When I was a child I wanted a
Spirograph but never had one, so I decided to simulate one.
Here is a movie of a wheel with an arm, rolling inside another wheel, drawing a hypotrochoid curve.
YouTube - Mac Grapher Examples: Spirograph2
I did the simulation and made the movie with Grapher, a free math and graphing package that comes with every Mac. (It is in the Applications folder under Utilities.)
If you have a Mac and want to play with it to see how Grapher works, download this zipped file
View attachment Spirograph.gcx.zip
It should unzip automatically and the Spirgraph.gcx file should be in your Downloads folder. Double click on the file icon to start Grapher.
The simulation allows you to change the radius of both wheels, and the length of the arm to make all sorts of weird and wonderful curves. You can also make rolling wheel roll around the outside of the fixed wheel for a whole different family of curves.
If the image may be stretched when Grapher opens, click on the "Equalize Axis" button at the top.
To start the animation, click on the start button to the left.
To change a parameter, click on one of them and change the value like you would in a spreadsheet.
The simulations run slowly and jerkily because they require a lot of calculations. To see a simulation run smoothly, click on the film icon at the upper right to make a movie.
If you have any trouble, PM me. And yes, I know that I am a hopeless Geek.