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Mathematics LibreTexts

15.3: Space Model

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Let us identify the Euclidean plane with a plane Π in the Euclidean space R3 that does not pass thru the origin O. Denote by ˆΠ the projective completion of Π.

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

Further denote by Ψ the set of all planes in the space thru O. In a similar fashion, we can define a bijection ˙ between lines in ˆΠ and Ψ. If a line is not ideal, then take the plane ˙ that contains and O; if the line is ideal, then take ˙ to be the plane thru O that is parallel to Π (that is, ˙Π=).

Observation 15.3.1

P and be a point and a line in the real projective plane. Then P if and only if ˙P˙, where ˙P and ˙ 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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