brachistochrone curve | Helen of Geometers | tautochrone curve

x(t) = a (t - sin(t)) y(t) = a (1 - cos(t))

abs(x/a - 2 π floor(x/(2 π a) + 1/2)) = cos^(-1)(1 - y/a) - sqrt((2 y)/a - y^2/a^2)

parametric | roulette

A = 3 π a^2

evolute | cycloid involute | cycloid