Rahul Sharma (Editor)

Epicycloid

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Epicycloid

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.

Contents

Equations

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 + 2r.

  • 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.

    Proof

    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 )

    References

    Epicycloid Wikipedia
    (Text) CC BY-SA


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