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.
It will be two images of the nearly-circular ellipse of the Earth's orbit (one for the north pole, one for the south), distorted by being projected onto the interior of a sphere, which means that the side of each orbit that has the axis tilted "towards" it will appear enlarged and closer to the ecliptic, and the side of each orbit that has the axis tilted "away" from it will be smaller and closer to the pole.

To a first approximation the pattern will retrace itself every year. The Earth's orbit is stable with respect to the Sun and the fixed stars. Over a period of tens of thousands to hundreds of thousands of years, the Earth's orbit will precess (causing the figures to rotate around the sphere), and nutation will also cause small wiggles at a much higher frequency.