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I'm trying to develop some intuition for projective space, and have encountered the following ways of thinking of the space (for simplicity, say we are looking at two dimensional real projective space)

1) The affine plane together with all "points at infinity," or all directions one could approach infinity.

2) The half-sphere in 3 dimensional space, with antipodal points identified. This characterization makes more sense with the algebraic notion that 2D projective space is all points in 3 dimensions up to multiplication by a scalar, i.e. all lines through the origin, where we simply pick "nice" representatives on the half sphere.

3) Some other type of "projection" visualization, such as seen at the top of page 2 at this link: http://elib.mi.sanu.ac.rs/files/journals/tm/18/tm1014.pdf

In particular, I am wondering the most practical manner in which to visualize the inclusion of regular 2 dimensional curves into 2 dimensional projective space. For instance, I suppose a normal parabola in R2 could be imagined to "connect" at infinity and form a closed curve. Could this visualization work by "projecting" the parabola onto a sphere, or by some other method?

Thanks for any help in making sense of these 3 different methods of visualization.

kfriend
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  • The (real) projective plane is topologically equivalent to the Roman surface, so there isn't actually a way to turn it into a sphere without ruining the open set structure. (The complex projective line, on the other hand, is identical to the Riemann sphere.) – Chappers Jun 09 '15 at 17:52
  • Thanks so much for the info, but I'm trying to grasp some intuition on the space having never encountered it, and am not familiar with Roman/Riemann spheres. Is there any more elementary intuition you could help me with? – kfriend Jun 10 '15 at 04:42
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    For many applications, I tend to think about this in terms of pairs of antipodal points on the sphere. Cinderella can display such a view, and you can indeed see that a parabola touches the line at infinity. – MvG Jun 10 '15 at 20:09

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