For ten points, name this curve generated by a point on a rolling circle.
Its area is three times that of its generating shape, and its length is four times that of the aforementioned shape's diameter. Christiaan Huygens used its isochronous properties, namely that a particle on it will descend it constant time regardless of its starting point) to fashion guides for the first pendulum clock that guaranteed a regular swing for any pendulum height, and these properties led to its proposal by Bernoulli of its inverse being the brachistochrone or curve of fastest descent. The solution to the differential equation (y')^2 = (2r-y)/y, its parametrization is x= r(t-sint) and y= r(1-cos t). For ten points, name this curve generated by a point on a rolling circle.
Its area is three times that of its generating shape, and its length is four times that of the aforementioned shape's diameter.
{cycloid}