Thursday, November 5, 2015

Cycloid – The Helen of Geometry

   The cycloid is the curve traced by a point on the circumference of a circle which rolls along a straight line without slipping.
   The cycloid has been called "The Helen of Geometry" in reference to the beauty of Helen of Troy (her face launched a thousand ships) and the infighting between mathematicians as they fought over the properties of the curve and who did what and when.
   In this Scratch project I code the cycloid using two different methods.
   The first method (see graphic below) uses the parametric equations for the cycloid,
           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.
   The second method (seen in the bottom half of the graphic) shows how a point on the rim of the circles traces a path (locus) identical to the shape generated by the parametric equations. Click on the video to see the Scratch program in action.
   For readers interested in the history and mathematics of the cycloid a free PDF about the Cycloid can be downloaded at this link:
   Apparently Galileo worked on the problem of the area under the curve for 50 years!
   The cycloid, when inverted, is called a brachistochrone. This curve has many surprising properties and will be the topic of a future Scratch project.

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