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Hypocycloid ( parametric equation Y- 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

 y_θ Y-coordinate (m) r The radius of the small circle (m) k Number of times the r (dimensionless) θ Parameter ( 0 - 2π ) (radians)