What is the projection of Earth's axis on the sphere with the Sun in center (see explanation)

Seems like it is the most convenient SE community for this question..

Imagine the sphere, witch center is the Sun and radius is, say, ~1.5 distance from Sun to earth (in arbitrary time, precise number doesn't matter much).
EDIT: the center of sphere is 'attached' to the Sun.

Then, imagine, the Earth's axis projection on that sphere in time. It leaves some trace.

What will be the picture? Will it repeat itself through some period? What will be the "scansion" for different time periods?

EDIT: as reply to comments I'll add, that my interest is purely out of curiosity, and degree of required accuracy is low. I don't want high precision here, just rough pattern. As I understand this could be done, using mentioned Milankovitch cycles, but I'm not much into mathematics. Though, if I'll not find answer here, I will be forced (by myself) to figure this out, when i'll get some spare time heh.