Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case).

There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field K, commonly denoted P¹(K), as the set of one-dimensional subspaces of a two-dimensional K-vector space. This definition is a special instance of the general definition of a projective space.

An arbitrary point in the projective line P¹(K) may be represented by an equivalence class of homogeneous coordinates, which take the form of a pair

of elements of K that are not both zero. Two such pairs are equivalent if they differ by an overall nonzero factor λ:

The projective line may be identified with the line K extended by a point at infinity. More precisely, the line K may be identified with the subset of P¹(K) given by

This subset covers all points in P¹(K) except one, which is called the point at infinity:

This allows to extend the arithmetic on K to P¹(K) by the formulas

Translating this arithmetic in term of homogeneous coordinates gives, when [0 : 0] does not occur:

The projective line over the real numbers is called the real projective line. It may also be thought of as the line K together with an idealised point at infinity ∞ ; the point connects to both ends of K creating a closed loop or topological circle.

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