HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, or, more generally, an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
with
points at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
, in such a way that there is one point at infinity of each
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
of
parallel lines In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

parallel lines
. This definition of a projective space has the disadvantage of not being
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by ...
, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, ''point'' and ''lines'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the following definition, which is more often encountered in modern textbooks. Using
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, a projective space of dimension is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of dimension . Equivalently, it is the
quotient set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of by the
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
"being on the same vector line". As a vector line intersects the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
of in two
antipodal points Antipodal points on a circle are 180 degrees apart. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, projective spaces can be equivalently defined as spheres in which antipodal points are identified. A projective space of dimension 1 is a
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line (geometry), 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 o ...
, and a projective space of dimension 2 is a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Projective spaces are widely used in
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, as allowing simpler statements and simpler proofs. For example, in
affine geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, two distinct lines in a plane intersect in at most one point, while, in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
, they intersect in exactly one point. Also, there is only one class of
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, which can be distinguished only by their intersections with the line at infinity: two intersection points for
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

hyperbola
s; one for the
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

parabola
, which is tangent to the line at infinity; and no real intersection point of
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
s. In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, and more specifically in
manifold theory In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or ''-manifold'' for short, is a topological space with the property that each point has a Neig ...
, projective spaces play a fundamental role, being typical examples of
non-orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclo ...
s.


Motivation

As outlined above, projective spaces were introduced for formalizing statements like "two
coplanar lines In geometry, a set of points in space are coplanar if there exists a geometric Plane (mathematics), plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear points, non-collinear, ...
intersect in exactly one point, and this point is at infinity if the lines are
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...

parallel
." Such statements are suggested by the study of perspective, which may be considered as a
central projection In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the
three dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., Point (mathematics), point). This is the info ...
onto a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
(see
Pinhole camera model . The pinhole camera model describes the mathematical relationship between the coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithm ...
). More precisely, the entrance pupil of a camera or of the eye of an observer is the ''center of projection'', and the image is formed on the ''projection plane''. Mathematically, the center of projection is a point of the space (the intersection of the axes in the figure); the projection plane (, in blue on the figure) is a plane not passing through , which is often chosen to be the plane of equation , when
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian coordinates
are considered. Then, the central projection maps a point to the intersection of the line with the projection plane. Such an intersection exists if and only if the point does not belong to the plane (, in green on the figure) that passes through and is parallel to . It follows that the lines passing through split in two disjoint subsets: the lines that are not contained in , which are in one to one correspondence with the points of , and those contained in , which are in one to one correspondence with the directions of parallel lines in . This suggests to define the ''points'' (called here ''projective points'' for clarity) of the projective plane as the lines passing through . A ''projective line'' in this plane consists of all projective points (which are lines) contained in a plane passing through . As the intersection of two planes passing through is a line passing through , the intersection of two distinct projective lines consists of a single projective point. The plane defines a projective line which is called the ''line at infinity'' of . By identifying each point of with the corresponding projective point, one can thus say that the projective plane is the
disjoint union In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

disjoint union
of and the (projective) line at infinity. As an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
with a distinguished point may be identified with its associated
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
(see ), the preceding construction is generally done by starting from a vector space and is called
projectivization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Also, the construction can be done by starting with a vector space of any positive dimension. So, a projective space of dimension can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension . A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines. This set can be the set of
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es under the equivalence relation between vectors defined by "one vector is the product of the other by a nonzero scalar". In other words, this amounts to defining a projective space as the set of vector lines in which the zero vector has been removed. A third equivalent definition is to define a projective space of dimension as the set of pairs of
antipodal points Antipodal points on a circle are 180 degrees apart. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
in a sphere of dimension (in a space of dimension ).


Definition

Given a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, the ''projective space'' is the set of
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es of under the equivalence relation defined by if there is a nonzero element of such that . If is a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
, the quotient space is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, endowed with the
quotient topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
. This is the case when is the field \mathbb R of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or the field \mathbb C of the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. If is finite dimensional, the ''dimension'' of is the dimension of minus one. In the common case where , the projective space is denoted (as well as or , although this notation may be confused with exponentiation). The space is often called ''the'' projective space of dimension over , or ''the projective -space'', since all projective spaces of dimension are
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphic
to it (because every vector space of dimension is isomorphic to ). The elements of a projective space are commonly called ''
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
''. If a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of has been chosen, and, in particular if , the
projective coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of a point ''P'' are the coordinates on the basis of any element of the corresponding equivalence class. These coordinates are commonly denoted , the colons and the brackets being used for distinguishing from usual coordinates, and emphasizing that this is an equivalence class, which is defined
up to Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
the multiplication by a non zero constant. That is, if are projective coordinates of a point, then are also projective coordinates of the same point, for any nonzero in . Also, the above definition implies that are projective coordinates of a point if and only if at least one of the coordinates is nonzero. If is the field of real or complex numbers, a projective space is called a
real projective space In mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n ...
or a
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
, respectively. If is one or two, a projective space of dimension is called a
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line (geometry), 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 o ...
or a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, respectively. The complex projective line is also called the
Riemann sphere In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Riemann sphere
. All these definitions extend naturally to the case where is a
division ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
; see, for example,
Quaternionic projective space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''n'' ...
. The notation is sometimes used for . If is a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with elements, is often denoted (see
PG(3,2) In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. It also has the following properties: * Each point is contained in ...
).


Related concepts


Subspace

Let be a projective space, where is a vector space over a field , and :p:V\to \mathbf P(V) be the ''canonical map'' that maps a nonzero vector to its equivalence class, which is the vector line containing with the zero vector removed. Every
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of is a union of lines. It follows that is a projective space, which can be identified with . A ''projective subspace'' is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines . If and are two different points of , the vectors and are
linearly independent In the theory of vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are con ...
. It follows that: :''There is exactly one projective line that passes through two different points of'' and :''A subset of'' ''is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.'' In
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.


Span

Every
intersection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

intersection
of projective subspaces is a projective subspace. It follows that for every subset of a projective space, there is a smallest projective subspace containing , the intersection of all projective subspaces containing . This projective subspace is called the ''projective span'' of , and is a spanning set for it. A set of points is ''projectively independent'' if its span is not the span of any proper subset of . If is a spanning set of a projective space , then there is a subset of that spans and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of is , such an independent spanning set has elements. Contrarily to the cases of
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s and
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
s, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.


Frame

A ''projective frame'' is an ordered set of points in a projective space that allows defining coordinates. More precisely, in a -dimensional projective space, a projective frame is a tuple of points such that any of them are independent—that is are not contained in a hyperplane. If is a -dimensional vector space, and is the canonical projection from to , then (p(e_0),\dots, p(e_)) is a projective frame if and only if (e_0, \dots, e_n) is a basis of , and the coefficients of e_ on this basis are all nonzero. By rescaling the first vectors, any frame can be rewritten as (p(e'_0),\dots, p(e'_)) such that e'_=e'_0+\dots+ e'_n; this representation is unique up to the multiplication of all e'_i with a common nonzero factor. The ''projective coordinates'' or ''homogeneous coordinates'' of a point on a frame (p(e_0),\dots, p(e_)) with e_=e_0+\dots+ e_n are the coordinates of on the basis (e_0, \dots, e_n). They are again only defined up to scaling with a common nonzero factor. The ''canonical frame'' of the projective space consists of images by of the elements of the canonical basis of (the
tuples In mathematics, a tuple is a finite ordered list (sequence) of Element (mathematics), elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tup ...
with only one nonzero entry, equal to 1), and the image by of their sum.


Projective transformation


Topology

A projective space is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, as endowed with the
quotient topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
of the topology of a finite dimensional real vector space. Let be the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
in a normed vector space , and consider the function :\pi: S \to \mathbf P(V) that maps a point of to the vector line passing through it. This function is continuous and surjective. The inverse image of every point of consist of two
antipodal point In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. As spheres are
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, it follows that: :''A (finite dimensional) projective space is compact''. For every point of , the restriction of to a neighborhood of is a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
onto its image, provided that the neighborhood is small enough for not containing any pair of antipodal points. This shows that a projective space is a manifold. A simple
atlas An atlas is a collection of map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages betw ...
can be provided, as follows. As soon as a basis has been chosen for , any vector can be identified with its coordinates on the basis, and any point of may be identified with its
homogeneous coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. For , the set :U_i = \ is an open subset of , and :\mathbf P(V) = \bigcup_^nU_i since every point of has at least one nonzero coordinate. To each is associated a
chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and c ...
, which is the
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
s :\begin \mathbb \varphi_i: R^n &\to U_i\\ (y_0,\dots,\widehat,\dots y_n)&\mapsto _0:\cdots:y_:1:y_:\cdots:y_n \end such that :\varphi_i^\left( _0:\cdots x_nright) =\left (\frac, \dots, \widehat, \dots, \frac \right ), where hats means that the corresponding term is missing. These charts form an
atlas An atlas is a collection of map A map is a symbol A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages betw ...
, and, as the
transition map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, it results that projective spaces are
analytic manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
s. For example, in the case of , that is of a projective line, there are only two , which can each be identified to a copy of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. In both lines, the intersection of the two charts is the set of nonzero real numbers, and the transition map is :x\mapsto \frac 1x in both directions. The image represents the projective line as a circle where antipodal points are identified, and shows the two homeomorphisms of a real line to the projective line; as antipodal points are identified, the image of each line is represented as an open half circle, which can be identified with the projective line with a single point removed.


CW complex structure

Real projective spaces have a simple
CW complex A CW complex is a kind of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...
structure, as can be obtained from by attaching an -cell with the quotient projection as the attaching map.


Algebraic geometry

Originally,
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
was the study of common zeros of sets of
multivariate polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. These common zeros, called
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
belong to an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the
fundamental theorem of algebra The fundamental theorem of algebra also known as d'Alembert's theorem or the d'Alembert-Gauss theorem states that every non-constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constan ...
asserts that a univariate
square-free polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of degree has exactly complex roots. In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider ''zeros at infinity''. For example, Bézout's theorem asserts that the intersection of two plane
algebraic curve In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s of respective degrees and consists of exactly points if one consider complex points in the projective plane, and if one counts the points with their multiplicity. Another example is the
genus–degree formula In classical algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are conta ...
that allows computing the genus of a plane
algebraic curve In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
from its singularities in the ''complex projective plane''. So a
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective spaces, projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n' ...
is the set of points in a projective space, whose
homogeneous coordinates In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
are common zeros of a set of
homogeneous polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. Any affine variety can be ''completed'', in a unique way, into a projective variety by adding its
points at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
, which consists of
homogenizing Homogeneity and heterogeneity are concepts often used in the Science, sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform ...
the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by saturating with respect to the homogenizing variable. An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...
(that is, it is an
algebraic set Algebraic may refer to any subject related to algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geome ...
). This is a generalization to every ground field of the compactness of the real and complex projective space. A projective space is itself a projective variety, being the set of zeros of the zero polynomial.


Scheme theory

Scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicity (mathematics), multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebra ...
, introduced by
Alexander Grothendieck Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative ...

Alexander Grothendieck
during the second half of 20th century, allows defining a generalization of algebraic varieties, called schemes, by gluing together smaller pieces called affine schemes, similarly as
manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

manifold
s can be built by gluing together open sets of \mathbb R^n. The
Proj construction In algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...
is the construction of the scheme of a projective space, and, more generally of any projective variety, by gluing together affine schemes. In the case of projective spaces, one can take for these affine schemes the affine schemes associated to the charts (affine spaces) of the above description of a projective space as a manifold.


Synthetic geometry

In
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, a projective space ''S'' can be defined axiomatically as a set ''P'' (the set of points), together with a set ''L'' of subsets of ''P'' (the set of lines), satisfying these axioms: * Each two distinct points ''p'' and ''q'' are in exactly one line. * Veblen's axiom: If ''a'', ''b'', ''c'', ''d'' are distinct points and the lines through ''ab'' and ''cd'' meet, then so do the lines through ''ac'' and ''bd''. * Any line has at least 3 points on it. The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
consisting of a set ''P'' of points, a set ''L'' of lines, and an
incidence relation In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an Incidence (geometry), incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one r ...
''I'' that states which points lie on which lines. The structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the Veblen–Young theorem, there is no difference. However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the
Theorem of Desargues In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective (geometry), perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertex (geometry), v ...
and are known as Non-Desarguesian planes. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically. It is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. gives such an extension due to Bachmann. To ensure that the dimension is at least two, replace the three point per line axiom above by; * There exist four points, no three of which are collinear. To avoid the non-Desarguesian planes, include Pappus's hexagon theorem, Pappus's theorem as an axiom; * If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear. And, to ensure that the vector space is defined over a field that does not have even Characteristic (field), characteristic include ''Fano's axiom''; * The three diagonal points of a complete quadrangle are never collinear. A subspace of the projective space is a subset ''X'', such that any line containing two points of ''X'' is a subset of ''X'' (that is, completely contained in ''X''). The full space and the empty space are always subspaces. The geometric dimension of the space is said to be ''n'' if that is the largest number for which there is a strictly ascending chain of subspaces of this form: : \varnothing = X_\subset X_\subset \cdots X_=P. A subspace X_i in such a chain is said to have (geometric) dimension i. Subspaces of dimension 0 are called ''points'', those of dimension 1 are called ''lines'' and so on. If the full space has dimension n then any subspace of dimension n-1 is called a hyperplane.


Classification

*Dimension 0 (no lines): The space is a single point. *Dimension 1 (exactly one line): All points lie on the unique line. *Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for is equivalent to a
projective plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. These are much harder to classify, as not all of them are isomorphic with a . The Desarguesian planes (those that are isomorphic with a satisfy Desargues's theorem and are projective planes over division rings, but there are many non-Desarguesian planes. *Dimension at least 3: Two non-intersecting lines exist. proved the Veblen–Young theorem that every projective space of dimension is isomorphic with a , the ''n''-dimensional projective space over some
division ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
''K''.


Finite projective spaces and planes

A ''finite projective space'' is a projective space where ''P'' is a finite set of points. In any finite projective space, each line contains the same number of points and the ''order'' of the space is defined as one less than this common number. For finite projective spaces of dimension at least three, Wedderburn's little theorem, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, GF(''q''), whose order (that is, number of elements) is ''q'' (a prime power). A finite projective space defined over such a finite field has points on a line, so the two concepts of order coincide. Notationally, is usually written as . All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are : 1, 1, 1, 1, 0, 1, 1, 4, 0, … finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the Bruck–Ryser–Chowla theorem, Bruck–Ryser theorem. The smallest projective plane is the Fano plane, with 7 points and 7 lines. The smallest 3-dimensional projective spaces is
PG(3,2) In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. It also has the following properties: * Each point is contained in ...
, with 15 points, 35 lines and 15 planes.


Morphisms

Injective linear maps between two vector spaces ''V'' and ''W'' over the same field ''k'' induce mappings of the corresponding projective spaces via: ::[''v''] → [''T''(''v'')], where ''v'' is a non-zero element of ''V'' and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If ''T'' is not injective, it has a null space larger than ; in this case the meaning of the class of ''T''(''v'') is problematic if ''v'' is non-zero and in the null space. In this case one obtains a so-called rational map, see also birational geometry). Two linear maps ''S'' and ''T'' in induce the same map between P(''V'') and P(''W'') if and only if they differ by a scalar multiple, that is if for some . Thus if one identifies the scalar multiples of the identity function, identity map with the underlying field ''K'', the set of ''K''-linear morphisms from P(''V'') to P(''W'') is simply . The automorphisms can be described more concretely. (We deal only with automorphisms preserving the base field ''K''). Using the notion of ample line bundle, sheaves generated by global sections, it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e., coming from a (linear) automorphism of the vector space ''V''. The latter form the group (mathematics), group general linear group, GL(''V''). By identifying maps that differ by a scalar, one concludes that :Aut(P(''V'')) = Aut(''V'')/''K''× = GL(''V'')/''K''× =: PGL(''V''), the quotient group of GL(''V'') modulo the matrices that are scalar multiples of the identity. (These matrices form the center of a group, center of Aut(''V'').) The groups PGL are called projective linear groups. The automorphisms of the complex projective line P1(C) are called Möbius transformations.


Dual projective space

When the construction above is applied to the dual space ''V'' rather than ''V'', one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of ''V''. That is, if ''V'' is ''n'' dimensional, then P(''V'') is the Grassmannian of planes in ''V''. In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles. One would like to be able to associate a projective space to ''every'' quasi-coherent sheaf ''E'' over a scheme ''Y'', not just the locally free ones. See Éléments de géométrie algébrique, EGAII, Chap. II, par. 4 for more details.


Generalizations

;dimension: The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space ''V'' is generalized to Grassmannian manifold, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of ''V''. ;sequence of subspaces: More generally flag manifold is the space of flags, i.e., chains of linear subspaces of ''V''. ;other subvarieties: Even more generally, moduli spaces parametrize objects such as elliptic curves of a given kind. ;other rings: Generalizing to associative ring (mathematics), rings (rather than only fields) yields, for example, the projective line over a ring. ;patching: Patching projective spaces together yields projective space bundles. Severi–Brauer variety, Severi–Brauer varieties are
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
over a field ''k'', which become isomorphic to projective spaces after an extension of the base field ''k''. Another generalization of projective spaces are weighted projective spaces; these are themselves special cases of toric variety, toric varieties.


See also


Generalizations

*Grassmannian manifold *Projective line over a ring *Space (mathematics)


Projective geometry

*projective transformation *projective representation


Related

* Geometric algebra


Notes


References

* * * * * * * Greenberg, M.J.; ''Euclidean and non-Euclidean geometries'', 2nd ed. Freeman (1980). * , esp. chapters I.2, I.7, II.5, and II.7 * Hilbert, D. and Cohn-Vossen, S.;
Geometry and the imagination
', 2nd ed. Chelsea (1999). * * (Reprint of 1910 edition)


External links

* *
Projective Planes of Small Order
{{Dimension topics, state=uncollapsed Projective geometry