Path of an Ant on the Diameter of a Circle Rotating on the Unit Circle

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An ant, indicated by the blue dot, begins at the point (1,0) on the circumference of the red circle, which rotates at a constant speed either inside or outside the unit circle. As the circle rotates, the ant starts moving along the green diameter, also at some constant speed, until it reaches the other end. At that moment, the ant turns and immediately begins to traverse the diameter in the opposite direction. The ant repeats this action in an assigned number of circuits of the unit circle. The path of the ant is shown as the blue curve traced behind it from the point (1,0) to the blue dot.


Set "type of curve" to "epicycloid" for the red circle to be outside or to "hypocycloid" for the red circle to be inside the unit circle. The next three sliders control the radius of the rotating circle, the maximum number of times the rotating circle travels around the unit circle and the speed at which the ant moves back and forth along the diameter in units of diameters per rotation around the unit circle. The final slider controls how far along its path the ant has journeyed, expressed as the radian measure for the rotation of the center circle around the unit circle. Set this slider to automatic or manually control it to view the ant's trajectory. When the speed is set to zero, the ant does not move along the diameter and the curves reduce to an ordinary epicycloid or hypocycloid.


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



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