How Are Parametric Equations for a Hypocycloid Derived?

[conquerer7]

Homework Statement

Find parametric equations for the hypocycloid that is produced when we track a point on a circle of radius 1/4 that rotates inside a circle of radius 1. Show that these equations are equivalent to (sin^3 t, cos^3 t).

Homework Equations

N/A

The Attempt at a Solution

I have the intended solution except for one step. The book claims that the small circle rotates 3 times every time it rotates once inside the big circle. That makes sense because of how the points line up... But the circumference of the big circle is 4 times that of the small one, and the surfaces are always touching, so why isn't it 4 times?

 

[NascentOxygen]

If it generates an astroid, then radii ratios are 1:4 http://en.wikipedia.org/wiki/Hypocycloid"