You know Pilou you can draw an ellipse directly on an egg surface, which is yet less planar than my piece was here ! (I'm joking don't worry ;-)

"not really planar" means "not planar" : my surface is rounded as an extruded circle and a bit like a conic one, also.
Yes, I know, a very complex surface for a so simple piece. O:-)
Anyway as you can see on my image - showing your method at the left and Michael's method at the right - this worked well enough :-)

Well, as you have imagined to be exact the ellipse was not all round touching on the "non-planar" surface, but as you understood I just simply drawn my ellipse with its hot points in touch on the surface, then this was enough to obtain my "perpendicular axis" as well as I needed for perpendicularly cutting as it was done.

Cordially,
Guy