# Projective aircraft and also its twin

So the projective aircraft $\mathbb{RP}^2$ is not a vector room. Is it still isomorphic to its twin? Otherwise, exists at the very least an invertible map that takes $\mathbb{RP}^2$ to its twin?

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2019-05-04 16:29:22
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Yes, though words "twin" is rather suspicious. If you suggest is $\mathbb{P}(\mathbb{R}^3)$, the typical projective aircraft, isomorphic to $\mathbb{P}((\mathbb{R}^3)^*)$, the projectivization of the twin, after that of course, it adheres to from the isomorphism of vector rooms.

Far more surprisingly, the duality permits you to switch over factors and also lines in theories, such as the Mystic Hexagon and also Brianchon's Theorem.

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2019-05-08 00:40:04
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This is a comment greater than a solution, mostly in feedback to Qiaochu's inquiry, yet I do not have enough rep. to comment.

The twin to a projective aircraft is the set of all lines in the aircraft, which itself is a. projective aircraft (as meant in Charles Siegel's solution ). This is a vital principle in timeless projective geometry. (Concretely, the formula for a line has the kind. a x + b y + c z = 0, where a, b, and also c are some parameters, not all absolutely no, and also x, y, z are uniform coords. for the factors in the proj. aircraft. The set of all such lines can hence. be taken the set of all (a, b, c ) not all absolutely no; yet keep in mind that all at once increasing a, b, and also c by a non-zero scalar does not transform the remedy set, i.e. does not transform the line, so the line needs to actually be taken representing the uniform works with (a :b :c ); hence the set of all lines is once more a projective aircraft. )

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2019-05-08 00:28:02
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