Equation of Epicycloid

Theorem

Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$.

Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.

Let $H$ be the epicycloid traced out by the point $P$.

Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.

The point $P = \tuple {x, y}$ is described by the equations:

Proof

Let $C_1$ have rolled so that the line $OC$ through the radii of $C_1$ and $C_2$ is at angle $\theta$ to the $x$-axis.

Let $C_1$ have turned through an angle $\phi$ to reach that point.

By definition of sine and cosine, $P = \tuple {x, y}$ is defined by:

The arc of $C_1$ between $P$ and $B$ is the same as the arc of $C_2$ between $A$ and $B$.

Thus by Arc Length of Sector:

$a \theta = b \phi$

Thus:

$\phi + \theta = \paren {\dfrac {a + b} b} \theta$

whence the result.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Epicycloid: $11.18$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $12$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): epicycloid
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): epicycloid
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Epicycloid: $9.18.$