Editing Cycloid (section)

{{Short description|Curve traced by a point on a rolling circle}}
{{Other uses}}
[[File:Cycloid f.gif|right|frame|A cycloid generated by a rolling circle]]

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

The cycloid, with the [[Cusp (singularity)|cusps]] pointing upward, is the curve of fastest descent under uniform [[gravity]] (the [[brachistochrone curve]]). It is also the form of a curve for which the [[Frequency|period]] of an object in [[simple harmonic motion]] (rolling up and down repetitively) along the curve does not depend on the object's starting position (the [[tautochrone curve]]). In physics, when a charged particle at rest is put under a uniform [[Electric field|electric]] and [[magnetic field]] perpendicular to one another, the particle’s trajectory draws out a cycloid.

[[File:Brachistochrone_curve.gif|thumb|right| Balls rolling under uniform gravity without friction on a cycloid (black) and straight lines with various gradients.  It demonstrates that the ball on the curve always beats the balls travelling in a straight line path to the intersection point of the curve and each straight line.]]