Relation between Equations for Hypocycloid and Epicycloid

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

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