Relation between Equations for Hypocycloid and Epicycloid
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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:
- $x = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}$
- $y = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}$
Proof
\(\ds x\) | \(=\) | \(\ds \paren {a - \paren {-b} } \cos \theta + \paren {-b} \map \cos {\paren {\dfrac {a - \paren {-b} } {\paren {-b} } } \theta}\) | putting $-b$ for $b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + b} \cos \theta - b \map \cos {-\paren {\dfrac {a + b} b} \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}\) | Cosine Function is Even |
\(\ds y\) | \(=\) | \(\ds \paren {a - \paren {-b} } \sin \theta - \paren {-b} \map \sin {\paren {\dfrac {a - \paren {-b} } {\paren {-b} } } \theta}\) | putting $-b$ for $b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + b} \sin \theta + b \map \sin {-\paren {\dfrac {a + b} b} \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}\) | Sine Function is Odd |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $13$