Summary

The basis of this concept came from this video: https://www.youtube.com/watch?v=2PR-vSNrazM (I opted not to flatten the base of my eggs, though). DISCLAIMER: I did not make the video. I am not Maker's Muse.

The thing I added was that I used a curved line to make (most of) these patterns. There's a couple steps to doing this. First, create a sketch on either the 'right' or 'front' plane, and then sketch the line that you wish to follow the egg's surface. Spline curves, arcs, conic sections, and even lines at angles are all fun ideas to try. Just make sure it lines up with the egg itself. Then, make it into an enclosed shape by drawing additional lines attached to the 'target' line. The whole shape should turn off-white when enclosed.

Extrude your shape in both directions such that it intersects the egg, making sure to select 'New Body' for the extrusion. Select the egg and copy it, to create another new body in the same place as the original egg. Then go to 'combine,' select the extruded path and the extra egg, and select 'intersect.' This will create a body that has one edge that follows your path along the surface of the egg.

Then just create your pattern as the Maker's Muse video instructs, but select the curved path for the pattern to follow. Be sure to hide the extra 'path' body before hitting 'okay' to actually create the pattern, and there you go!

Disclaimer: As with most things I do, there may be a simpler way to do this. But it works.