Characterizations of cycloid There are several motions that create a cycloid. I have some examples here. Are there any others?


*

*Trace of a fixed point on a rolling circle

*Evolute of another cycloid (the locus of all its centers of curvature)

*Involute of another cycloid (trace of a pendulum constrained to another cycloid)

*Envelope of a family of lines with uniformly varying angle and intercept

 A: May I interest y'all in a short cartoon?


If you look carefully at the cartoon, you'll see two cycloids being generated by the same rolling circle. The first one is the usual case, where a point on the rolling circle's circumference traces out the cycloid.
The other, smaller cycloid is being generated by a related mechanism: it is the envelope of the diameter of the rolling circle!
Skipping the details, it can be shown that if the larger cycloid has the parametric equation $\left(t-\sin t\quad 1-\cos t\right)^\top$ the smaller cycloid has the corresponding equation $\left(\frac{2t-\sin 2t}{2}\quad\frac{1-\cos 2t}{2}\right)^\top$.
A: The Brachistochrone curve between two points at the same height is a cycloid.
A: In dynamics, time taken for rolling oscillation of a small heavy marble irrespective of amplitude in such a shaped trough.. is constant $( = 2 \pi \sqrt {\frac{4 a}{g}}) $.. Tautochrone property.
EDIT1:
Distance of any cycloid point to x-axis ( on which the circle rolls) along its normal is half the radius of its curvature...one of its properties. 
