In a previous paper (2019 Eur. J. Phys. 40 035005) we showed how to design a discrete brachistochrone with an arbitrary number of segments. We have proved, numerically and graphically, that in the limit of a large number of segments, N ≫ 1, the discrete brachistochrone converges into the continuous brachistochrone, i.e. into a cycloid. Here we show this convergence analytically, in two different ways, based upon the results we obtained from investigating the characteristics of the discrete brachistochrone. We prove that at any arbitrary point, the sliding bead has the same velocity on both the continuous and discrete paths, and the radius of the curvature of both paths is the same at corresponding points. The proofs are based on the well-known fact that the curve of a cycloid is generated by a point attached to the circumference of a rolling wheel. We also show that the total acceleration magnitude of the bead along the cycloid is constant and equal to g, whereas the acceleration vector is directed toward the center of the wheel, and it rotates with a constant angular velocity.

在先前的论文（2019欧元。J.的PHY。 40 035005），我们显示如何设计的离散速降与段的任意数量。我们已经通过数字和图形的方式证明了，在大量分段的限制下，N≫ 1，离散腕线收敛到连续腕线，即摆线。在这里，我们根据调查离散腕线虫的特征得到的结果，以两种不同的方式分析性地显示了这种收敛。我们证明，在任意点上，连续和离散路径上的滑动珠都具有相同的速度，并且两个路径的曲率半径在相应点处相同。证明是基于众所周知的事实，即摆线的曲线是由附着在滚轮圆周上的点产生的。我们还表明，沿摆线的磁珠的总加速度大小是恒定的，等于g，而加速度矢量指向车轮的中心，并且以恒定的角速度旋转。