Simply click on the green flag. The top cycloid is drawn using the parametric equations for a cycloid (see Notes and Credits). The cycloid at the bottom of the screen is the locus of a point on the rim of a circle as the circle rotates (without slipping) one complete revolution (and a bit more).
This project was inspired by Pi2345's project that can be seen at https://scratch.mit.edu/projects/10710404/145 and the project of dapontes that can be seen at https://scratch.mit.edu/projects/172893/ but it is not a remix of either project. The cycloid has been called "The Helen of Geometry" in reference to the beauty of Helen of Troy (her face launched a thousand ships). In parametric form, the equations for the cycloid are: x = r(theta –sin(theta)) y = r(1 - cos (theta)) where r is the radius of the generating circle. In the coding of the equation for x, there is a problem. If theta is given in degrees (º) then the value of (theta – sin(theta)) will not vary much since the sine varies from 0 to 1. Therefore it's necessary to change the first theta in the parenthesis from degrees to radians. The formula for converting degrees to radians is radians = degrees * (pi/180) and this formula can be seen in the script for x in the parametric form. A free PDF about the Cycloid can be downloaded at this link: http://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/Martin2011.pdf It contains much interesting and useful information. Apparently Galileo worked on the problem of the area under the curve for 50 years!
