In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P¹(K), may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K² (it does carry other geometric structures). The projective line may also be thought of as the line K together with an idealised point at infinity. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a projective space is a fundamental construction from any vector space. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Screenshot (from SSCX Star Warzone). ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

For the generalisation to the projective line over an associative ring, see inversive ring geometry. In mathematics, inversive ring geometry is the extension, to the context of associative rings, of the concepts of Projective line, homogeneous coordinates, projective transformations, and Cross-ratio, concepts usually built upon rings that happen to be fields. ...

Homogeneous coordinates

An arbitrary point in the projective line P¹(K) may be given in homogeneous coordinates by a pair In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

[x₁:x₂]

of points in K which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor λ:

[x₁:x₂] = [λx₁:λx₂].

The line K may be identified with the subset of P¹(K) given by

This subset covers all points in P¹(K) except one: the point at infinity, which may be given as

Examples

Real projective line

The projective line over the real numbers is called the real projective line. It is given 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}. Topologically, it is again a circle. One may also think of gluing the two "ends" of the real line onto a new point ∞ resulting in a circle. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Illustration of a unit circle. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. ... Group theory is that branch of mathematics concerned with the study of groups. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...

Compare the extended real number line, which distinguishes ∞ and −∞. The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

Complex projective line: the Riemann sphere

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. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. ...

For a finite field

The case of K a finite field F is also simple to understand. In this case if F has q elements, the projective line has In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

q + 1

elements. We can write all but one of the subspaces as

y = ax

with a in F; this leaves out only the case of the line x = 0. For a finite field there is a definite loss if the projective line is taken to be this set, rather than an algebraic curve — one should at least see the underlying infinite set of points in an algebraic closure as potentially on the line. In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

Symmetry group

Quite generally, the group of Möbius transformations with coefficients in K acts on the projective line P¹(K). This group action is transitive, so that P¹(K) is a homogeneous space for the group, often written PGL₂(K) to emphasise its definition as a projective linear group. Transitivity says that any point Q may be transformed to any other point R by a Möbius transformation. The point at infinity on P¹(K) is therefore an artefact of choice of coordinates: homogeneous coordinates In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, the word transitive admits at least two distinct meanings: A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear... In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

[X:Y] = [tX:tY]

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 i = 1,2,3 to any other 3-tuple R_i of distinct points (triple transitivity). This amount of specification 'uses up' the three dimensions of PGL₂(K). The computational aspect of this is the cross-ratio. Two or more things are distinct if no two of them are the same thing. ... In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ...

As algebraic curve

The projective line is a fundamental example of an algebraic curve. From the point of view of algebraic geometry, P¹(K) is a non-singular curve of genus 0. If K is algebraically closed, it is the unique such curve over K, up to isomorphism. In general (non-singular) curves of genus 0 are isomorphic over K to a conic C, which is the 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 clear the correspondence using lines through P. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ...

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 PGL₂(K) discussed above. In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

One reason for the great importance of the projective line is that any function field K(V) of an algebraic variety V over K, other than a single point, will have 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 P¹(K), that is not constant. The image will omit only finitely many points of P¹(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. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... Complex analysis is the branch of mathematics investigating functions of complex numbers. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...

If V is now taken to be of dimension 1, we get a picture of a typical algebraic curve C presented 'over' P¹(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 P¹(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. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, ramification is a geometric term used for branching out, in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. ...

Many curves, for example hyperelliptic curves, are best 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. In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ... In mathematics, branched covering is a term mainly used in algebraic geometry, to describe morphisms f from an algebraic variety V to another one W, the two dimensions being the same, and the typical fibre of f being of dimension 0. ... In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ...

A rational curve is a curve of genus 0, so any curve in the birational class of the projective line (see rational variety). A rational normal curve in projective space Pⁿ is a rational curve that lies in no proper linear subspace; it is known that there is essentially one example, given parametrically in homogeneous coordinates as In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K. This is a question on its function field: is it up to isomorphism the field of all rational functions for some set of indeterminates... In mathematics, the rational normal curve is a smooth, rational curve of degree n in projective n-space . ...

[1:t:t²:...:tⁿ].

See twisted cubic for the first interesting case. In mathematics, a twisted cubic is a smooth, rational curve of degree three in projective 3-space . ...