Epicycloid

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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

If the rolling circle has radius r {\displaystyle r} {\displaystyle r}, and the fixed circle has radius R = k r {\displaystyle R=kr} {\displaystyle 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 ) {\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}}} {\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:

x ( θ ) = r ( k + 1 ) cos ⁡ θ − r cos ⁡ ( ( k + 1 ) θ ) y ( θ ) = r ( k + 1 ) sin ⁡ θ − r sin ⁡ ( ( k + 1 ) θ ) . {\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}}} {\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]

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

where

  • the angle θ ∈ [ 0 , 2 π ] , {\displaystyle \theta \in [0,2\pi ],} {\displaystyle \theta \in [0,2\pi ],}
  • the rolling circle has radius r {\displaystyle r} {\displaystyle r}, and
  • the fixed circle has radius k r {\displaystyle kr} {\displaystyle kr}.

Area and arc length

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

A = ( k + 1 ) ( k + 2 ) π r 2 , {\displaystyle A=(k+1)(k+2)\pi r^{2},} {\displaystyle A=(k+1)(k+2)\pi r^{2},}
s = 8 ( k + 1 ) r . {\displaystyle s=8(k+1)r.} {\displaystyle s=8(k+1)r.}

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

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

If k {\displaystyle k} {\displaystyle k} is a rational number, say k = p / q {\displaystyle k=p/q} {\displaystyle k=p/q} expressed as irreducible fraction, then the curve has p {\displaystyle p} {\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 k {\displaystyle k} {\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 R + 2 r {\displaystyle R+2r} {\displaystyle R+2r}.

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

R ≤ O P ¯ ≤ R + 2 r {\displaystyle R\leq {\overline {OP}}\leq R+2r} {\displaystyle R\leq {\overline {OP}}\leq R+2r}

where

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

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

sketch for proof

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

Since there is no sliding between the two cycles, then

ℓ R = ℓ r {\displaystyle \ell _{R}=\ell _{r}} {\displaystyle \ell _{R}=\ell _{r}}

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

ℓ R = θ R {\displaystyle \ell _{R}=\theta R} {\displaystyle \ell _{R}=\theta R}

and

ℓ r = α r {\displaystyle \ell _{r}=\alpha r} {\displaystyle \ell _{r}=\alpha r}.

From these two conditions, the following identity is obtained

θ R = α r {\displaystyle \theta R=\alpha r} {\displaystyle \theta R=\alpha r}.

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

α = R r θ {\displaystyle \alpha ={\frac {R}{r}}\theta } {\displaystyle \alpha ={\frac {R}{r}}\theta }.

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

x = ( R + r ) cos ⁡ θ − r cos ⁡ ( θ + α ) = ( R + r ) cos ⁡ θ − r cos ⁡ ( R + r r θ ) {\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)} {\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)}
y = ( R + r ) sin ⁡ θ − r sin ⁡ ( θ + α ) = ( R + r ) sin ⁡ θ − r sin ⁡ ( R + r r θ ) {\displaystyle y=\left(R+r\right)\sin \theta -r\sin \left(\theta +\alpha \right)=\left(R+r\right)\sin \theta -r\sin \left({\frac {R+r}{r}}\theta \right)} {\displaystyle y=\left(R+r\right)\sin \theta -r\sin \left(\theta +\alpha \right)=\left(R+r\right)\sin \theta -r\sin \left({\frac {R+r}{r}}\theta \right)}

See also

Animated gif with turtle in MSWLogo (Cardioid)[4]

References