 # Epicycloid

Updated on 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.

## 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

R = r

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

R = θ R , 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

Similar Topics