Relation between Equations for Hypocycloid and Epicycloid

Theorem
Consider the hypocycloid defined by the equations:
 * $x = \left({a - b}\right) \cos \theta + b \cos \left({\left({\dfrac {a - b} b}\right) \theta}\right)$
 * $y = \left({a - b}\right) \sin \theta - b \sin \left({\left({\dfrac {a - b} b}\right) \theta}\right)$

By replacing $b$ with $-b$, this converts to the equations which define an epicycloid:
 * $x = \left({a + b}\right) \cos \theta - b \cos \left({\left({\dfrac {a + b} b}\right) \theta}\right)$
 * $y = \left({a + b}\right) \sin \theta - b \sin \left({\left({\dfrac {a + b} b}\right) \theta}\right)$