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Hypocycloid

Hypocycloid is a special kind of hypotrochoid. Hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

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:

If k is an integer, then the curve is closed, and has k cusps, If k is a rational number then the curve has p cusps (where k = p/q), 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.

A hypocycloid curve with 3 cusps is known as a deltoid, with 4 cusps is known as an astroid.

To draw function Hypocycloid use the folowing script:

# Example
# 2D. Hypocycloid

# z must be zero
z = 0

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

# Constant
r = 0.8
k = 10

# Calculations
x = r * k * cos(t) + r * cos(k * t)
y = r * k * sin(t) - r * sin(k * t)