Epicycloid
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 , 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
x = (a + b) cosθ − mb cos(a + b/b)θ, y = (a + b) sinθ − mb sin(a + b/b)θ,
The equations to the hypocycloid and its corresponding trochoidal curves are derived from the two preceding equations by changing the sign of b. Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of (a ± b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii. These propositions may be derived from the formulae given above, or proved directly by purely geometrical methods.
The tangential polar equation to the epicycloid, as given above, is p = (a + 2b) sin(a/a + 2b)ψ, while the intrinsic equation is s = 4(b/a)(a + b) cos(a/a + 2b)ψ and the pedal equation is r = a + (4b·a + b)p/(a + 2b). Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin Bψ or p = A cos Bψ, s = A sin Bψ or s = A cos Bψ, or r = A + Bp, the constants A and B being readily determined by the above considerations.
If the radius of the rolling circle be one-half of the fixed circle, the hypocycloid becomes a diameter of this circle; this may be confirmed from the equation to the hypocycloid. If the ratio of the radii be as 1 to 4, we obtain the four-cusped hypocycloid, which has the simple cartesian equation x [2/3] + y2/3 = a2/3. This curve is the envelope of a line of constant length, which moves so that its extremities are always on two fixed lines at right angles to each other, i.e. of the line x/α + y/β = 1, with the condition α + β = 1/a, a constant. The epicycloid when the radii of the circles are equal is the cardioid (q.v.), and the corresponding trochoidal curves are limaçons (q.v.). Epicycloids are also examples of certain caustics (q.v.).
For the methods of determining the formulae and results stated above see J. Edwards, Differential Calculus, and for geometrical constructions see T.H. Eagles, Plane Curves.
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
