Epicycloid

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.

An epicycloid with a minor radius (R2) of 0 is a circle.

If the smaller circle has radius

, and the larger circle has radius

, then the parametric equations for the curve can be given by either: or: This can be written in a more concise form using complex numbers as[2] where (Assuming the initial point lies on the larger circle.)

is a positive integer, the area

larger in area than the original stationary circle.

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

expressed as irreducible fraction, then the curve has

Count the animation rotations to see p and q If

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

on the small circle varies up and down as where 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.

Since there is no sliding between the two cycles, then we have that By the definition of angle (which is the rate arc over radius), then we have that and From these two conditions, we get the identity By calculating, we get the relation between

, which is From the figure, we see the position of the point