Formula of a hypotrochoid (spirograph pattern)

[mrgreenankle]

Hi!

I am writing about hypotrochoids, and need to explain the equation behind them. You can read about them here: https://en.wikipedia.org/wiki/Spirograph and here: https://en.wikipedia.org/wiki/Hypotrochoid


The equation is:

Look at the wikipedia links for further details.

I understand the first part (before the +/- sign), and that it describes the position of the midpoint or the smaller circle while rotating in the larger circle.

I also know that the second part (after the +/- sign) defines the position of the "pen spot" in relation to the centre of the smaller circle, but how does this work? I need to understand why this specific equation defines this point, so that I can explain it in my own words.

I would be thankful for help!

 

[mrgreenankle]

I need to be finished with this on friday, so if anyone has any ideas I'd be glad to hear them. Thought I had figured it out a few days ago but then I realized that I had mistaken myself so I'm back to square one sort of haha.

Please help out if you can understand how this works!

 

[Misanthropist]

Hi!

I am writing about hypotrochoids, and need to explain the equation behind them. You can read about them here: https://en.wikipedia.org/wiki/Spirograph and here: https://en.wikipedia.org/wiki/Hypotrochoid


The equation is:

Look at the wikipedia links for further details.

I understand the first part (before the +/- sign), and that it describes the position of the midpoint or the smaller circle while rotating in the larger circle.

I also know that the second part (after the +/- sign) defines the position of the "pen spot" in relation to the centre of the smaller circle, but how does this work? I need to understand why this specific equation defines this point, so that I can explain it in my own words.

I would be thankful for help!

I'm obviously a bit late, and I hope you figured it out. I only just registered here myself, so I didn't see this until a few hours ago.
Anyways, d is the radius of the circle that the "pen point" describes around the centre of the small circle.

And (R-r)/r is the ratio between how many times the small circle rotates around its own centre and the number of revolutions it makes around the centre of the large circle.

For example in the top animation on the Wikipedia page, R = 5 and r = 3. This gives (R-r)/r = (5-3)/3 = 2/3. This means that every three times the small circle revolves around the centre of the large circle it rotates twice around its own centre

If you marked the points on the two circles where they touch before they start moving and closely followed them, you'd see that, in the Wikipedia case, the small circle would do 1080° around the centre of the large circle, while the point you marked on the small circle would only do 720° around its own centre before lining up with the point you made on the large circle. Think of the "pen point" as the end of a pole that is attached to the marked point on the small circle.