Trochoid

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A cycloid (a common trochoid) generated by a rolling circle Cycloid f.gif
A cycloid (a common trochoid) generated by a rolling circle

In geometry, a trochoid (from the Greek word for wheel, "trochos") is a roulette formed by a circle rolling along a line. In other words, it is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. [1] If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval.[ citation needed ]

Contents

Basic description

A prolate trochoid with b/a = 5/4 TrohoidH1,25.gif
A prolate trochoid with b/a = 5/4
A curtate trochoid with b/a = 4/5 TrohoidH0,8.gif
A curtate trochoid with b/a = 4/5

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. Parametric equations of the trochoid for which L is the x-axis are

where θ is the variable angle through which the circle rolls.

Curtate, common, prolate

If P lies inside the circle (b<a), on its circumference (b = a), or outside (b>a), the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively. [2] A curtate trochoid is traced by a pedal when a normally geared bicycle is pedaled along a straight line. [3] A prolate 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.

General description

A more general approach would define a trochoid as the locus of a point orbiting at a constant rate around an axis located at ,

which axis is being translated in the x-y-plane at a constant rate in either a straight line,

or a circular path (another orbit) around (the hypotrochoid/epitrochoid case),

The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions, , is a rational number, say , where & are coprime, in which case, one period consists of orbits around the moving axis and orbits of the moving axis around the point . The special cases of the epicycloid and hypocycloid, generated by tracing the locus of a point on the perimeter of a circle of radius while it is rolled on the perimeter of a stationary circle of radius , have the following properties:

where is the radius of the orbit of the moving axis. The number of cusps given above also hold true for any epitrochoid and hypotrochoid, with "cusps" replaced by either "radial maxima" or "radial minima".

See also

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Cycloid Curve traced by a point on a rolling circle

In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

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Hypocycloid

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Cardioid

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Epicycloid

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Hypotrochoid

A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.

Epitrochoid

An epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.

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Cyclocycloid

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References

  1. Weisstein, Eric W. "Trochoid". MathWorld .
  2. "Trochoid". Xah Math. Retrieved October 4, 2014.
  3. The Bicycle Pulling Puzzle. YouTube . Archived from the original on 2021-12-11.