trochoid

A cycloid (a common trochoid) generated by a rolling circle

Trochoid is the word created by Gilles de Roberval for the curve described by a fixed point as a circle rolls along a straight line. As a circle of radius a rolls without slipping along a line L, the center C moves parallel to L, and every other point P in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let CP = b. If P lies inside the circle (b < a), on its circumference (b = a), or outside (b > a), the trochoid is described as being curtate, common, or prolate, respectively. Parametric equations of the trochoid, which assume L is the x-axis, are

where θ is the variable angle through which the circle rolls. A curtate trochoid is traced by a pedal when a bicycle is pedaled along a straight line. A prolate, or extended trochoid is traced by the tip of a paddle when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where P touches the L.

A hypotrochoid is formed by a wheel rolling around the inside of a fixed circle.

External links

• http://www.xahlee.org/SpecialPlaneCurves_dir/Trochoid_dir/trochoid.html
• Weisstein, Eric W., "Trochoid" from MathWorld.
• Online experiments with the Trochoid using JSXGraph