Epicycloid

Plane curve traced by a point on a circle rolled around another circle
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 smaller circle has radius r {\displaystyle r} , and the larger circle has radius R = k r {\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}}}

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}}}

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)}

where

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

Area

(Assuming the initial point lies on the larger circle.) When k {\displaystyle k} is a positive integer, the area of this epicycloid is

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

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

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

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

The distance O P ¯ {\displaystyle {\overline {OP}}} from the origin to the point 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}

where

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

sketch for proof

We assume that the position of p {\displaystyle p} is what we want to solve, α {\displaystyle \alpha } is the angle from the tangential point to the moving point p {\displaystyle p} , and θ {\displaystyle \theta } is the angle 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 {\displaystyle \ell _{R}=\ell _{r}}

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

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

and

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

From these two conditions, we get the identity

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

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

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

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

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)}
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)}

See also

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

References

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.
  1. ^ [1]
  2. ^ Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye
  3. ^ Epicycloid Evolute - from Wolfram MathWorld
  4. ^ Pietrocola, Giorgio (2005). "Tartapelago". Maecla.

External links