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)

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.

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.
Select point F in the list and then select "Backward (CW)" at the
bottom of the diaolg to set F to travel clock-wise around the
circle. Finally, multi-select points C and F in the list (hold
the Hsift key down when selecting) and click the "Start" button to
start the animation. You should see something like the figure
below. (Make sure to click "Stop" when F gets close to B)

While part of the
cycloid
was generated, it appears that we could improve our cycloid trace by
lengthening
AB and shortening DE. Let's try again.

That looks
better.
Try
experimenting
with different types of cycloids. If we draw a ray from C through F and
then attach a point G to the ray we can trace G as the circle rotates.
What do you think will happen if G is outside the circle? inside the
circle?
Try this.