The remarkable properties of the discrete brachistochrone

We present a new solution to the discrete brachistochrone problem, based on the variational principle. There are two interesting and surprising properties of the solution. First, the sliding times along all the straight segments are equal and are independent of the initial velocity. Secondly, we find a simple relation between the angle of the kth segment to the vertical and the angle to the vertical of the first segment. Based on these results, a Matlab code was written that calculates efficiently and accurately the solution for N segments between arbitrary end points.