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There are several constructions of a cycloid. I have some examples below. 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
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    $\begingroup$ Have you tried mathcurve.com/courbes2d/cycloid/cycloid.shtml? It's in French, but the website is full of relationships between curves, maybe you'll find «new» characterizations. $\endgroup$ Aug 7, 2015 at 14:49
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    $\begingroup$ There are various characterisations of the cycloid in terms of mechanical properites if curvies. For example, it is both a brachistochrone and a tautochrone for a constant force law parallel to the $y$-axis and a trajectory for such a force law, with force proportional to $\frac 1 {y^2}$. $\endgroup$
    – priel
    Aug 7, 2015 at 15:39

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