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Concept
Projective line
Summary
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.
The projective line over the reals is a manifold; see real projective line for details.
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 terms of homogeneous coordinates gives, when [0 : 0] does not occur:
real projective line
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.
An example is obtained by projecting points in R2 onto the unit circle and then identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup {1, −1}.
Compare the extended real number line, which distinguishes ∞ and −∞.
Official source
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