This article needs additional citations for
verification. (December 2009) |

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 **P**^{1}(*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 **P**^{1}(*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 **P**^{1}(*K*) given by

This subset covers all points in **P**^{1}(*K*) except one, which is called the *point at infinity*:

This allows to extend the arithmetic on *K* to **P**^{1}(*K*) by the formulas

Translating this arithmetic in terms 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.

An example is obtained by projecting points in **R**^{2} 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 −∞.

Adding a point at infinity to the
complex plane results in a space that is topologically a
sphere. Hence the complex projective line is also known as the **
Riemann sphere** (or sometimes the *Gauss sphere*). It is in constant use in
complex analysis,
algebraic geometry and
complex manifold theory, as the simplest example of a
compact Riemann surface.

The projective line over a
finite field *F*_{q} of *q* elements has *q* + 1 points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates *x* : *y*, *q* of these points have the form:

*a*: 1] for each*a*in*F*_{q},

and the remaining
point *at infinity* may be represented as [1 : 0].

Quite generally, the group of
homographies with
coefficients in *K* acts on the projective line **P**^{1}(*K*). This
group action is
transitive, so that **P**^{1}(*K*) is a
homogeneous space for the group, often written PGL_{2}(*K*) to emphasise the projective nature of these transformations. *Transitivity* says that there exists a homography that will transform any point *Q* to any other point *R*. The *point at infinity* on **P**^{1}(*K*) is therefore an *artifact* of choice of coordinates:
homogeneous coordinates

express a one-dimensional subspace by a single non-zero point (*X*, *Y*) lying in it, but the symmetries of the projective line can move the point ∞ = [1 : 0] to any other, and it is in no way distinguished.

Much more is true, in that some transformation can take any given
distinct points *Q _{i}* for

The projective line is a fundamental example of an
algebraic curve. From the point of view of algebraic geometry, **P**^{1}(*K*) is a
non-singular curve of
genus 0. If *K* is
algebraically closed, it is the unique such curve over *K*, up to
rational equivalence. In general a (non-singular) curve of genus 0 is rationally equivalent over *K* to a
conic *C*, which is itself birationally equivalent to projective line if and only if *C* has a point defined over *K*; geometrically such a point *P* can be used as origin to make explicit the birational equivalence..

The
function field of the projective line is the field *K*(*T*) of
rational functions over *K*, in a single indeterminate *T*. The
field automorphisms of *K*(*T*) over *K* are precisely the group PGL_{2}(*K*) discussed above.

Any function field *K*(*V*) of an
algebraic variety *V* over *K*, other than a single point, has a subfield isomorphic with *K*(*T*). From the point of view of
birational geometry, this means that there will be a
rational map from *V* to **P**^{1}(*K*), that is not constant. The image will omit only finitely many points of **P**^{1}(*K*), and the inverse image of a typical point *P* will be of dimension dim *V* − 1. This is the beginning of methods in algebraic geometry that are inductive on dimension. The rational maps play a role analogous to the
meromorphic functions of
complex analysis, and indeed in the case of
compact Riemann surfaces the two concepts coincide.

If *V* is now taken to be of dimension 1, we get a picture of a typical algebraic curve *C* presented 'over' **P**^{1}(*K*). Assuming *C* is non-singular (which is no loss of generality starting with *K*(*C*)), it can be shown that such a rational map from *C* to **P**^{1}(*K*) will in fact be everywhere defined. (That is not the case if there are singularities, since for example a *
double point* where a curve *crosses itself* may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is
ramification.

Many curves, for example hyperelliptic curves, may be presented abstractly, as ramified covers of the projective line. According to the Riemann–Hurwitz formula, the genus then depends only on the type of ramification.

A **rational curve** is a curve that is
birationally equivalent to a projective line (see
rational variety); its
genus is 0. A
rational normal curve in projective space P^{n} is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence),^{
[2]} given parametrically in homogeneous coordinates as

- [1 :
*t*:*t*^{2}: ... :*t*^{n}].

See twisted cubic for the first interesting case.

- Algebraic curve
- Cross-ratio
- Möbius transformation
- Projective line over a ring
- Projectively extended real line
- Projective range
- Wheel theory

**^**Action of PGL(2) on Projective Space – see comment and cited paper.**^**Harris, Joe (1992),*Algebraic Geometry: A First Course*, Graduate Texts in Mathematics, vol. 133, Springer, ISBN 9780387977164.