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Comment by vinchuco | Hacker News Reader

There's a one-line formula that is supposed to capture most of these motions. https://en.wikipedia.org/wiki/Hypotrochoid

What always struck me as interesting is how some of them 'seem' like they're a projection of a surface onto the plane.

Edit: In fact, it would be nice to see a 3d version where the z-axis is the angle of incidence between gears (taking one as reference).

Especially in crazier patterns with gears that are not perfect circles. Too sad this is far in the past: https://www.kickstarter.com/projects/465068187/wild-gears-20...

Note: in the starting position when you open the page, fixed 96, rotating 52. 13 revolutions for the curve to close. 52=4x13=gcd(52,96)x13

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You could also have a 3D spirograph where you rotate an object with an embedded "pen" around the inside of a sphere, say. It draws a 3D shape. This would be very hard to construct, but easy to simulate. Would it give you anything interesting, either as a line drawing or a surface?

I'm not sure, because you would have an additional degree of freedom that needs to be constrained. With a 2D spirograph, you only have one axis of rotation. You can rotate forwards or backwards, but there are no other options. With a 3D object, there are always two possible axes of rotation. The simplest way to handle it would be to only rotate around one axis, but then you are back to just a normal 2D spirograph. There would need to be some non-linear correlation between the rotations, which adds a great deal of complexity.

I think the commenter meant to have it "keep going forward" in the same manner as 2D but with some rotation involved. It's doable.

See also John Freedman's Cycloid Drawing Machine: https://www.youtube.com/watch?v=ygcGfnVM6Ho

May be slightly off-topic, but for a creative feat using similar mechanisms, see https://www.youtube.com/watch?v=DfznnKUwywQ

While it is in the past, chasing links ( https://www.kickstarter.com/projects/465068187/wild-gears-20... to https://www.facebook.com/WildGearsPlay to http://www.wildgears.com ) brings you to http://www.ponoko.com/showroom/wonders which has the products for sale. Not in the cheap realm, but if one is a hypotrochoid fan, they are out there.

I once made a tool using this: https://github.com/zhirzh/spirograph-gen

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