[symmetricsaurus]

Interesting case!

You can see it is R/r local revolutions of the small circle. Then you need to add one global revolution from going around the large circle. So R/r + 1.

This is of course what the article is saying pretty much.

Are there other situations that require a similar reasoning?

[xpe]

> You can see it is R/r local revolutions of the small circle. Then you need to add one global revolution from going around the large circle. So R/r + 1.

I understand the article, but I don't I think I quite get or buy your explanation.

I'm curious about your linguistic separation of "local" and "global". What is local and what is global in this situation? I wonder if you are the frame of reference concept? Could you unpack what you mean?

I don't think the terms quite fit. You did when you wrote it; do you still? By this I mean: do you think the way you're using the terms local and global would be intuitive to, say, an audience with a high-school level background in geometry? This is an empirical question, but my inclination would be to say 'probably not'.

I find the article's emphasis on decomposing sliding (i.e. translation) from rotating to be much more intuitive. (Of course people will vary.)

Still, I'm curious. I'm searching for a sense in which this linguistic global/local distinction adds explanatory power. Care to elaborate?

[symmetricsaurus]

Yes I’m thinking in terms of different frames of reference.

Imagine that you’re a tiny ant that lives on the large circle. When you push the small circle around the large circle you will see it make four rotations before you get back to the starting point. This is the local frame of reference.

Now imagine you’re a giant living in the space with the circles. You see the small circle do five full rotations. Four are the same that the ant sees but you also see the ant itself do one rotation simultaneously as it walks along the large circle making five in total.

It would be more accurate to say that you have 4 local rotations and that the local frame rotates one full turn in the global frame.

Does this make sense to someone with a high-school level of geometry knowledge? Not as I wrote it I initially (but there was no such goal). The analogy with the giant and the ant together with some nice illustrations maybe?

[mannykannot]

Here's another way of putting it, without explicitly mentioning frames of reference:

Before starting, imagine drawing a radius on the moving coin from its center to its point of contact with the stationary coin. It is pointing down.

As the moving coin goes clockwise around the fixed coin, this radius will also rotate clockwise around the center of the moving coin.

The next time the end of the radius is in contact with the fixed coin is after the moving coin has gone 1/4 of the way around the fixed one - but now the radius is pointing to the left. It has passed the point where it is pointing down again (which is when the moving coin has made one full revolution) and gone a 1/4 turn beyond that.

Repeat until you are back at the top again.

[adammarples]

Yes, the missing phrase from the article was clearly "frame of reference". Ie the coin makes 4 rotations if we stretch the larger circle out into a straight line, which we do implicitly when doing our mathematical calculation, but we should be aware that this frame of reference is also making a full rotation.

[xpe]

typo fix: "I wonder if you are using the frame of reference concept? Could you unpack what you mean?"

[jojobas]

Sidereal day vs solar day.