(math) hypocycloid = fixed point on a circle rolls within a larger circle
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 k is a rational number, say k = p/q expressed in simplest terms, 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.
 k=3 - a deltoid |
 k=4 - an astroid |
 k=5 |
 k=6 |
 k=2.1 |
 k=3.8 |
 k=5.5 |
 k=7.2 |
The hypocycloid is a special kind of hypotrochoid, which are a particular kind of roulette.
A hypocycloid with three cusps is known as a deltoid.
A hypocycloid curve with four cusps is known as an astroid.
The red curve is a hypocycloid traced as the smaller black circle rolls around inside the larger blue circle (parameters are R=3.0, r=1.0, and so k=3), giving a deltoid.R=k ??waht if
R=3.0, r=2.0, and so k=??? ... parabolic?
Derived curves
The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself.
The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve.
The isoptic of a hypocycloid is a hypocycloid.
