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?
Answer
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.
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