Path of an Ant on the Diameter of a Rolling Circle

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An ant (indicated by the blue dot) begins at the origin on the circumference of the red circle of radius one, which rotates at a constant speed in the direction of the positive axis. As the circle rotates, the ant moves along the green diameter, also at some constant speed. When the ant reaches the end of the diameter, it turns and immediately goes along the diameter in the opposite direction. This Demonstration shows the position of the ant at any time and the path it traces.


You can vary the length of the track on which the circle can roll, the speed of the ant in terms of diameters per revolution and the position of the circle (number of revolutions). Because the width of the graph is fixed, the choice of a longer track forces the circle to get smaller in order to maintain the aspect ratio. When the speed of the ant is zero, the path is the familiar cycloid. Note what happens when the speed equals two.


Contributed by: Carl McCarty (December 2020)
Open content licensed under CC BY-NC-SA



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