This is a post to explain how to make rolling cycloids, with the outer and the middle cycloid standing still. (See for example, https://plus.google.com/100749485701818304238/posts/6UMwHbZTnou).
Referring to the figure, we have 5 circles, a circle of radius N, N-1, N-2, N-3, N-4 (red, blue, black, magenta, green).
At t=0, the circles are arranged so that they touch in a special alignment, that is probably clear from the picture. This alignment ensures that the black circle is centred on the red centre, as is the green circle.
Now we start rolling the blue circle, and inside it the black one. We can choose the rolling rate of the blue circle, but the black rolling rate needs to be coupled to the blue rolling rate in such a way that its centre does not move. The Magenta circle can now roll inside the black one, at a rate that we can still choose later. But again, the green one needs its rolling rate to be coupled to the magenta rolling rate so that it remains centred.
OK, so now we have the circles rolling, with the black and green ones centred with the red one. To make the cycloids, just choose a point on a circle, and trace out its path relative to the circle in which it rolls. That will be a cycloid. We then get a set of rolling cycloids, with the middle one still centred, but not necessarily stationary. To make it stationary, remember we were still free to choose the rolling rate of the magenta circle. We can use that degree of freedom so that the green cycloid stands still.
This picture shows hypocycloids, but the epicycloids work the same way. In fact, I only have to change on minus sign in the code for the cycloids to flip from hypo- to epi-.
Referring to the figure, we have 5 circles, a circle of radius N, N-1, N-2, N-3, N-4 (red, blue, black, magenta, green).
At t=0, the circles are arranged so that they touch in a special alignment, that is probably clear from the picture. This alignment ensures that the black circle is centred on the red centre, as is the green circle.
Now we start rolling the blue circle, and inside it the black one. We can choose the rolling rate of the blue circle, but the black rolling rate needs to be coupled to the blue rolling rate in such a way that its centre does not move. The Magenta circle can now roll inside the black one, at a rate that we can still choose later. But again, the green one needs its rolling rate to be coupled to the magenta rolling rate so that it remains centred.
OK, so now we have the circles rolling, with the black and green ones centred with the red one. To make the cycloids, just choose a point on a circle, and trace out its path relative to the circle in which it rolls. That will be a cycloid. We then get a set of rolling cycloids, with the middle one still centred, but not necessarily stationary. To make it stationary, remember we were still free to choose the rolling rate of the magenta circle. We can use that degree of freedom so that the green cycloid stands still.
This picture shows hypocycloids, but the epicycloids work the same way. In fact, I only have to change on minus sign in the code for the cycloids to flip from hypo- to epi-.
