# Definition:Cycloid

## Curve

Consider a circle rolling without slipping along a straight line.

Consider a point $P$ on the circumference of this circle.

The curve traced out by $P$ is called a **cycloid**.

### Arc

An **arc of the cycloid** is defined as being a part of the cycloid traced out by $P$ between the points where it meets the straight line.

### Cusp

A **cusp of the cycloid** is defined as a point where the cycloid meets the straight line.

### Generating Circle

The circle which rolls along the straight line is called the **generating circle** of the cycloid.

## Also see

- Equation of Cycloid
- Length of Arc of Cycloid
- Area under Arc of Cycloid
- Results about
**cycloids**can be found here.

## Historical Note

Some sources suggest that the **cycloid** may first have been investigated by Charles de Bouvelles, in the course of developing a technique of Squaring the Circle.

Galileo subsequently raised interest in it in the early $1600$s.

He named the curve, but seems not to have actually discovered any of its interesting properties.

He passed on his interest to Marin Mersenne, among others.

Mersenne in turn suggested it to Descartes and others of his friends in the Académie Parisienne as an object worth investigating.

Subsequently it was studied in detail over the next couple of centuries by such as Roberval, Torricelli, Pascal, Huygens, Johann Bernoulli, Newton, Leibniz, Christopher Wren, Euler, Abel and many others.

For various fanciful reasons, the cycloid has been called the **Helen of Geometry**.

## Sources

- Weisstein, Eric W. "Cycloid." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Cycloid.html - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.6$: The Brachistochrone. Fermat and the Bernoullis - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.12$: Mersenne ($1588$ – $1648$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.15$: Torricelli ($1608$ – $1647$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid