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Thread: Out of Phase Epitrochoid

  1. #1

    Unhappy Out of Phase Epitrochoid

    Hi, I'm having trouble generating the accurate trace of an epitrochoid by rotating a circle of radius 'r' around a larger circle of radius '2r'. On the face of it the animation appears to be working with the smaller circle running smoothly around the circumference of the larger circle. However, the trace is just an offset circle. On closer inspection, a point on the circumference of the smaller circle is only making one revolution for each rotation of the small circle around the large - i.e., there is some sliding going on somewhere but I can't work it out. It should be making two revolutions. Does anyone know if it is possible to get this to work in Sketchpad. I know it is in Geogebra though setting up the algebra is quite complex.

  2. #2
    Hi VicR,
    The trick with trochoids is to get the rolling circle to roll at the correct rate so that there's no slipping. Without built-in friction, we have to actually figure out the relationships between the location of the rolling circle and the angle through which it's rolled.

    I've just posted an example sketch here:

    The first several pages show animations of cycloids, epicycloids, and hypocycloids; by animating the rolling circle's center along its path at the same rate as a point on the rolling circle is animated to "roll" the circle, we take care of the friction problem. But this uses two animating points, and so cannot be used to construct a locus. For that, the locus page has a calculation to determine the relative angular velocities of the center and the radius point.

    I hope this helps!

  3. #3
    Thanks for clearing that up Scott.

    That's great.

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