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