Abstract:

Although there are many articles about the origin principle of the brachistochrone ( i. e.，the cycloid) at present，most of them directly apply the law of optical refraction to analogical reasoning，lacking of detailed introduction to the source of method and discussion on the details of motion，or just do local numerical calculations and lack of physical connotation. As for the more basic problem associated with the cycloid that the time when the particle falls along two connected straight lines fixed at the beginning and end，few papers have been written to study it. This paper first derives the " refraction law" in the problem of prompt drop of particle，and then uses it to deduce the equation of the brachistochrone ( i.e. the cycloid) . It is proved that the dropping motion of a particle along a cycloid is equivalent to the absolute motion of the corresponding point on a uniform rolling circle. The equation of the steepest descent motion and the methods of calculating the specific parameters of the cycloid and the descent time of the particle are given，and some examples are given to illustrate them. It is pointed out that there are different formulas for calculating the parameters of cycloid when the landing end is at the left and right half arch of the cycloid. It is simply proved that the cycloid is only shortcut. Finally，through the method of finding extremum， the position of the fastest descending break point of two connected straight lines with fixed ends，and the positions of the fastest descending break points with given longitudinal or abscissa coordinates are calculated. Through listing and drawing，the change rule of landing time with breaking point is analyzed.

Key words: cycloid, prompt drop, time, broken line, break point