NavList:

Ignoring changing declination, the source of light (the Sun!) traces out a circle in the sky. It's a small circle, as opposed to a great circle, unless the declination is zero (at the equinoxes for the Sun). Now the end of your stick is a fixed point. If you connect the points on the small circle in the sky with that point and then project through to the shadow of the end of the stick, you get a cone. And what shapes does a cone make when it intersects a plane surface? The shadow of the end of the stick will trace out various conic sections. In most latitudes, you will get hyperbolas, except at the equinoxes when you get a straight line. At some latitudes, you will get circles, ellipses, and somewhere there's a parabola..

-FER
PS: while changing declination will change things slightly, it's probably worth mentioning the refraction also slightly distorts the traced shadow paths, flattening the cones where they are near the horizon. Far more important than these factors is the fact that the surface of the Earth is rarely close to being a true plane.