Relation between Equations for Hypocycloid and Epicycloid

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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:

$x = \left({a + b}\right) \cos \theta - b \cos \left({\left({\dfrac {a + b} b}\right) \theta}\right)$
$y = \left({a + b}\right) \sin \theta - b \sin \left({\left({\dfrac {a + b} b}\right) \theta}\right)$


Proof

\(\displaystyle x\) \(=\) \(\displaystyle \left({a - \left({-b}\right)}\right) \cos \theta + \left({-b}\right) \cos \left({\left({\dfrac {a - \left({-b}\right)} {\left({-b}\right)} }\right) \theta}\right)\) putting $-b$ for $b$
\(\displaystyle \) \(=\) \(\displaystyle \left({a + b}\right) \cos \theta - b \cos \left({-\left({\dfrac {a + b} b}\right) \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({a + b}\right) \cos \theta - b \cos \left({\left({\dfrac {a + b} b}\right) \theta}\right)\) Cosine Function is Even


\(\displaystyle y\) \(=\) \(\displaystyle \left({a - \left({-b}\right)}\right) \sin \theta - \left({-b}\right) \sin \left({\left({\dfrac {a - \left({-b}\right)} {\left({-b}\right)} }\right) \theta}\right)\) putting $-b$ for $b$
\(\displaystyle \) \(=\) \(\displaystyle \left({a + b}\right) \sin \theta + b \sin \left({-\left({\dfrac {a + b} b}\right) \theta}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({a + b}\right) \sin \theta - b \sin \left({\left({\dfrac {a + b} b}\right) \theta}\right)\) Sine Function is Odd

$\blacksquare$


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