Inmathematics,projective geometry is the study of geometric properties that are invariant with respect toprojective transformations. This means that, compared to elementaryEuclidean geometry, 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 thanEuclidean space, for a given dimension, and thatgeometric 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 atransformation matrix andtranslations (theaffine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike inEuclidean geometry, the concept of anangle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen inperspective 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 whichparallel lines can be said to meet in apoint 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. SeeProjective 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 ofcomplex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (includinginvariant theory, theItalian school of algebraic geometry, andFelix Klein'sErlangen programme resulting in the study of theclassical groups) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, assynthetic geometry. Another topic that developed from axiomatic studies of projective geometry isfinite geometry.
The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study ofprojective varieties) andprojective differential geometry (the study ofdifferential invariants of the projective transformations).
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 ofconfigurations ofpoints andlines. That there is indeed some geometric interest in this sparse setting was first established byDesargues and others in their exploration of the principles ofperspective art.[1] Inhigher dimensional spaces there are consideredhyperplanes (that always meet), and other linear subspaces, which exhibitthe 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 astraight-edge alone, excludingcompass constructions, common instraightedge and compass constructions.[2] As such, there are no circles, no angles, no measurements, no parallels, and no concept ofintermediacy (or "betweenness").[3] It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the differentconic 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 ofJean-Victor Poncelet,Lazare Carnot and others established projective geometry as an independent field ofmathematics.[3] Its rigorous foundations were addressed byKarl von Staudt and perfected by ItaliansGiuseppe Peano,Mario Pieri,Alessandro Padoa andGino Fano during the late 19th century.[4] Projective geometry, likeaffine andEuclidean geometry, can also be developed from theErlangen program of Felix Klein; projective geometry is characterized byinvariants undertransformations of theprojective group.
After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. Theincidence structure and thecross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by theaffine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary".[5] An algebraic model for doing projective geometry in the style ofanalytic geometry is given by homogeneous coordinates.[6][7] On the other hand, axiomatic studies revealed the existence ofnon-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 andordered geometry are elementary since they each involve a minimal set ofaxioms and either can be used as the foundation foraffine andEuclidean geometry.[8][9] Projective geometry is not "ordered"[3] and so it is a distinct foundation for geometry.
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Projective geometry is less restrictive than eitherEuclidean geometry oraffine geometry. It is an intrinsically non-metrical geometry, meaning that facts are independent of any metric structure. Under the projective transformations, theincidence structure and the relation ofprojective harmonic conjugates are preserved. Aprojective range is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: thatparallel lines meet atinfinity, 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 aEuclidean geometry 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" usinghomogeneous coordinates.
Additional properties of fundamental importance includeDesargues' Theorem and theTheorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to proveDesargues' Theorem. But for dimension 2, it must be separately postulated.
UsingDesargues' 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 requiresPappus's hexagon theorem. As a result, the points of each line are in one-to-one correspondence with a given field,F, supplemented by an additional element, ∞, such thatr ⋅ ∞ = ∞,−∞ = ∞,r + ∞ = ∞,r / 0 = ∞,r / ∞ = 0,∞ −r =r − ∞ = ∞, except that0 / 0,∞ / ∞,∞ + ∞,∞ − ∞,0 ⋅ ∞ and∞ ⋅ 0 remain undefined.
Projective geometry also includes a full theory ofconic sections, a subject also extensively developed in Euclidean geometry. There are advantages to being able to think of ahyperbola and anellipse as distinguished only by the way the hyperbolalies across the line at infinity; and that aparabola is distinguished only by being tangent to the same line. The whole family of circles can be considered asconics passing through two given points on the line at infinity — at the cost of requiringcomplex coordinates. Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering thelinear 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 byH. F. Baker.
The first geometrical properties of a projective nature were discovered during the 3rd century byPappus of Alexandria.[3]Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425[10] (seePerspective (graphical) § History for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry).Johannes Kepler (1571–1630) andGirard Desargues (1591–1661) independently developed the concept of the "point at infinity".[11] 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 madeEuclidean geometry, 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-oldBlaise Pascal and helped him formulatePascal's theorem. The works ofGaspard 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 untilMichel 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 concretepole and polar relation with respect to a circle, established a relationship between metric and projective properties. Thenon-Euclidean geometries discovered soon thereafter were eventually demonstrated to have models, such as theKlein model ofhyperbolic space, relating to projective geometry.
In 1855A. F. Möbius wrote an article about permutations, now calledMöbius transformations, ofgeneralised circles in thecomplex plane. These transformations represent projectivities of thecomplex projective line. In the study of lines in space,Julius Plücker usedhomogeneous coordinates in his description, and the set of lines was viewed on theKlein quadric, one of the early contributions of projective geometry to a new field calledalgebraic geometry, an offshoot ofanalytic geometry with projective ideas.
Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerninghyperbolic geometry by providingmodels for thehyperbolic plane:[12] for example, thePoincaré disc model where generalised circles perpendicular to theunit circle correspond to "hyperbolic lines" (geodesics), and the "translations" of this model are described by Möbius transformations that map theunit disc to itself. The distance between points is given by aCayley–Klein metric, known to be invariant under the translations since it depends oncross-ratio, a key projective invariant. The translations are described variously asisometries inmetric space theory, aslinear fractional transformations formally, and as projective linear transformations of theprojective linear group, in this caseSU(1, 1).
The work ofPoncelet,Jakob Steiner and others was not intended to extend analytic geometry. Techniques were supposed to besynthetic: in effectprojective 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 theprojective plane alone, the axiomatic approach can result inmodels not describable vialinear algebra.
This period in geometry was overtaken by research on the generalalgebraic curve byClebsch,Riemann,Max Noether and others, which stretched existing techniques, and then byinvariant theory. Towards the end of the century, theItalian 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 inenumerative geometry in particular, by Schubert, that is now considered as anticipating the theory ofChern classes, taken as representing thealgebraic topology ofGrassmannians.
Projective geometry later proved key toPaul Dirac's invention ofquantum mechanics. At a foundational level, the discovery thatquantum measurements could fail to commute had disturbed and dissuadedHeisenberg, 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.[13]
There are many projective geometries, which may be divided into discrete and continuous: adiscrete geometry comprises a set of points, which may or may not befinite in number, while acontinuous 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 ofDesargues' Theorem.
The smallest 2-dimensional projective geometry (that with the fewest points) is theFano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
withhomogeneous coordinatesA = (0,0,1),B = (0,1,1),C = (0,1,0),D = (1,0,1),E = (1,0,0),F = (1,1,1),G = (1,1,0), or, in affine coordinates,A = (0,0),B = (0,1),C = (∞),D = (1,0),E = (0),F = (1,1)andG = (1). 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, afinite projective geometry is writtenPG(a,b) where:
Thus, the example having only 7 points is writtenPG(2, 2).
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 ofhomogeneous coordinates, and in whichEuclidean geometry may be embedded (hence its name,Extended Euclidean plane).
The fundamental property that singles out all projective geometries is theellipticincidence property that any two distinct linesL andM in theprojective plane intersect at exactly one pointP. The special case inanalytic geometry ofparallel lines is subsumed in the smoother form of a lineat infinity on whichP lies. Theline 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 theErlangen programme one could point to the way thegroup of transformations can move any line to theline at infinity).
The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
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.
In 1825,Joseph Gergonne noted the principle ofduality characterizing projective plane geometry: given any theorem or definition of that geometry, substitutingpoint forline,lie on forpass through,collinear forconcurrent,intersection forjoin, 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 swappingpoint andplane,is contained by andcontains. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimensionR and dimensionN −R − 1. ForN = 2, this specializes to the most commonly known form of duality—that between points and lines.The duality principle was also discovered independently byJean-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 adual correspondence between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in aconic curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetricalpolyhedron in a concentric sphere to obtain the dual polyhedron.
Another example isBrianchon's theorem, the dual of the already mentionedPascal'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):
Any given geometry may be deduced from an appropriate set ofaxioms. Projective geometries are characterised by the "elliptic parallel" axiom, thatany 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).
These axioms are based onWhitehead, "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying thesethree axioms either have at most one line, or are projective spaces of some dimension over adivision ring, or arenon-Desarguesian planes.
One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter'sProjective Geometry,[14] references Veblen[15] 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.
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:
For two distinct 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 equivalentform as follows. A projective space is of:
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:
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.
Inincidence geometry, most authors[16] give a treatment that embraces theFano planePG(2, 2) as the smallest finite projective plane. An axiom system that achieves this is as follows:
Coxeter'sIntroduction to Geometry[17] gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, addingPappus's theorem to the list of axioms above (which eliminatesnon-Desarguesian planes) and excluding projective planes over fields of characteristic 2 (those that do not satisfyFano's axiom). The restricted planes given in this manner more closely resemble thereal projective plane.
Given three non-collinear 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 thecomplete quadrangle configuration.
Anharmonic 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.[18]
A spatialperspectivity of aprojective 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 aprojectivity.
While corresponding points of a perspectivity all converge at a point, this convergence isnot true for a projectivity that isnot 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 aprojective conic, and in acknowledgement of the work ofJakob Steiner, it is referred to as aSteiner conic.
Suppose a projectivity is formed by two perspectivities centered on pointsA andB, relatingx toX by an intermediaryp:
The projectivity is then Then given the projectivity the induced conic is
Given a conicC and a pointP not on it, two distinctsecant lines throughP intersectC in four points. These four points determine a quadrangle of whichP is a diagonal point. The line through the other two diagonal points is called thepolar ofP andPis thepole of this line.[19] Alternatively, the polar line ofP is the set ofprojective harmonic conjugates ofP on a variable secant line passing throughP andC.
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