Animation - Cycloids

One of the most interesting geometric figures in analytic geometry is the cycloid. A cycloid is defined as the set of points which are created by a point that is attached to the rim of a circle as the circle rolls along a line. Creating a cycloid using Geometry Explorer is really quite simple.

First we create a line segment AB that will serve as a guide for a circle to roll on.  There is no easy way to roll a circle on AB, but we can get the same effect by 1) attaching a point C to AB, 2)creating a circle with center at C, and 3) moving C along the segment as we rotate the circle.  We will need our circle to stay rigid as we "roll" it, so we construct a circle with center C and fixed radius equal to the segment length of another fixed segment DE. (Do this by selecting C and then DE and then using the circle construction tool in the center of the Tool Panel)

cycloid example1
Now if we animate point C along AB the circle will be translated as well. To create the cycloid we attach a new point F to the circle. To better see the point as it moves we change its color. Now we are ready to generate the cycloid.
cycloid example2
To "see" the cycloid being traced out, we need to turn on a trace for point F.  Do this by selecting F and then choosing "Trace On" under the View menu.  Now, select points C and F and choose "Show Animation Panel..." under the View menu. A dialog similar to the one below will appear.

Animation Dialog