Going Down Fast

RGB has fond memories of his days long ago as an occasional mathematician. So here's a light mathematical entertainment that should be of some interest. Imagine you're rolling a hoop down a sidewalk, which some of us of sufficient age may remember having actually done. You've painted a red dot on the hoop, and someone across the street is watching you. What kind of path or curve do they see the red dot trace out? Glad you asked.

                                                       THE CYCLOID

The curve shown in red below is called a cycloid, and the schematic right below it illustrates how it can be generated and why it is so named. One hump is traced out by a point on a circle rolling without slipping along a straight horizontal line. Further rolling continues to trace out additional humps.

For the mathematically inclined (just a little trigonometry), it's not too difficult to obtain equations for the cycloid. If you're allergic to even a little math, you can skip this paragraph, but you shouldn't. In an xy-coordinate system, suppose the circle, with radius say r, starts with its center on the y-axis and the fixed point we're watching is the one at the origin, (0,0). If we visualize a radius from the center of the circle to our observed point (as in the schematic above), the angle measured from the vertical to this radius will begin at zero and will increase as the circle rolls. For example, when the point is about halfway up the hump this angle is 90deg.; when the point is at the top of the hump the angle is 180deg. As the circle continues to roll, the point will descend along the hump and when the angle is 360deg will again be on the x-axis.

If we call the angle say t, then with a little analysis of the situation we can get the following equations for the x and y coordinates of our observed point in terms of the angle t.
                                                    x = r(t - sint)
                                                    y = r(1 - cost).

Okay, pretty nice. But, you say, so what? Well, actually the cycloid curve is more famous for something other than a rolling circle.

 THE BRACHISTOCHRONE PROBLEM     

The word "brachistochrone" comes from two ancient Greek words: βραχιστος, meaning quickest or fastest (think of "brachicardia", rapid beating of the heart); and χρονος, meaning time ("chronological").

Now say we have two points, call them A and B, in a vertical plane, with B lower than but not directly below A. The brachistochrone is the curve connecting the two points along which a bead would slide frictionlessly under the force of gravity from A to B in the fastest time. What is this curve? Surely it's a straight line from A down to B, since this is the shortest path? Well, no.

The problem was posed by Johann Bernoulli in 1696 as a challenge to "the most brilliant mathematicians in the world." At least five people, all definitely in the group of the most brilliant mathematicians of the time, solved it, although typically taking several months to do so. Except for ISAAC NEWTON, who, informed of the problem one day, solved it that night and tossed it off the next day.

By now you must suspect that the desired curve has something to do with the cycloid. It is a cycloid, but upside-down. Think of a bowl instead of a hump; that is, think of the schematic above with the circle rolling along the underside of the x-axis, with the point A at the origin and point B at the bottom of the first upside-down hump. See drawing below. This is the curve of most rapid descent; the brachistochrone is a cycloid!

                                              THE BRACHISTOCHRONE IS A CYCLOID.

Check out this demonstration on youtube.