Final Project :: One Final Exploration

The Cycloid

By Jamie K. York

The cycloid is the locus of a point on the rim of a circle of radius a rolling along a straight line. It was investigated and name by Galileo, with many additional researchers to follow. Gear teeth were initally constructed using the cycloid configuration.

Graphing Calculator

In parametric form, as discussed in assignment 10, the cycloid can be graphed as shown below:

The cycloid can also be written in a Cartesian form:

To construct a cycloid, following these instructions:

1. Define a Cartesian coordinate system.

2. Create a circle of a given radius, preferrably in the upper right corner of the window.

3. Label the center of this circle O. Create a movable point on the circle and label it A.

4. Graph the function f(x)=1. This is the equation of the horizontal line y=1.

5. Create a movable point O' on this line.

6. Highlight the center of circle O and the movable point O'. Then choose from the menu bar transform > mark vector. This is where we want to translate the circle on the right upper corner.

7. Now highlight the center O, the movable point A, and the circle. Then choose from the menu bar transform > translate (use the defaults and click translate). In this way we get a copy of the circle on the y=1 line.

8. Select the point O' and the point A'. Edit one animation button for these points. Note: We want to animate the center of mass point O' forward and the point A clockwise.

9. Mark sure that you enable the trace option for point A. This point will generate the cycloid.

The construction should resemble this GSP file.