In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, projective geometry is the study of geometric properties that are invariant with respect to
projective transformations. This means that, compared to elementary
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, projective geometry has a different setting (''
projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, for a given dimension, and that
geometric transformations are permitted that transform the extra points (called "
points at infinity") to Euclidean points, and vice versa.
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a
transformation matrix and
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s (the
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More general ...
s). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the concept of an
angle
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in
perspective drawing from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which
parallel lines can be said to meet in a
point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See ''
Projective plane'' for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of
complex projective space, the coordinates used (
homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (including
invariant theory, the
Italian school of algebraic geometry, and
Felix Klein's
Erlangen programme resulting in the study of the
classical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as
synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is
finite geometry.
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of
projective varieties) and
projective differential geometry (the study of
differential invariants of the projective transformations).
Overview
Projective geometry is an elementary non-
metrical form of geometry, meaning that it does not support any concept of distance. In two dimensions it begins with the study of
configurations of
points and
lines. That there is indeed some geometric interest in this sparse setting was first established by
Desargues and others in their exploration of the principles of
perspective art. In
higher dimensional spaces there are considered
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s (that always meet), and other linear subspaces, which exhibit
the principle of duality. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a
straight-edge alone, excluding
compass constructions, common in
straightedge and compass constructions. As such, there are no circles, no angles, no measurements, no parallels, and no concept of
intermediacy (or "betweenness"). It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different
conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems.
During the early 19th century the work of
Jean-Victor Poncelet,
Lazare Carnot
Lazare Nicolas Marguerite, Comte Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist, military officer, politician and a leading member of the Committee of Public Safety during the French Revolution. His military refor ...
and others established projective geometry as an independent field of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. Its rigorous foundations were addressed by
Karl von Staudt and perfected by Italians
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...
,
Mario Pieri,
Alessandro Padoa and
Gino Fano during the late 19th century. Projective geometry, like
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substi ...
and
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, can also be developed from the
Erlangen program of Felix Klein; projective geometry is characterized by
invariants under
transformations of the
projective group.
After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The
incidence structure and the
cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the
affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing projective geometry in the style of
analytic geometry is given by homogeneous coordinates. On the other hand, axiomatic studies revealed the existence of
non-Desarguesian planes, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.

In a foundational sense, projective geometry and
ordered geometry are elementary since they each involve a minimal set of
axioms and either can be used as the foundation for
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substi ...
and
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. Projective geometry is not "ordered" and so it is a distinct foundation for geometry.
Description
Projective geometry is less restrictive than either
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
or
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is i ...
. It is an intrinsically non-
metrical geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the
incidence structure and the relation of
projective harmonic conjugates are preserved. A
projective range is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that
parallel lines meet at
infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
, and therefore are drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines meet on a horizon line by virtue of their incorporating the same direction.
Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or planes in this regard—those at infinity are treated just like any others.
Because a
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using
homogeneous coordinates.
Additional properties of fundamental importance include
Desargues' Theorem and the
Theorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to prove
Desargues' Theorem. But for dimension 2, it must be separately postulated.
Using
Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations satisfy the axioms of a field – except that the commutativity of multiplication requires
Pappus's hexagon theorem. As a result, the points of each line are in one-to-one correspondence with a given field, , supplemented by an additional element, ∞, such that , , , , , , except that , , , , and remain undefined.
Projective geometry also includes a full theory of
conic sections, a subject also extensively developed in Euclidean geometry. There are advantages to being able to think of a
hyperbola and an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
as distinguished only by the way the hyperbola ''lies across the line at infinity''; and that a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
is distinguished only by being tangent to the same line. The whole family of circles can be considered as ''conics passing through two given points on the line at infinity'' — at the cost of requiring
complex coordinates. Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering the ''linear system'' of all conics passing through those points as the basic object of study. This method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by
H. F. Baker.
History
The first geometrical properties of a projective nature were discovered during the 3rd century by
Pappus of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
.
Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see ' for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry).
Johannes Kepler
Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
(1571–1630) and
Girard Desargues (1591–1661) independently developed the concept of the "point at infinity". Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year-old
Blaise Pascal
Blaise Pascal (19June 162319August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest ...
and helped him formulate
Pascal's theorem. The works of
Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until
Michel Chasles chanced upon a handwritten copy during 1845. Meanwhile,
Jean-Victor Poncelet had published the foundational treatise on projective geometry during 1822. Poncelet examined the projective properties of objects (those invariant under central projection) and, by basing his theory on the concrete
pole and polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into i ...
relation with respect to a circle, established a relationship between metric and projective properties. The
non-Euclidean geometries discovered soon thereafter were eventually demonstrated to have models, such as the
Klein model of
hyperbolic space, relating to projective geometry.
In 1855
A. F. Möbius wrote an article about permutations, now called
Möbius transformations, of
generalised circles in the
complex plane. These transformations represent projectivities of the
complex projective line. In the study of lines in space,
Julius Plücker used
homogeneous coordinates in his description, and the set of lines was viewed on the
Klein quadric, one of the early contributions of projective geometry to a new field called
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, an offshoot of
analytic geometry with projective ideas.
Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning
hyperbolic geometry by providing
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
s for the
hyperbolic plane: for example, the
Poincaré disc model where generalised circles perpendicular to the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
correspond to "hyperbolic lines" (
geodesics), and the "translations" of this model are described by Möbius transformations that map the
unit disc to itself. The distance between points is given by a
Cayley–Klein metric, known to be invariant under the translations since it depends on
cross-ratio, a key projective invariant. The translations are described variously as
isometries in
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
theory, as
linear fractional transformations formally, and as projective linear transformations of the
projective linear group, in this case .
The work of
Poncelet,
Jakob Steiner and others was not intended to extend analytic geometry. Techniques were supposed to be ''
synthetic'': in effect
projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the
projective plane alone, the axiomatic approach can result in
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
s not describable via
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 (mathemat ...
.
This period in geometry was overtaken by research on the general
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
by
Clebsch,
Riemann,
Max Noether and others, which stretched existing techniques, and then by
invariant theory. Towards the end of the century, the
Italian school of algebraic geometry (
Enriques,
Segre,
Severi) broke out of the traditional subject matter into an area demanding deeper techniques.
During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in
enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
es, taken as representing the
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
of
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
s.
Projective geometry later proved key to
Paul Dirac's invention of
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. At a foundational level, the discovery that
quantum measurements could fail to commute had disturbed and dissuaded
Heisenberg, but past study of projective planes over noncommutative rings had likely desensitized Dirac. In more advanced work, Dirac used extensive drawings in projective geometry to understand the intuitive meaning of his equations, before writing up his work in an exclusively algebraic formalism.
Classification
There are many projective geometries, which may be divided into discrete and continuous: a ''discrete'' geometry comprises a set of points, which may or may not be ''finite'' in number, while a ''continuous'' geometry has infinitely many points with no gaps in between.
The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of
Desargues' Theorem.

The smallest 2-dimensional projective geometry (that with the fewest points) is the
Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
*
BC*
DE*
FG*
DG*
EF*
DF*
EG
with
homogeneous coordinates , , , , , , , or, in affine coordinates, , , , , , and . The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways.
In standard notation, a
finite projective geometry is written where:
: is the projective (or geometric) dimension, and
: is one less than the number of points on a line (called the ''order'' of the geometry).
Thus, the example having only 7 points is written .
The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of
homogeneous coordinates, and in which
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
may be embedded (hence its name,
Extended Euclidean plane).
The fundamental property that singles out all projective geometries is the ''elliptic''
incidence property that any two distinct lines and in the
projective plane intersect at exactly one point . The special case in
analytic geometry of ''parallel'' lines is subsumed in the smoother form of a line ''at infinity'' on which lies. The ''line at infinity'' is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the
Erlangen programme one could point to the way the
group of transformations can move any line to the ''line at infinity'').
The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
: Given a line and a point not on the line,
::; ''
Elliptic'' : there exists no line through that does not meet
::; ''
Euclidean'' : there exists exactly one line through that does not meet
::; ''
Hyperbolic'' : there exists more than one line through that does not meet
The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.
Duality
In 1825,
Joseph Gergonne noted the principle of
duality characterizing projective plane geometry: given any theorem or definition of that geometry, substituting ''point'' for ''line'', ''lie on'' for ''pass through'', ''collinear'' for ''concurrent'', ''intersection'' for ''join'', or vice versa, results in another theorem or valid definition, the "dual" of the first. Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping ''point'' and ''plane'', ''is contained by'' and ''contains''. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension ''R'' and dimension . For , this specializes to the most commonly known form of duality—that between points and lines.
The duality principle was also discovered independently by
Jean-Victor Poncelet.
To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).
In practice, the principle of duality allows us to set up a ''dual correspondence'' between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a
conic curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
in a concentric sphere to obtain the dual polyhedron.
Another example is
Brianchon's theorem, the dual of the already mentioned
Pascal's theorem, and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):
* Pascal: If all six vertices of a hexagon lie on a
conic, then the intersections of its opposite sides ''(regarded as full lines, since in the projective plane there is no such thing as a "line segment")'' are three collinear points. The line joining them is then called the Pascal line of the hexagon.
* Brianchon: If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the Brianchon point of the hexagon.
: (If the conic degenerates into two straight lines, Pascal's becomes
Pappus's theorem, which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)
Axioms of projective geometry
Any given geometry may be deduced from an appropriate set of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s. Projective geometries are characterised by the "elliptic parallel" axiom, that ''any two planes always meet in just one line'', or in the plane, ''any two lines always meet in just one point''. In other words, there are no such things as parallel lines or planes in projective geometry.
Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).
Whitehead's axioms
These axioms are based on
Whitehead, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
* G1: Every line contains at least 3 points
* G2: Every two distinct points, A and B, lie on a unique line, AB.
* G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).
The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying these
three axioms either have at most one line, or are projective spaces of some dimension over a
division ring, or are
non-Desarguesian planes.
Additional axioms
One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's ''Projective Geometry'', references Veblen in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not 2.
Axioms using a ternary relation
One can pursue axiomatization by postulating a ternary relation,
BCto denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
* C0:
BA* C1: If A and B are distinct points such that
BCand
BDthen
DC* C2: If A and B are distinct points then there exists a third distinct point C such that
BC* C3: If A and C are distinct points, and B and D are distinct points, with
CEand
DEbut not
BE then there is a point F such that
CFand
DF
For two distinct points, A and B, the line AB is defined as consisting of all points C for which
BC The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.
The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set of points is independent,
B...Zif is a minimal generating subset for the subspace AB...Z.
The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent
form as follows. A projective space is of:
* (L1) at least dimension 0 if it has at least 1 point,
* (L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),
* (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),
* (L4) at least dimension 3 if it has at least 4 non-coplanar points.
The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:
* (M1) at most dimension 0 if it has no more than 1 point,
* (M2) at most dimension 1 if it has no more than 1 line,
* (M3) at most dimension 2 if it has no more than 1 plane,
and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.
It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.
Axioms for projective planes
In
incidence geometry, most authors
[, , , , , , , and among the references given.] give a treatment that embraces the
Fano plane as the smallest finite projective plane. An axiom system that achieves this is as follows:
* (P1) Any two distinct points lie on a line that is unique.
* (P2) Any two distinct lines meet at a point that is unique.
* (P3) There exist at least four points of which no three are collinear.
Coxeter's ''Introduction to Geometry'' gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, adding
Pappus's theorem to the list of axioms above (which eliminates
non-Desarguesian planes) and excluding projective planes over fields of characteristic 2 (those that do not satisfy
Fano's axiom). The restricted planes given in this manner more closely resemble the
real projective plane.
Perspectivity and projectivity
Given three non-
collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the
complete quadrangle configuration.
An
harmonic quadruple of points on a line occurs when there is a complete quadrangle two of whose diagonal points are in the first and third position of the quadruple, and the other two positions are points on the lines joining two quadrangle points through the third diagonal point.
A spatial
perspectivity of a
projective configuration in one plane yields such a configuration in another, and this applies to the configuration of the complete quadrangle. Thus harmonic quadruples are preserved by perspectivity. If one perspectivity follows another the configurations follow along. The composition of two perspectivities is no longer a perspectivity, but a projectivity.
While corresponding points of a perspectivity all converge at a point, this convergence is ''not'' true for a projectivity that is ''not'' a perspectivity. In projective geometry the intersection of lines formed by corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a projective conic, and in acknowledgement of the work of
Jakob Steiner, it is referred to as a
Steiner conic.
Suppose a projectivity is formed by two perspectivities centered on points ''A'' and ''B'', relating ''x'' to ''X'' by an intermediary ''p'':
:
The projectivity is then
Then given the projectivity
the induced conic is
:
Given a conic ''C'' and a point ''P'' not on it, two distinct
secant line
In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points..
The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inter ...
s through ''P'' intersect ''C'' in four points. These four points determine a quadrangle of which ''P'' is a diagonal point. The line through the other two diagonal points is called the
polar of ''P'' and ''P ''is the pole of this line. Alternatively, the polar line of ''P'' is the set of
projective harmonic conjugates of ''P'' on a variable secant line passing through ''P'' and ''C''.
See also
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Projective line
In projective geometry and mathematics more generally, 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 ...
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Projective plane
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Incidence (mathematics)
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Fundamental theorem of projective geometry
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Projective line over a ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field (mathematics), field. Given a ring (mathematics), ring ''A'' (with 1), the projective line P1(''A'') over ''A'' consists of points iden ...
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Grassmann–Cayley algebra
Notes
References
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Santaló, Luis (1966) ''Geometría proyectiva'', Editorial Universitaria de Buenos Aires
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External links
Projective Geometry for Machine Vision— tutorial by Joe Mundy and Andrew Zisserman.
based on Coxeter's ''The Real Projective Plane''.
— free tutorial by Roger Mohr and Bill Triggs.
Projective Geometry.— free tutorial by Tom Davis.
The Grassmann method in projective geometryA compilation of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"(English translation of book)
E. Kummer, "General theory of rectilinear ray systems"(English translation)
M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes"(English translation)
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