Epicycloid, in Geometry, a curve generated by a point in one circle, which revolves about the circumference of another circle. They are distinguished into exterior and interior epicycloids. An exterior epicycloid is formed by revolution of a generating circle upon the convexity of the quiescent circle; and an interior epicycloid is formed by its revolution along the concavity of the quiescent circle. A curious property of the latter description of epicycloid is, that when the diameter of the generating circle is equal to the radius of the quiescent circle, the epicycloid described is a right line equal and coincident with the diameter of the latter. Mr. Murray, of Leeds, applied this property to obtain a rotatory motion of the fly-wheel of a steam engine directly from the piston rod, without the intervention of a connecting rod. Epicycloids have been recommended by many eminent mathematicians as the best curve for the teeth of wheels; but in practice they are usually formed of circular arcs, as these work very well, and are easier of construction.