Hypocycloid ( parametric equation X- coordinate)

Description

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 shape of the curve is depended on the value of k. If k is an integer, then the curve is closed, and has k cusps. Specially for k=2 the curve is a straight line and the circles are called Cardano circles.
If k is a rational number,say 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.
Each hypocycloid (for any value of r) is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius R.

Related formulas

Variables

x_θX-coordinate (m)
rThe radius of the small circle (m)
kNumber of times the r (dimensionless)
θParameter ( 0 - 2π) (radians)