vertex of the normal section is either a generating point of the trochoid or has the form of a circular arc whose center is the generating point of the trochoid and whose radius is a fixed distance. Thus all the edges of the piston slide continuously on the inside face of engine cavity. The piston describes a planetary motion about a point at the same time as it revolves around an origin. Thus the volume of the working chambers changes. The opening and closing of connecting ducts for gas exchange are controlled. The resulting engine has only rotating moving parts, and they are in complete balance. The engine has no constrictions on the minor axis of the contour. Thus the movement of combustion gases is not impeded near top dead center. The engine has a high compression ratio.