here are the orbit travels of Earth and Venus done on a spirograph. There ia more to orbits than formulae http://img-9gag-fun.9cache.com/photo/am9xWbv_460sv.mp4

Not sure what your point is. Gravity dictates the orbit of the earth and venus. Equations describe the motion of the planets. The pattern in the video (which is kinda cool) is not surprising at all since the planets move in a relatively constant speed in a relatively circular orbit.

It rather looks as if your trace describes a caustic of some kind. However I am not enough of a mathematician to confirm that it does or, if so, why it does. Perhaps someone else can comment.

On the other hand, if you'd LOOKED at the formulae you'd have known that your drawing isn't the orbits: neither orbit is circular. Please Register or Log in to view the hidden image!

Yes, Earth and Venus orbits are among the most circular we have, and the "Tango" supplied by Dywydyr, (thank you) gives the dance of the ~zero gravity center anti-Langrangian point between the two sister planets?? These would be traces left of the gradients in gravity, space warpings? BSW: I always liked to dance the waltz (in the 1940s) with the most weight-endowed lady, because I would be in orbit around her for extra spin. ha.

If you look at these traces you could almost imagine a standing wave pattern, and the eccentricities would add an interesting additional slant.

Appealing to you learned contributors to find and post the dance pattern of different moons, the epicycles, cycloid curves they trace against the stars. thank you.

What you're seeing is the formulae - plotted over time. Please Register or Log in to view the hidden image!

Yes, one formula can reproduce the same spirograph- orbital pattern for a very long time. The help I am asking to find a pattern that would trace the cycloid/epicentre movement of the moon Europa or a similar situation.