The involute of the cycloid
is given by
As can be seen in the above figure, the involute is simply a shifted copy of the original cycloid, so the
cycloid is its own involute!
More things to try:
Weisstein, Eric W. "Cycloid Involute."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CycloidInvolute.html