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Cycloid
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Main Concept
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A cycloid is the path that is traced out by a point on the circumference of the circle as it rolls along a straight line (without slipping).
The cycloid can be defined by the following two parametric equations:
where r is the radius of the circle that is rolled, and  is the angle through which the circle was rolled.
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Curtate and prolate cycloids
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Two related curves result when we include another parameter, L, which represents the ratio of pen length to the radius of the circle:
When  , the curve is called a curtate cycloid; when  , the curve is called a prolate cycloid.
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