- Projective geometry
**Projective geometry**is a non-metrical form ofgeometry , notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during theRenaissance period, and was first systematically developed byDesargues in the 17th century, although it did not achieve prominence as a field ofmathematics until the early 19th century through the work of Poncelet and others. Projective geometry involves affine, metrical (with similitude), and Euclidean geometries as its special and more restrictive cases. Consequently, the projective geometry is a natural unifying frame for a large class of geometries, a point of view emphasized byFelix Klein in hisErlangen program . Theincidence structure and thecross-ratio are fundamental invariants under the projective transformations.**Description**Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e. Euclidean - metric (similarity) - affine - projective. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the

incidence structure and thecross-ratio are preserved. It is anon-Euclidean geometry . In particular, it formalizes one of the central principles of perspective art: that parallel lines meet atinfinity 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 geometry 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" usinghomogeneous 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 proveDesargues' Theorem . But for dimension 2, it must be separately postulated.Under

Desargues' Theorem , 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 fields — except that the commutativity of multiplication will requirePappus'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, 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 ahyperbola and anellipse as distinguished only by the way the hyperbola "lies 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 seen 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 approach proved very attractive to talented geometers, and the field was thoroughly worked over. An example of this approach is the multi-volume treatise byH. F. Baker .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' Theorem .According to Greenberg (1999) and others, the simplest 2-dimensional projective geometry 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}. For an image review the

* [ABC]

* [ADE]

* [AFG]

* [BDG]

* [BEF]

* [CDF]

* [CEG]Fano plane . 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 coordinates , and in whichEuclidean geometry 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 plane intersect at exactly one point "P". The special case inanalytic geometry 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 theErlangen programme one could point to the way the group 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:

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**:"See main article -

"Duality (projective geometry) 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 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 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 line 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 symmetricalpolyhedron in a concentric sphere to obtain the dual polyhedron.**Axioms of projective geometry**Any given geometry may be deduced from an appropriate set of

axiom 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). Those given here are based on Whitehead, "The Axioms of Projective Geometry":

* 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 apparent when thinking of the original motivating example of a Euclidean space supplemented by the lines and points at infinity. The 3rd point is the line's direction.

One can pursue axiomatization in greater depth by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. A relatively simple 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 equivalentform 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 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.

**History**Projective geometry originated through the efforts of a French artist and mathematician,

Gerard Desargues (1591–1661), as 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-years oldBlaise Pascal and helped him formulatePascal's theorem . Important for the subsequent development of projective geometry are the works ofGaspard Monge at the end of 18 and beginning of 19 century. The work of Desargues was totally ignored untilMichel Chasles chanced upon a handwritten copy in 1845. Meanwhile,Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. The main contributions of Poncelet were the separation of projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as theKlein model ofhyperbolic space , relating to projective geometry.This early 19th century projective geometry was a stepping stone from

analytic geometry toalgebraic geometry . When treated in terms ofhomogeneous coordinates , projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems toalgebra , an extension reducing the number of special cases. The detailed study ofquadric s and the "line geometry " ofJulius Plücker still form a rich set of examples for geometers working with more general concepts.The work of Poncelet, Steiner and others was not intended to extend

analytic geometry . Techniques were supposed to be "synthetic": 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 in models not describable vialinear algebra .This period in geometry was overtaken by research on the general

algebraic 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.In the later part of the 19th century, the detailed study of projective geometry became less important, although the literature is voluminous. Some important work was done in

enumerative geometry in particular, by Schubert, that is now seen as anticipating the theory ofChern class es, taken as representing thealgebraic topology ofGrassmannian s.Hermann von Baravalle has explored the pedagogical potential of projective geometry for school mathematics.

**Forms of the living world**In the spirit of projective geometry's origins in

synthetic geometry , some mathematicians have investigated projective geometry as a useful way of describing natural phenomena. The first research in this direction was stimulated by a suggestion by the philosopherRudolf Steiner (not to be confused with the mathematicianJakob Steiner , mentioned above). Fact|date=May 2007In the first half of the twentieth century, both George Adams, and Louis Locher-Ernst independently explored the tension between central forces and peripheral influences.

Lawrence Edwards (1912–2004) discovered significant applications ofKlein path curves to organic development. In the spirit ofD'Arcy Thompson 's "On Growth and Form", but with more mathematical rigor, Edwards demonstrated that such forms as the buds of leaves and flowers, pine cones, eggs, and the human heart can be simply described by certain [*http://www.nct.anth.org.uk/path.htm path curves*] . Varying a single parameter, lambda, metamorphoses the interaction of what are known in projective geometry as growth measures into surprisingly accurate representations of many organic forms not otherwise easily describable mathematically; negative values of the same parameter produce inversions representing vortices of both water and of air.**ee also**

*Projective line

*Projective plane

*Projective space

*Incidence

*Cross-ratio

*Möbius transformation

*Projective transformation

*Homogeneous coordinates

*Duality (projective geometry)

*Fundamental theorem of projective geometry

*Projective configuration

*Complete quadrangle

*Desargues' theorem

*Pappus's hexagon theorem

*Pascal's theorem

*Inversive ring geometry

*Joseph Wedderburn

*Grassmann-Cayley algebra **References*** Coxeter, H. S. M., 1995. "The Real Projective Plane", 3rd ed. Springer Verlag.

* Coxeter, H. S. M., 2003. "Projective Geometry", 2nd ed. Springer Verlag.

* Hartshorne, Robin, 2000. "Geometry: Euclid and Beyond". Springer.

* Edwards, Lawrence, 1985. "Projective Geometry", Rudolph Steiner Inst. 2nd ed. 2003, Floris (ppbk).

* Edwards, Lawrence, "The Vortex of Life".

*Howard Eves , 1997. "Foundations and Fundamental Concepts of Mathematics", 3rd ed. Dover.

* Greenberg, M.J., 1980. "Euclidean and non-Euclidean geometries", 2nd ed. Freeman.

* Hilbert, D. and Cohn-Vossen, S., 1999. "Geometry and the imagination", 2nd ed. Chelsea.

* Locher-Ernst, Louis, "Space and Counterspace".

*Oswald Veblen andJ. W. A. Young , 1938–46. "Projective Geometry", 2 vols. New York: Blaisdell.

* Richard Hartley and Andrew Zisserman , 2003. "Multiple view geometry in computer vision", 2nd ed. Cambridge University Press. ISBN 0-521-54051-8**External links*** [

*http://xahlee.org/projective_geometry/projective_geometry.html Notes*] based on Coxeter's "The Real Projective Plane".

* [*http://www.nct.anth.org.uk Projective Geometry*] — one of the best non-algebraic introductions on the web by Nick Thomas.

* [*http://www.archive.org/details/117714799_001 Projective Geometry*] — free ebook by Oswald Veblen.

* [*http://lear.inrialpes.fr/people/triggs/pubs/isprs96/isprs96.html Projective Geometry for Image Analysis*] — free tutorial by Roger Mohr and Bill Triggs.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**projective geometry**— n. the branch of geometry dealing with those properties of a figure (projective properties) that do not vary when the figure is projected … English World dictionary**projective geometry**— the geometric study of projective properties. [1880 85] * * * Branch of mathematics that deals with the relationships between geometric figures and the images (mappings) of them that result from projection. Examples of projections include motion… … Universalium**projective geometry**— noun the geometry of properties that remain invariant under projection • Syn: ↑descriptive geometry • Topics: ↑mathematics, ↑math, ↑maths • Hypernyms: ↑geometry … Useful english dictionary**projective geometry**— noun the study of the projective properties of geometric figures … English new terms dictionary**projective geometry**— /prədʒɛktɪv dʒiˈɒmətri/ (say pruhjektiv jee omuhtree) noun the geometric study of projective properties … Australian English dictionary**projective geometry**— noun Date: 1885 a branch of geometry that deals with the properties of configurations that are unaltered by projection … New Collegiate Dictionary**projective geometry**— noun a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane … Wiktionary**projective geometry**— study of geometric properties that are not changed by projection … English contemporary dictionary**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 (plane) duality is the formalization of this metamathematical concept. There are two approaches to the subject of … Wikipedia**Oriented projective geometry**— is an oriented version of real projective geometry. Whereas the real projective plane describes the set of all unoriented lines through the origin in R3, the oriented projective plane describes lines with a given orientation. There are… … Wikipedia