Number of Cusps of Hypocycloid from Rational Ratio of Circle Radii

Theorem

Consider the hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$.

Let $k = \dfrac a b$ be a rational number.

Let $k$ be expressed in canonical form:

$k = \dfrac p q$

where $p$ and $q$ are coprime.

Then $H$ has $p$ cusps.

Proof

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Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid