Equation of Cycloid

Theorem

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

Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.

Consider the cycloid 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:

$x = a \paren {\theta - \sin \theta}$

$y = a \paren {1 - \cos \theta}$

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

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

From the diagram above, we see:

The $x$-coordinate is to the left of the center of the circle by $a \sin \theta$

The $y$-coordinate is below the center of the circle by $a \cos \theta$

It follows that:

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

$y = a - a \cos \theta$

whence the result.

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Cycloid: $11.5$
1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cycloid
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cycloid
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cycloid
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Cycloid: $9.5.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cycloid