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Motivation of standard construction of projective space P(V) from a vector space V

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Projective space is the moduli space of LINES, not the moduli space of DIRECTIONS.
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Here’s one good reason that only requires high school geometry. Remember Euclid’s first axiom: given any two distinct points, there exists exactly one line that contains both of them. If you don’t identify antipodal points on the sphere, then this axiom fails. “Lines” on the sphere are great circles, but, for example, there are infinitely many great circles containing the north and south poles. It turns out that this is fixed once you identify antipodal points. Of course, not all geometric spaces need to satisfy Euclid’s axioms, but this is a desirable property for a space that we want to have some sort of linear structure.
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I like to imagine the plane as being tangent to the sphere, rather than going through the origin. Then ignoring lines through the equator, every line through the origin goes through exactly one point on the plane, so you have a correspondence between most lines and points on the plane. Then the lines through the equator correspond to the points at infinity.
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I like to think of projective space as a natural compactification of the affine space of the same dimension (the picture in my head is the affine plane {z=1} sitting inside R\^3 and representing a copy of R\^2 inside P\^2). Adding in the projective points at infinity is natural because all you are doing is adding in enough points to generalize natural geometric properties. As another commentor mentioned, any pair of lines in projective space intersect at exactly one point, while only non-parallel lines intersecting at a point in affine space. More generally the zeroes of a generic polynomial of degree d intersects any line at exactly d points, while in the affine plane it can be either d or d-1.
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I think the idea of constructing P(V) as the space of lines through the origin is perhaps the best geometric intuition behind the problem, but if you're still looking for another angle or motivation for it, I think the idea of P(V) as generalized ratios is worth thinking about.

If we assume our vector space is k^n for some field k and natural n, then P( k^2 ) is the space of ratios between two elements of k, and P( k^n ) is the space of ratios between n elements. This has many nice analogies, for example in cooking you may mix three ingredients in a ratio of 1:1:2, so in some sense projective space models the space of "recipes" without regard for the exact quantity of ingredients.
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I like to think of it as the set of all lines thru the origin. Then its almost the same as the unit sphere but you identify antipodal points

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