Epicycloid


In geometry, an epicycloid or hypercycloid 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:
or:
If k is a positive integer, then the curve is closed, and has k cusps.
If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has 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 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.
The distance OP from origin to varies up and down as
R <= OP <=
R = radius of large circle and
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.

Proof

We assume that the position of is what we want to solve, is the radian from the tangential point to the moving point, and is the radian from the starting point to the tangential point.
Since there is no sliding between the two cycles, then we have that
By the definition of radian, then we have that
From these two conditions, we get the identity
By calculating, we get the relation between and, which is
From the figure, we see the position of the point on the small circle clearly.