Hypocycloid

In geometry, a 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:

or:

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable). Specially for k=2 the curve is a straight line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids and their applications to high-speed printing.

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.

Each hypocycloid (for any value of r) is a for the gravitational potential inside a homogeneous sphere of radius R.

k=3 — a deltoid

k=4 — an astroid

k=5

k=6

k=2.1 = 21/10

k=3.8 = 19/5

k=5.5 = 11/2

k=7.2 = 36/5

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 hypocycloid with two cusps is a degenerate but still very interesting case, known as the Tusi couple.

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Wikipedia