In this MO question, I stated that a circle rolls down the fastest along a cycloid curve - as shown by Johan Bernoulli, thereby solving the Brachistochrone problem.
However, as user Manfred Weis pointed out, his solution involved finding the curve of fastest descent not for a circle, but for a point.
Therefore, I was wondering if the solution has also been found for an actual circle. The solution probably is not very different from the cycloid found by Bernoulli for the point, but I wonder how it differs and how it takes into account the circle's radius $r$.