your ellipsoid invention is brilliant, but out of curiosity, i can't seem to get the other invention of yours to fulfill the requirements. I am going to show a gif to see if its what you had in mind. You specified a 1:1 ratio for the gear so I made the gear the same size, idk why the original diagram had a smaller gear in the middle.
https://gifyu.com/image/Oq1q
https://gifyu.com/image/Oq1q