Equation of Trochoid

Theorem

Consider a circle $C$ of radius $a$ rolling without slipping along the x-axis of a cartesian plane.

Consider the point $P$ on on the line of a radius of $C$ at a distance $b$ from the center of $C$.

Let $P$ be on the y-axis when the center of $C$ is also on the y-axis.

Consider the trochoid 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:

$x = a \theta - b \sin \theta$

$y = a - b \cos \theta$

Let $C$ have rolled so that the radius to the point $P = \tuple {x, y}$ is at angle $\theta$ to the vertical.

The center of $C$ is at $\tuple {a \theta, a}$.

Then it follows from the definition of sine and cosine that:

$x = a \theta - b \sin \theta$

$y = a - b \cos \theta$

whence the result.

$\blacksquare$