# 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**.

### Graph

### 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.

This is a single cusp of the first kind.

### Generating Circle

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

## Also known as

The **cycloid** is sometimes seen referred to as a **common cycloid**, so as to distinguish it from:

- the
**prolate trochoid**, also known as the**extended cycloid** - the
**curtate trochoid**, also known as the**contracted cycloid**.

## Also see

- Definition:Trochoid, of which a
**cycloid**is a special case

- 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 AcadÃ©mie 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

- 1926: Henry Ernest Dudeney:
*Modern Puzzles*... (previous) ... (next): Solutions: $209$. -- A Wheel Fallacy - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1968: Henry Ernest Dudeney:
*536 Puzzles & Curious Problems*... (previous) ... (next): Answers: $293$. A Wheel Fallacy - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Cycloid: $11.7$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 11$: Special Plane Curves: Trochoid: $11.20$ - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**cycloid** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.12$: Mersenne ($\text {1588}$ – $\text {1648}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.15$: Torricelli ($\text {1608}$ – $\text {1647}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.21$: The 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** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cycloid**

- Weisstein, Eric W. "Cycloid." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Cycloid.html