**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

*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 12(*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* + 2*b*) sin(*a*/*a* + 2*b*)ψ, while the intrinsic equation is
*s* = 4(*b*/*a*)(*a* + *b*) cos(*a*/*a* + 2*b*)ψ and the pedal equation is
*r* ^{2} = *a*^{2} + (4*b*·*a* + *b*)*p*^{2}/(*a* + 2*b*)^{2}. 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* ^{2} = A + B*p*^{2}, 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} + *y*^{2/3} = *a*2/3. 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^{2} + β^{2} = 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*.