In geometry, an **epicycloid** is a plane curve produced by tracing the path of a chosen point on the circumference of a circle — called an *epicycle* — which rolls without slipping around a fixed circle. It is a particular kind of roulette.

If the smaller circle has radius *r*, and the larger circle has radius *R* = *kr*, then the parametric equations for the curve can be given by either:

x
(
θ
)
=
(
R
+
r
)
cos
θ
−
r
cos
(
R
+
r
r
θ
)
y
(
θ
)
=
(
R
+
r
)
sin
θ
−
r
sin
(
R
+
r
r
θ
)
,
or:

x
(
θ
)
=
r
(
k
+
1
)
cos
θ
−
r
cos
(
(
k
+
1
)
θ
)
y
(
θ
)
=
r
(
k
+
1
)
sin
θ
−
r
sin
(
(
k
+
1
)
θ
)
.
If *k* is an integer, then the curve is closed, and has *k* cusps (i.e., sharp corners, where the curve is not differentiable).

If *k* is a rational number, say *k=p/q* expressed in simplest terms, then the curve has *p* cusps.

If *k* is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius *R* + 2*r*.

Epicycloid examples
The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid, two cusps is a nephroid.

An epicycloid and its evolute are similar.

We assume that the position of
p
is what we want to solve,
α
is the radian from the tangential point to the moving point
p
, and
θ
is the radian from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that

ℓ
R
=
ℓ
r
By the definition of radian (which is the rate arc over radius), then we have that

ℓ
R
=
θ
R
,
ℓ
r
=
α
r
From these two conditions, we get the identity

θ
R
=
α
r
By calculating, we get the relation between
α
and
θ
, which is

α
=
R
r
θ
From the figure, we see the position of the point
p
clearly.

x
=
(
R
+
r
)
cos
θ
−
r
cos
(
θ
+
α
)
=
(
R
+
r
)
cos
θ
−
r
cos
(
R
+
r
r
θ
)
y
=
(
R
+
r
)
sin
θ
−
r
sin
(
θ
+
α
)
=
(
R
+
r
)
sin
θ
−
r
sin
(
R
+
r
r
θ
)