It is commonly known that the solution to the brachistochrone problem is a cycloid. However, in order for a solution curve to be a cycloid, the ratio between points A on the y-axis and B on the x-axis has to be r/pi*r, since that is the ratio between the "height" of the cycloid and half of its "length". However, what is the solution curve to the brachistochrone problem if points A and B share a different ratio to each other - say, 1/2 ? Two possible solutions that I have considered: The curve is an affine function of the cycloid; the curve is stretched by a factor k along one of the directions. The curve is a segment of a larger cycloid. How would I approach this problem?