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Mathematics LibreTexts

15.3: Space Model

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Let us identify the Euclidean plane with a plane \Pi in the Euclidean space \mathbb{R}^3 that does not pass thru the origin O. Denote by \hat{\Pi} the projective completion of \Pi.

Denote by \Phi the set of all lines in the space thru O. Let us define a bijection P \leftrightarrow \dot P between \hat \Pi and \Phi. If P\in \Pi, then take the line \dot P=(OP); if P is an ideal point of \hat \Pi, so it is defined by a parallel pencil of lines, then take the line \dot P thru O parallel to the lines in this pencil.

Further denote by \Psi the set of all planes in the space thru O. In a similar fashion, we can define a bijection \ell\leftrightarrow \dot \ell between lines in \hat \Pi and \Psi. If a line \ell is not ideal, then take the plane \dot \ell that contains \ell and O; if the line \ell is ideal, then take \dot \ell to be the plane thru O that is parallel to \Pi (that is, \dot{\ell} \cap \Pi=\emptyset).

Observation \PageIndex{1}

P and \ell be a point and a line in the real projective plane. Then P \in \ell if and only if \dot{P} \subset \dot{\ell}, where \dot{P} and \dot{\ell} denote the line and plane defined by the constructed bijections.


This page titled 15.3: Space Model is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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