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Epicycloid

Epicycloid is a plane curve produced by tracing the path of a chosen point of a circle (called epicycle) which rolls without slipping around a fixed circle. It is a particular kind of curve.

If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the Epicycloid can be given by either:

If k is an integer, then the curve is closed, and has k cusps, if k is a rational number (k=p/q), then the curve has p cusps, if k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R + 2r.

You can plot function online using the folowing script for Epicycloid:

# Example
# 2D. Epicycloid

# z must be zero
z = 0

# t-parameter
tmin = 0
tmax = 2*pi
tgrid = 400

# Constant
a = 7.0
b = 1.0
d = b

k = a + b
m = (a + b) / b

# Calculations
x = k * cos(t) - d * cos(m * t)
y = k * sin(t) - d * sin(m * t)