Date: Mon, 30 Dec 1996 18:58:39 -0500
From: Alex Bogomolny
It's my hope you have just missed my Cycloids page:
because, otherwise, I have very little to add. Please have a look at this page. Some of your questions may be resolved right away. Come back with the remaining ones.
Attach a pen to a wheel standing on a floor by a wall so that the pen touches the wall. Now roll the wheel and watch the movement of the pen. See that at all times it touches the wall. While in motion the pen will draw a line known as cycloid. In more rigorous terms, we are talking about a curve traced by a point which takes part in a rigid motion of a circle. A circle may roll along a straight line or another circle. Do see my page. The page has a Java applet that simulates several such motions.
Another name for cycloids is brachistochrones. These are the curves traced by a free sliding ball to a point not directly beneath its starting position.
Hope this helps.