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Field: Geometry
Image Created By: Wolfram MathWorld
Website: Wolfram MathWorld


Four different roulettes formed by rolling four different shapes through one fixed point.

Basic Description

Suppose you see a nickel rolling on the sidewalk. Imagine a pen traced the path path of one fixed point on the coin as it rolled. A curve would be created. This curve is called a roulette. The example is depicted below:

Cycloid animated.gif     

However, a roulette is not restricted to straight lines and circles. The rolling curve can range from a line to a parabola to a decagon. Similarly, the surface on which this curve rolls on does not have to be a line. It can be a parabola as well, or a circle, among many others. There are a few restrictions that apply:

  • The curve that is not rolling must remain fixed.
  • The point on the rolling curve must remain fixed.
  • Both curves must be differentiable.
  • The curves must be tangent at one point at all times.

There are numerous cases of different roulettes; the most common ones have been named. A few of these have been listed below:


A trochoid is any roulette in which the fixed curve is a line and the rolling curve is a circle.When the pole is on the edge of the rolling circle, then the roulette is called a cycloid. If the pole is inside the rolling circle, then the curve produced is called a curtate cycloid. If the pole is on the outside of the rolling circle, then the curve produced is called a prolate cycloid.


A hypotrochoid is any roulette in which both the rolling curve and the fixed curve are circles and the rolling circle is on the INSIDE of the fixed circle. The pole can be on the inside , outside or edge of the rolling circle.

A roulette is a curve traced by one single point on a shape as it rolls without slipping along another shape, curve, or line.

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Video of a man riding a bike with square-shaped wheels.

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