(W. W.; X.)
EPICYCLE (Gr. ἐπί, upon, and κύκλος, circle), in ancient astronomy, a small circle the centre of which describes a larger one. It was especially used to represent geometrically the periodic apparent retrograde motion of the outer planets, Mars, Jupiter and Saturn, which we now know to be due to the annual revolution of the earth around the sun, but which in the Ptolemaic astronomy were taken to be real.
EPICYCLOID, the curve traced out by a point on the circumference of a circle rolling externally on another circle. If the moving circle rolls internally on the fixed circle, a point on the circumference describes a “hypocycloid” (from ὑπό, under). The locus of any other carried point is an “epitrochoid” when the circle rolls externally, and a “hypotrochoid” when the circle rolls internally. The epicycloid was so named by Ole Römer in 1674, who also demonstrated that cog-wheels having epicycloidal teeth revolved with minimum friction (see [Mechanics]: Applied); this was also proved by Girard Desargues, Philippe de la Hire and Charles Stephen Louis Camus. Epicycloids also received attention at the hands of Edmund Halley, Sir Isaac Newton and others; spherical epicycloids, in which the moving circle is inclined at a constant angle to the plane of the fixed circle, were studied by the Bernoullis, Pierre Louis M. de Maupertuis, François Nicole, Alexis Claude Clairault and others.
In the annexed figure, there are shown various examples of the curves named above, when the radii of the rolling and fixed circles are in the ratio of 1 to 3. Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself. In the particular case when the radii are in the ratio of 1 to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference. Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve c. It is interesting to compare the forms of these curves with the three forms of the cycloid (q.v.). The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e. The epicycloid shown is termed the “three-cusped epicycloid” or the “epicycloid of Cremona.”
The cartesian equation to the epicycloid assumes the form
x = (a + b) cosθ − b cos(a + b/b)θ, y = (a + b) sinθ − b sin(a + b/b)θ,
when the centre of the fixed circle is the origin, and the axis of x passes through the initial point of the curve (i.e. the original position of the moving point on the fixed circle), a and b being the radii of the fixed and rolling circles, and θ the angle through which the line joining the centres of the two circles has passed. It may be shown that if the distance of the carried point from the centre of the rolling circle be mb, the equation to the epitrochoid is