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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.




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.


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

  • 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.