In geometry, an epicycloid (also called hypercycloid)[1] 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.
An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form.
If the smaller circle has radius
r
R=kr
\begin{align} &x(\theta)=(R+r)\cos\theta -r\cos\left(
| R+r | |
| r |
\theta\right)\\ &y(\theta)=(R+r)\sin\theta -r\sin\left(
| R+r | |
| r |
\theta\right) \end{align}
\begin{align} &x(\theta)=r(k+1)\cos\theta-r\cos\left((k+1)\theta\right)\\ &y(\theta)=r(k+1)\sin\theta-r\sin\left((k+1)\theta\right). \end{align}
This can be written in a more concise form using complex numbers as[2]
z(\theta)=r\left((k+1)e-ei(k+1)\theta\right)
where
\theta\in[0,2\pi],
r
kr
(Assuming the initial point lies on the larger circle.) When
k
A
s
A=(k+1)(k+2)\pir2,
s=8(k+1)r.
It means that the epicycloid is
| (k+1)(k+2) | |
| k2 |
If
k
If
k
k=p/q
p
| To close the curve and | |
| complete the 1st repeating pattern : | |
| to rotations | |
| to rotations | |
| total rotations of outer rolling circle = rotations |
If
k
R+2r
The distance
\overline{OP}
p
R\leq\overline{OP}\leqR+2r
R
2r
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.[3]
We assume that the position of
p
\alpha
p
\theta
Since there is no sliding between the two cycles, then we have that
\ellR=\ellr
\ellR=\thetaR
\ellr=\alphar
\thetaR=\alphar
\alpha
\theta
\alpha=
| R | |
| r |
\theta
From the figure, we see the position of the point
p
x=\left(R+r\right)\cos\theta-r\cos\left(\theta+\alpha\right)=\left(R+r\right)\cos\theta-r\cos\left(
| R+r | |
| r |
\theta\right)
y=\left(R+r\right)\sin\theta-r\sin\left(\theta+\alpha\right)=\left(R+r\right)\sin\theta-r\sin\left(
| R+r | |
| r |
\theta\right)