The question Why not explore Cruithne? includes the GIF below. I've extracted two frames and extracted the red pixels of Cruithne's orbit where
red = (R > 0.7) * (G < 0.3) * (B < 0.3) and show this below for comparison.
It looks to me like the orbit itself changes size and shape throughout the GIF, which is supposed to be for only one of Cruithne's orbital periods of essentially "one of your Earth years".
Does this really happen? Does it even make sense to talk about shape of a complete orbit changing within one period? Or is this some projection artifact resulting from the way it is projected and the Earth is forced to remain at a fixed distance from the Sun?
Ellipse on right seems to be about 10% larger than the one on the left:
The orbits of Cruithne and Earth over the course of a year (from September 2007 to August 2008). Cruithne's location is indicated by the red box as it is too small to be seen at this distance. Earth is the white dot moving along the blue circle. The pale yellow circle in the centre is our Sun. The blue circle represents the path of Earth as it revolves counter-clockwise around the sun. The "camera" revolves with Earth, so Earth appears to not be moving.
The red path represents the path that Cruithne takes as it revolves around the sun. Since the camera is rotating with Earth, Cruithne appears to move along a bean-shaped path. This illusion is known as a Horseshoe orbit. Source Frames from the GIF were acquired from the GFDL-licensed Celestia program in the form of screen shots using the Mac OS X screen-capture feature. Screen captures were converted into an animated GIF using GraphicConverter's batch edit feature by User:Jecowa.