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Mathematics LibreTexts

15.1: Projective Completion

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    In the Euclidean plane, two distinct lines might have one or zero points of intersection (in the latter case the lines are parallel). Our aim is to extend the Euclidean plane by ideal points so that any two distinct lines will have exactly one point of intersection.

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    A collection of lines in the Euclidean plane is called concurrent if they all intersect at a single point or all of them pairwise parallel. A maximal set of concurrent lines in the plane is called pencil. There are two types of pencils: central pencils contain all lines passing thru a fixed point called the center of the pencil and parallel pencil contain pairwise parallel lines.

    Each point in the Euclidean plane uniquely defines a central pencil with the center in it. Note that any two lines completely determine the pencil containing both.

    Let us add one ideal point for each parallel pencil, and assume that all these ideal points lie on one ideal line. We also assume that the ideal line belongs to each parallel pencil.

    We obtain the so-called real projective plane, or projective completion of the original plane. It comes with an incidence structure — we say that three points lie on one line if the corresponding pencils contain a common line. Projective geometry studies this incidence structure.

    Let us describe points of projective completion in coordinates. A parallel pencil contains the ideal line and the lines \(y = m \cdot x+b\) with fixed slope \(m\); if \(m=\infty\), we assume that the lines are given by equations \(x=a\). Therefore the real projective plane contains every point \((x,y)\) in the coordinate plane plus the ideal line containing one ideal point \(P_m\) for every slope \(m \in \mathbb{R} \cup \{\infty\}\).