Voozh

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The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

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.

Equations

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If the rolling circle has radius 👁 {\displaystyle r}
, and the fixed circle has radius 👁 {\displaystyle R=kr}
, then the parametric equations for the curve can be given by either:

👁 {\displaystyle {\begin{aligned}&x(\theta )=(R+r)\cos \theta \ -r\cos \left({\frac {R+r}{r}}\theta \right)\\&y(\theta )=(R+r)\sin \theta \ -r\sin \left({\frac {R+r}{r}}\theta \right)\end{aligned}}}

or:

👁 {\displaystyle {\begin{aligned}&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{aligned}}}

This can be written in a more concise form using complex numbers as^[2]

👁 {\displaystyle z(\theta )=r\left((k+1)e^{i\theta }-e^{i(k+1)\theta }\right)}

where

the angle 👁 {\displaystyle \theta \in [0,2\pi ],}
the rolling circle has radius 👁 {\displaystyle r}
, and
the fixed circle has radius 👁 {\displaystyle kr}
.

Area and arc length

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Assuming the initial point lies on the larger circle, when 👁 {\displaystyle k}
is a positive integer, the area 👁 {\displaystyle A}
and arc length 👁 {\displaystyle s}
of this epicycloid are

👁 {\displaystyle A=(k+1)(k+2)\pi r^{2},}

👁 {\displaystyle s=8(k+1)r.}

It means that the epicycloid is 👁 {\displaystyle {\frac {(k+1)(k+2)}{k^{2}}}}
larger in area than the original stationary circle.

If 👁 {\displaystyle k}
is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).

If 👁 {\displaystyle k}
is a rational number, say 👁 {\displaystyle k=p/q}
expressed as irreducible fraction, then the curve has 👁 {\displaystyle p}
cusps.

To close the curve and

complete the 1st repeating pattern :

θ = 0 to q rotations

α = 0 to p rotations

total rotations of outer rolling circle = p + q rotations

Count the animation rotations to see p and q

If 👁 {\displaystyle 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 👁 {\displaystyle R+2r}
.

The distance 👁 {\displaystyle {\overline {OP}}}
from the origin to the point 👁 {\displaystyle p}
on the small circle varies up and down as

👁 {\displaystyle R\leq {\overline {OP}}\leq R+2r}

where

👁 {\displaystyle R}
= radius of large circle and
👁 {\displaystyle 2r}
= diameter of small circle .

Epicycloid examples
👁 k = 1; a cardioid

k = 1; a cardioid
👁 k = 2; a nephroid

k = 2; a nephroid
👁 k = 3; a trefoiloid

k = 3; a trefoiloid
👁 k = 4; a quatrefoiloid

k = 4; a quatrefoiloid
👁 k = 2.1 = 21/10

k = 2.1 = 21/10
👁 k = 3.8 = 19/5

k = 3.8 = 19/5
👁 k = 5.5 = 11/2

k = 5.5 = 11/2
👁 k = 7.2 = 36/5

k = 7.2 = 36/5

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]

Proof

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👁 Image

sketch for proof

Assuming that the position of 👁 {\displaystyle p}
is what has to be solved, 👁 {\displaystyle \alpha }
is the angle from the tangential point to the moving point 👁 {\displaystyle p}
, and 👁 {\displaystyle \theta }
is the angle from the starting point to the tangential point.

Since there is no sliding between the two cycles, then

👁 {\displaystyle \ell _{R}=\ell _{r}}

By the definition of angle (which is the rate arc over radius), then

👁 {\displaystyle \ell _{R}=\theta R}

and

👁 {\displaystyle \ell _{r}=\alpha r}
.

From these two conditions, the following identity is obtained

👁 {\displaystyle \theta R=\alpha r}
.

By calculating, the relation between 👁 {\displaystyle \alpha }
and 👁 {\displaystyle \theta }
is obtained, which is

👁 {\displaystyle \alpha ={\frac {R}{r}}\theta }
.

From the figure, the position of the point 👁 {\displaystyle p}
on the small circle is clearly visible.

👁 {\displaystyle x=\left(R+r\right)\cos \theta -r\cos \left(\theta +\alpha \right)=\left(R+r\right)\cos \theta -r\cos \left({\frac {R+r}{r}}\theta \right)}