Trochoid

[
thumb|290px|right|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:$x\; =\; a\; heta\; -\; b\; sin(\; heta)\; ,$:$y\; =\; a\; -\; b\; cos(\; heta)\; ,$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 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
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