Equation of Cycloid

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

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

Let $$\left({x, y}\right)$$ be the coordinates of this point as it travels over the plane.

The point $$\left({x, y}\right)$$ is described by the equations:
 * $$x = a \left({\theta - \sin \theta}\right)$$
 * $$y = a \left({1 - \cos \theta}\right)$$

The curve this traces out is called the cycloid.

Arc
An arc of a cycloid is defined as being a part of the cycloid traced out when $$\theta$$ goes from $$2 n \pi$$ to $$2 \left({n + 1}\right) \pi$$ for $$n \in \Z$$.

Cusp
Any of the points of the cycloid where $$\theta = 2 n \pi$$ where $$n \in \Z$$ is a cusp.

Proof

 * Cycloid.png

Let the circle have rolled so that the radius to the point $$\left({x, y}\right)$$ is at angle $$\theta$$ to the vertical.

The center of the circle is at $$\left({a \theta, a}\right)$$.

Then it follows from the definition of sine and cosine that:
 * $$x = a \theta - a \sin \theta$$
 * $$y = a - a \cos \theta$$

whence the result.