What is the difference between the sphere and projective space?

Question

I know about the antipodal mapping.

What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship to one another.

I come at this from a coding theory/vector quantization perspective; I'm trying to understand the difference between quantization techniques that codes a set of vectors in $R^n$ based on their direction vs. based on their oriented direction.

Answer 1

• On the sphere, any two "lines" intersect twice; in the projective plane, just once.
• The sphere is oriented; the projective plane is not. The letter R has more asymmetry than other letters of the alphabet; move and "R" around the projective plane and keep it going in the same direction until it returns to where it started. It will be a backwards, mirror-image "R".
• The sphere is simply connected: Draw any curve that returns to its starting point, and it can be contracted to a point. A line in the projective plane cannot be so contracted.