The curve traced out by a point on the rim of a circle rolling along a straight line is called a cycloid. Let the radius of the circle be

. Allowing the tracing point to be either within or without the circle at a distance

from the center generates "curtate" or "prolate" cycloids, respectively. The variable

is

, limited to the range [0, 2]. For

, the result is a straight horizontal line. For

, the curve is a curtate cycloid. For

, it is an ordinary cycloid. For

, it is a prolate cycloid. Note that

is allowable but not used here.