What does a sphere moving close to the speed of light look like? What shape does the viewer in a reference frame with $v=0$ perceive? I suppose that since the  sphere moves in one direction only (oX only, not oY) its section would change into an ellipse, where the horizontal diameter would be shorter.
However, my textbook says that the viewer still perceives a regular spherical shape. How come?
 A: A sphere is always seen with an exactly circular outline, see
http://www.spacetimetravel.org/fussball/fussball.html
Reference: Mary L. Boas: Apparent Shape of Large Objects at Relativistic Speeds, 1961, American Journal of Physics 29, 283-286
A: This is just a footnote to Crazy Buddy's answer (which is correct! :-):
Length contraction is a real phenomenon, and indeed the RHIC observes this every day because the nuclei are moving so fast that the collision is between two disks not two spheres.
However to see something you need to have light emitted from the object reach your eye, and the light from different parts of the moving sphere takes different times to reach your eye. This distorts the image of the contracted object and has the apparently paradoxical effect of making it look spherical even though it is contracted.
So the moving sphere looks spherical even though it isn't spherical. The calculation of how light from the object reaches your eye is quite involved, and I'm afraid I don't know of a simple analogy to understand it. There are various animations showing this effect on the web. See for example this one.
A: The sphere is contracted in the horizontal axis and perceived as an ellipsoid. This is what we believe about length contraction and this happens only, when we take Einstein's simultaneity into account.
But, the stationary observer would see the sphere appearing as the sphere always (i.e) the circular outline would still be there at any velocity relative to the observer. This is because when we deal with space-time and as the sphere moves accordingly near the speed of light, the wave-fronts emitted from other parts (which don't show up when we stare at it) would reach the observer at that instant and the wavefront at the other end of its static appearance would pass out.
Thus, the observer would see the sphere being rotated as it passes past him.
