Cusps of Hypocycloid from Irrational Ratio of Circle Radii

Theorem

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

Let $k = \dfrac a b$ be an irrational number.

Then $H$ has an infinite number of cusps, which are evenly and densely distributed around the circumference of $C_2$.

Proof

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Sources

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