In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
,
projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting,
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
, and a selective set of basic geometric concepts. The basic intuitions are that projective space has
more points than
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, in a given dimension, and that geometric transformations are permitted that move the extra points (called "
points at infinity") to traditional points, and vice versa.
The properties that are meaningful in projective geometry are those that are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a
transformation matrix and
translationIn Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
s (the
affine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s); the first issue for geometers is what kind of geometric language would be adequate to the novel situation. It is not possible to talk about
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s in projective geometry as it is in
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which
parallel linesParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
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 planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
of that time. This was the theory of
complex projective spaceIn 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 complex projective space label the complex lines...
, since the coordinates used (
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
) were complex numbers. Several major strands of more abstract mathematics (including
invariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
, the
Italian school of algebraic geometryIn relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
, and
Felix KleinChristian Felix Klein was a German mathematician, known for his work in group theory, function theory, nonEuclidean geometry, and on the connections between geometry and group theory...
's Erlangen programme leading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of
synthetic geometrySynthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.Logical synthesis:The process...
. Another field that emerged from axiomatic studies of projective geometry is
finite geometryA finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many points as there are real numbers...
.
The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) and
projective differential geometryIn mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties that are invariant under the projective group. This is a mixture of attitudes from Riemannian geometry, and the Erlangen program....
(the study of differential invariants of the projective transformations).
Overview
Projective geometry is an elementary non
metricalIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of
configurationsIn mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.Although certain specific...
of
pointsIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zerodimensional; i.e., they do not have volume, area, length, or any other higherdimensional analogue. In branches of mathematics...
and
linesThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
. That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by
DesarguesGirard Desargues was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour.Born in Lyon, Desargues came from a family devoted to service to the French...
and others in their exploration of the principles of
perspective artPerspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...
. In higher dimensional spaces there are considered hyperplanes (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
straightedgeA straightedge is a tool with an edge free from curves, or straight, used for transcribing straight lines, or checking the straightness of lines...
alone. Since projective geometry excludes
compassA compass or pair of compasses is a technical drawing instrument that can be used for inscribing circles or arcs. As dividers, they can also be used as a tool to measure distances, in particular on maps...
constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy. It was realised that the theorems that do hold in projective geometry are simpler statements. For example the different
conic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s are all equivalent in (complex) projective geometry, and some theorems about circles can be seen as special cases of these general theorems.
In the early 19th century the work of
PonceletJeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
,
Lazare CarnotLazare Nicolas Marguerite, Comte Carnot , the Organizer of Victory in the French Revolutionary Wars, was a French politician, engineer, and mathematician.Education and early life:...
and others established projective geometry as an independent field of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians
Giuseppe PeanoGiuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
,
Mario PieriMario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...
,
Alessandro PadoaAlessandro Padoa was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive notion is truly independent of the other primitive notions...
and
Gino FanoGino Fano was an Italian mathematician. He was born in Mantua, Italy and died in Verona, Italy.Fano worked on projective and algebraic geometry; the Fano plane, Fano fibration, Fano surface, and Fano varieties are named for him....
late in the 19th century. Projective geometry, like
affineIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
and
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by
invariantsIn mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
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 structureIn mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...
and the
crossratioIn geometry, the crossratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
are fundamental invariants under projective transformations. Projective geometry can be modeled by the
affine planeIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
(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 geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
is given by homogeneous coordinates. On the other hand axiomatic studies revealed the existence of
nonDesarguesian planeIn mathematics, a nonDesarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...
s, 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 geometryOrdered geometry is a form of geometry featuring the concept of intermediacy but, like projective geometry, omitting the basic notion of measurement...
are elementary since they involve a minimum of axioms and either can be used as the foundation for
affineIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
and
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. Projective geometry is not "ordered" and so it is a distinct foundation for geometry.
History
The first geometrical properties of a projective nature were discovered in the third century by
Pappus of AlexandriaPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
.
Filippo BrunelleschiFilippo Brunelleschi was one of the foremost architects and engineers of the Italian Renaissance. He is perhaps most famous for inventing linear perspective and designing the dome of the Florence Cathedral, but his accomplishments also included bronze artwork, architecture , mathematics,...
(1404–1472) started investigating the geometry of perspective in 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts which motivated much of the development of projective geometry).
Johannes KeplerJohannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
(1571–1630) and
Gérard DesarguesGirard Desargues was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour.Born in Lyon, Desargues came from a family devoted to service to the French...
(1591–1661) independently developed the pivotal 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 geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, where parallel lines are truly parallel, into a special case of an allencompassing geometric system. Desargues's study on conic sections drew the attention of 16year old
Blaise Pascal Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...
and helped him formulate
Pascal's theoremIn projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration. Related results :This theorem...
. The works of
Gaspard MongeGaspard Monge, Comte de Péluse was a French mathematician, revolutionary, and was inventor of descriptive geometry. During the French Revolution, he was involved in the complete reorganization of the educational system, founding the École Polytechnique...
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 ChaslesMichel Floréal Chasles was a French mathematician.He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coalition he was drafted to fight in the defence of Paris in 1814...
chanced upon a handwritten copy in 1845. Meanwhile,
JeanVictor PonceletJeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
had published the foundational treatise on projective geometry in 1822. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The
nonEuclidean geometriesNonEuclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
discovered shortly thereafter were eventually demonstrated to have models, such as the
Klein modelIn geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of ndimensional hyperbolic geometry in which points are represented by the points in the interior of the ndimensional unit ball and lines are represented by the...
of
hyperbolic spaceIn mathematics, hyperbolic space is a type of nonEuclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
, relating to projective geometry.
This early 19th century projective geometry was a stepping stone from
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
to
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. When treated in terms of
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, an extension reducing the number of special cases. The detailed study of
quadricIn mathematics, a quadric, or quadric surface, is any Ddimensional hypersurface in dimensional space defined as the locus of zeros of a quadratic polynomial...
s and the "line geometry" of
Julius PlückerJulius Plücker was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. Early...
still form a rich set of examples for geometers working with more general concepts.
The work of
PonceletJeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
,
SteinerJakob Steiner was a Swiss mathematician who worked primarily in geometry.Personal and professional life:...
and others was not intended to extend
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
. Techniques were supposed to be
syntheticSynthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.Logical synthesis:The process...
: in effect
projective spaceIn mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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 planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
alone, the axiomatic approach can result in
modelIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
s not describable via
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
.
This period in geometry was overtaken by research on the general
algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections. Plane algebraic curves...
by Clebsch,
RiemannGeorg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
,
Max NoetherMax Noether was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".Biography:...
and others, which stretched existing techniques, and then by
invariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
. Towards the end of the century the
Italian school of algebraic geometryIn relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
(
EnriquesFederigo Enriques was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry....
, Segre,
SeveriFrancesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
) broke out of the traditional subject matter into an area demanding deeper techniques.
In 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 geometryIn mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.History:...
in particular, by Schubert, that is now seen as anticipating the theory of
Chern classIn mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .Basic idea and motivation:...
es, taken as representing the
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
of
GrassmannianIn mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
s.
Paul DiracPaul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
studied projective geometry and used it as a basis for developing his concepts of
Quantum MechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, although his published results were always in algebraic form. See
a blog article referring to an article and a book on this subject, also to a talk Dirac gave to a general audience in 1972 in Boston about projective geometry, without specifics as to its application in his physics.
Description
Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e.
EuclideanEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
– metric (similarity) –
affineIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
– projective. It is an intrinsically non
metricalIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
geometry, whose facts are independent of any metric structure. Under the projective transformations, the
incidence structureIn mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...
and the
crossratioIn geometry, the crossratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
are preserved. It is a
nonEuclidean geometryNonEuclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
. In particular, it formalizes one of the central principles of perspective art: that
parallelParallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
lines meet at
infinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
and therefore are to be 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 will meet on a horizon line in virtue of their possessing 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 plane in this regard — those at infinity are treated just like any others.
Because a
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
is contained within a Projective geometry, with Projective geometry having a simpler foundation, general results in Euclidean geometry may be arrived at in a more transparent fashion, where separate but similar theorems in 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 – we single out some arbitrary projective plane as the ideal plane and locate it "at infinity" using
homogeneous coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
.
Additional properties of fundamental importance include
Desargues' TheoremIn projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
and the
Theorem of PappusIn mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
. In projective spaces of dimension 3 or greater there is a construction that allows one to prove
Desargues' TheoremIn projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
. But for dimension 2, it must be separately postulated.
Under
Desargues' TheoremIn projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations will satisfy the axioms of a field — except that the commutativity of multiplication will require
Pappus's hexagon theoremIn mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
. As a result, the points of each line are in one to one correspondence with a given field, F, supplemented by an additional element, W, such that rW = W, −W = W, r+W = W, r/0 = W, r/W = 0, W−r = r−W = W. However, 0/0, W/W, W+W, W−W, 0W and W0 remain undefined.
Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidean geometry. There are clear advantages in being able to think of a
hyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
and an
ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
as distinguished only by the way the hyperbola
lies across the line at infinity; and that a
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
is distinguished only by being tangent to the same line. The whole family of circles can be seen as
conics passing through two given points on the line at infinity — at the cost of requiring
complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 approach proved very attractive to talented geometers, and the field was thoroughly worked over. An example of this approach is the multivolume treatise by
H. F. BakerHenry Frederick Baker was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations , and Lie groups....
.
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 carried out in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of
Desargues' TheoremIn projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
.
According to Greenberg (1999) and others, the simplest 2dimensional projective geometry is the
Fano planeIn finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.Homogeneous coordinates:...
, which has 3 points on every line, with 7 points and lines in all arranged with the following schedule of collinearities:
with the coordinates A = {0,0}, B = {0,1}, C = {0,W} = {1,W}, D = {1,0}, E = {W,0} = {W,1}, F = {1,1}, G = {W, W}. The coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) will generally not be unambiguously defined.
However this geometry is not sufficiently complex to be consistent with Coxeter's (2003) approach, where the simplest example has 31 points, 31 lines, and 6 points on each line, which he writes as PG[2,5].
In Coxeter's notation, a finite projective geometry is written PG[
a,
b] where:
 a is the number of dimensions, and
 given a point on a line, b is the number of other lines through the point.
Thus, the example having only 7 points is written PG[2,2].
The term "projective geometry" is sometimes used 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 coordinatesIn mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
, and in which
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
may be embedded (hence its name, Extended Euclidean geometry.
The fundamental property that singles out all projective geometries is the
elliptic incidence property that any two distinct lines
L and
M in the
projective planeIn mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
intersect at exactly one point
P. The special case in
analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
of
parallel lines is subsumed in the smoother form of a line
at infinity on which
P 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
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of transformations can move any line to the
line at infinity).
Given a line
l and a point
P not on the line, the elliptic parallel property contrasts with the Euclidean and hyperbolic parallel properties as follows:
EllipticElliptic geometry is a nonEuclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

: 
any line through P meets l in just one point. 
Euclidean Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

: 
just one line through P may be found, which does not meet l. 
Hyperbolic In mathematics, hyperbolic geometry is a nonEuclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

: 
more than one line through P may be found, which do not meet l. 
The elliptic parallel property is the key idea which leads to the principle of projective duality, possibly the most important property which all projective geometries have in common.
Duality
In 1825, Joseph Gergonne noted the principle of
dualityA striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
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 will exist a duality between the subspaces of dimension R and dimension N−R−1. For N = 2, this specializes to the most commonly known form of duality — that between points and lines.
The duality principle was also discovered independently by
JeanVictor PonceletJeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
.
To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3dimensional 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
polyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
in a concentric sphere to obtain the dual polyhedron.
Axioms of projective geometry
Any given geometry may be deduced from an appropriate set of
axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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 & CohnVossen 1999, Greenberg 1980).
Whitehead's axioms
These axioms are based on
WhiteheadAlfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
, "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 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 ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with...
, or are
nonDesarguesian planeIn mathematics, a nonDesarguesian plane, named after Gérard Desargues, is a projective plane that does not satisfy Desargues's theorem, or in other words a plane that is not a Desarguesian plane...
s.
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, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
 C0: [ABA]
 C1: If A and B are two points such that [ABC] and [ABD] then [BDC]
 C2: If A and B are two points then there is a third point C such that [ABC]
 C3: If A and C are two points, B and D also, with [BCE], [ADE] but not [ABE] then there is a point F such that [ACF] and [BDF].
For two different points, A and B, the line AB is defined as consisting of all points C for which [ABC]. 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 {A, B, …, Z} of points is independent, [AB…Z] if {A, B, …, Z} 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 noncollinear 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 noncoplanar 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 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 meant to be 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, some authors give a treatment that embraces the
Fano planeIn finite geometry, the Fano plane is the finite projective plane with the smallest possible number of points and lines: 7 each.Homogeneous coordinates:...
PG(2, 2) as the minimal finite projective plane. An axiom system that achieves this is as follows:
 (P1) Any two distinct points lie on a unique line.
 (P2) Any two distinct lines meet in a unique point.
 (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 the projective plane attributed to Bachmann, adding Pappus's theorem to the list of axioms above and excluding projective planes over fields of characteristic 2.
See also
 Projective line
In mathematics, a projective line is a onedimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of onedimensional subspaces of the twodimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
 Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
 Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
 Incidence
 Crossratio
In geometry, the crossratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
 Möbius transformation
 Projective transformation
 Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
 Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
 Fundamental theorem of projective geometry
 Projective configuration
 Complete quadrangle
 Desargues' theorem
In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
 Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear...
 Pascal's theorem
In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration. Related results :This theorem...
 Inversive ring geometry
In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and crossratio to the context of associative rings, concepts usually built upon rings that happen to be fields....
 Joseph Wedderburn
Joseph Henry Maclagan Wedderburn was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras...
 Grassmann–Cayley algebra
External links