Epicycloid

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations

(1) (2)

A polar equation can be derived by computing

(3) (4)

so

(5)

But

(6)

so

(7) (8)

Note that is the parameter here, not the polar angle. The polar angle from the center is

(9)

To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.

(10) (11) (12) (13)

so

(14) (15)

An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.

Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps times as large. After traveling around the circle once, the envelope of an -cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have torsion

(16)

and satisfy

(17)

where is the radius of curvature ( ).