Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii

Theorem

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

Let $a = n b$ where $n$ is an integer.

Then $H$ has $n$ cusps.

Proof

The length of the arc of $C_2$ between two adjacent cusps of $H$ is $2 \pi b$.

The total length of the circumference of $C_1$ is $2 \pi a$.

Thus the total number of cusps of $H$ is:

$\dfrac {2 \pi a} {2 \pi b} = \dfrac {2 \pi n b} {2 \pi b} = n$

$\blacksquare$

Sources

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