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 (i.e., sharp corners).
If "k" is a rational number, say "k = p / q" expressed as irreducible fraction, then the curve has "p" cusps. 
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" + 2"r". 
The distance OP from (x=0,y=0) origin to (the point formula_5 on the small circle) varies up and down as 
R <= OP <= (R + 2r) 
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 formula_5 is what we want to solve, formula_7 is the radian from the tangential point to the moving point formula_5, and formula_9 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 (which is the rate arc over radius), then we have that
From these two conditions, we get the identity
By calculating, we get the relation between formula_7 and formula_9, which is
From the figure, we see the position of the point formula_5 on the small circle clearly.