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In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...
that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem,
points at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adj ...
have been introduced, in such a way that in a
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified. An example of a real projective line is the projectively extended real line, which is often called ''the'' projective line. Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R). Each element of PGL(2, R) can be defined by a nonsingular 2×2 real matrix, and two matrices define the same element of PGL(2, R) if one is the product of the other and a nonzero real number. Topologically, real projective lines are homeomorphic to
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s. The complex analog of a real projective line is a
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
; that is, a Riemann sphere.


Definition

The points of the real projective line are usually defined as equivalence classes of an equivalence relation. The starting point is a real vector space of dimension 2, . Define on the
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
to hold when there exists a nonzero real number such that . The definition of a vector space implies almost immediately that this is an equivalence relation. The equivalence classes are the vector lines from which the zero vector has been removed. The real projective line is the set of all equivalence classes. Each equivalence class is considered as a single point, or, in other words, a ''point'' is defined as being an equivalence class. If one chooses a basis of , this amounts (by identifying a vector with its
coordinates vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
) to identify with the direct product , and the equivalence relation becomes if there exists a nonzero real number such that . In this case, the projective line is preferably denoted or \mathbb\mathbb^1. The equivalence class of the pair is traditionally denoted , the colon in the notation recalling that, if , the
ratio In mathematics, a ratio shows how many times one number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lan ...
is the same for all elements of the equivalence class. If a point is the equivalence class one says that is a pair of
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinate system, Cartesian coordinates are u ...
of . As is defined through an equivalence relation, the
canonical projection In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
from to defines a topology (the quotient topology) and a differential structure on the projective line. However, the fact that equivalence classes are not finite induces some difficulties for defining the differential structure. These are solved by considering as a Euclidean vector space. The
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
of the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
s is, in the case of , the set of the vectors whose coordinates satisfy . This circle intersects each equivalence classes in exactly two opposite points. Therefore, the projective line may be considered as the quotient space of the circle by the equivalence relation such that if and only if either or .


Charts

The projective line is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
. This can be seen by above construction through an equivalence relation, but is easier to understand by providing an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographi ...
consisting of two charts * Chart #1: y\ne 0, \quad : y\mapsto \frac * Chart #2: x\ne 0, \quad : y\mapsto \frac The equivalence relation provides that all representatives of an equivalence class are sent to the same real number by a chart. Either of or may be zero, but not both, so both charts are needed to cover the projective line. The
transition map In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
between these two charts is the multiplicative inverse. As it is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
, and even an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
(outside of zero), the real projective line is both a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
and an
analytic manifold In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic g ...
. The
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
of chart #1 is the map : x \mapsto : 1 It defines an embedding of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
into the projective line, whose complement of the image is the point . The pair consisting of this embedding and the projective line is called the projectively extended real line. Identifying the real line with its image by this embedding, one sees that the projective line may be considered as the union of the real line and the single point , called the point at infinity of the projectively extended real line, and denoted . This embedding allows us to identify the point either with the real number if , or with in the other case. The same construction may be done with the other chart. In this case, the point at infinity is . This shows that the notion of point at infinity is not intrinsic to the real projective line, but is relative to the choice of an embedding of the real line into the projective line.


Structure

The real projective line is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
projective range In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For inst ...
that is found in the real projective plane and in the complex projective line. Its structure is thus inherited from these superstructures. Primary among these structures is the relation of projective harmonic conjugates among the points of the projective range. The real projective line has a
cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. In ...
that extends the usual order of the real numbers.


Automorphisms


The projective linear group and its action

Matrix-vector multiplication defines a left action of on the space of column vectors: explicitly, : \begin a & b \\ c & d \end \begin x \\ y \end = \begin ax+by \\ cx+dy \end. Since each matrix in fixes the zero vector and maps proportional vectors to proportional vectors, there is an induced action of on : explicitly, : \begin a & b \\ c & d \end :y= ax+by : cx+dy (Here and below, the notation :y/math> for homogeneous coordinates denotes the equivalence class of the
column matrix In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
\textstyle \beginx\\y\end; it must not be confused with the
row matrix In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
\;y) The elements of that act trivially on are the nonzero scalar multiples of the identity matrix; these form a subgroup denoted . The projective linear group is defined to be the quotient group . By the above, there is an induced faithful action of on . For this reason, the group may also be called the ''group of linear automorphisms'' of .


Linear fractional transformations

Using the identification sending to and to , one obtains a corresponding action of on , which is by linear fractional transformations: explicitly, since : \begin a & b \\ c & d \end :1= ax+b : cx+d \quad \mathrm \quad \begin a & b \\ c & d \end :0= a : c the class of \begin a & b \\ c & d \end in acts as x \mapsto \frac and \infty \mapsto \frac, with the understanding that each fraction with denominator 0 should be interpreted as .Koblitz, ''Introduction to elliptic curves and modular forms'', Springer, 1993, III.§1.


Properties

*Given two ordered triples of distinct points in , there exists a unique element of mapping the first triple to the second; that is, the action is sharply 3-transitive. For example, the linear fractional transformation mapping to is the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform i ...
x\mapsto \frac. *The stabilizer in of the point is the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Rel ...
of the real line, consisting of the transformations x \mapsto ax+b for all and .


Notes

{{Reflist


References

* Juan Carlos Alvarez (2000
The Real Projective Line
course content from New York University. * Santiago Cañez (2014
Notes on Projective Geometry
from Northwestern University. Projective geometry Manifolds