Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in ".
Take a open half sphere e.g. a sphere but only the points with .
Each point in the half sphere identifies a unique line through the origin.
Then, the only lines missing are the lines in the x-y plane itself.
For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.
To see why this is called a plane, move he center of the sphere to , and project each line on the x-y plane. This works for all points of the sphere, except those at the equator . Those are the points at infinity. Note that there is one for each direction in the x-y plane.
It good to think about how Euclid's postulates look like in the real projective plane:
- two parallel lines on the plane meet at a point on the sphere!Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.
- points in the real projective plane are lines in
- For every two projective points there is a single projective line that passes through them.Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.