Characterizations of cycloid

Question

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

1. Trace of a fixed point on a rolling circle
2. Evolute of another cycloid (the locus of all its centers of curvature)
3. Involute of another cycloid (trace of a pendulum constrained to another cycloid)
4. Envelope of a family of lines with uniformly varying angle and intercept

Answer 1

The Brachistochrone curve between two points at the same height is a cycloid.

Answer 2

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.

Answer 3

May I interest y'all in a short cartoon?

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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 $\left(\frac{2t-\sin 2t}{2}\quad\frac{1-\cos 2t}{2}\right)^\top$.