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Projective geometry is a non-metrical form of geometry, notable for its principle of duality. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th century, although it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Girard Desargues ( February 21 or March 2, 1591 -October 1661 was a French Mathematician and engineer who is considered one of As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Jean-Victor Poncelet ( July 1, 1788 &ndash December 22, 1867) was a French Engineer and Mathematician who served

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Description

Projective geometry is a non-Euclidean geometry that formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness 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 line 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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. 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 coordinates. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

Additional properties of fundamental importance include Desargues' Theorem and the Theorem of Pappus. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of Collinear points A, B, C, and another In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles 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. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles The resulting operations will satisfy the axioms of a fields — except that the commutativity of multiplication will require Pappus's hexagon theorem. Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of Collinear points A, B, C, and another 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. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface There are clear advantages in being able to think of a hyperbola and an ellipse as distinguished only by the way the hyperbola lies across the line at infinity; and that a parabola is distinguished only by being tangent to the same line. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 by H. F. Baker. Henry Frederick Baker ( July 3, 1866 - March 17, 1956) was a British mathematician working mainly in Algebraic geometry

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. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles

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 Fano plane. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each 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. In Mathematics, a finite projective space S is a set P (the set of points together with a set of subsets of P (the set of lines all of which

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 which Euclidean geometry may be embedded (hence its name, Extended Euclidean geometry. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

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. In Geometry, the Relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L' and 'intersects' (as in 'line L1 See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry The line at infinity is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity). An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

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:

Elliptic  : any line through P meets l in just one point. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point
Euclidean  : just one line through P may be found, which does not meet l. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.
Hyperbolic  : more than one line through P may be found, which do not meet l. In

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. In the Geometry of the Projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving Joseph Diaz Gergonne ( 19 June 1771 Nancy, France -- 4 May 1859 Montpellier, France) was a French In the Geometry of the Projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving 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 Jean-Victor Poncelet. Jean-Victor Poncelet ( July 1, 1788 &ndash December 22, 1867) was a French Engineer and Mathematician who served

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). In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface A commonplace example is found in the reciprocation of a symmetrical polyhedron in a concentric sphere to obtain the dual polyhedron. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions

Axioms of projective geometry

Any given geometry may be deduced from an appropriate set of axioms. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject 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":

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:

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:

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 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. Girard Desargues ( February 21 or March 2, 1591 -October 1661 was a French Mathematician and engineer who is considered one of By generalizing the use of vanishing points to include the case when these are infinitely far away, he made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Desargues's study on conic sections drew the attention of 16-years old Blaise Pascal and helped him formulate Pascal's theorem. Blaise Pascal (blɛz paskal (June 19 1623 &ndash August 19 1662 was a French Mathematician, Physicist, and religious Philosopher In Projective geometry, Pascal's theorem (aka Hexagrammum Mysticum Theorem) states that if an arbitrary Hexagon is inscribed in any Conic section Important for the subsequent development of projective geometry are the works of Gaspard Monge at the end of 18 and beginning of 19 century. Gaspard Monge Comte de Péluse ( May 10, 1746 &ndash July 28, 1818) was a French Mathematician and inventor of The work of Desargues was totally ignored until Michel Chasles chanced upon a handwritten copy in 1845. Michel Chasles (15 November 1793 &ndash 18 December 1880 was a French Mathematician. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. Jean-Victor Poncelet ( July 1, 1788 &ndash December 22, 1867) was a French Engineer and Mathematician who served 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 the Klein model of hyperbolic space, relating to projective geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In geometry the Klein model, also called the projective model the Beltrami–Klein model the Klein–Beltrami model and the Cayley–Klein model is a model of n-dimensional Hyperbolic In Mathematics, hyperbolic n -space, denoted H n, is the maximally symmetric Simply connected, n -dimensional

This early 19th century projective geometry was a stepping stone from analytic geometry to algebraic geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with When treated in terms of homogeneous coordinates, projective geometry looks like an extension or technical improvement of the use of coordinates to reduce geometric problems to algebra, an extension reducing the number of special cases. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. The detailed study of quadrics and the "line geometry" of Julius Plücker still form a rich set of examples for geometers working with more general concepts. In mathematics a quadric, or quadric surface, is any D -dimensional Hypersurface defined as the locus of zeros of a Quadratic In Geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century are a way to assign six Homogenous coordinates to each line Julius Plücker ( June 16, 1801 &ndash May 22, 1868) was a German Mathematician and Physicist.

The work of Poncelet, Steiner and others was not intended to extend analytic geometry. Jean-Victor Poncelet ( July 1, 1788 &ndash December 22, 1867) was a French Engineer and Mathematician who served Jakob Steiner ( 18 March, 1796 &ndash April 1, 1863) was a Swiss Mathematician. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Linear algebra is the branch of Mathematics concerned with

This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one Rudolf Friedrich Alfred Clebsch ( 19 January 1833 – 7 November 1872) was a German Mathematician who made important contributions Max Noether ( 24 September 1844 - 13 December 1921) was a German Mathematician who worked on Algebraic geometry and Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect Towards the end of the century the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques. In relation with the history of Mathematics, the Italian school of Algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935 Federigo Enriques ( 5 January 1871 – 14 June 1946) was an Italian mathematician now known principally as the first to give a Francesco Severi ( 13 April 1879, Arezzo, Italy - 8 December 1961, Rome) was an Italian Mathematician

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 of Chern classes, taken as representing the algebraic topology of Grassmannians. See also Intersection theory (mathematics In Mathematics, enumerative geometry is the branch of Algebraic geometry concerned with counting numbers In Mathematics, in particular in Algebraic topology and differential geometry, the Chern classes are a particular type of Characteristic class Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, a Grassmannian is a space which parameterizes all Linear subspaces of a Vector space V of a given Dimension.

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. Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry The first research in this direction was stimulated by a suggestion by the philosopher Rudolf Steiner (not to be confused with the mathematician Jakob Steiner, mentioned above). Rudolf Steiner ( 25 February 1861 – 30 March 1925) was an Austrian philosopher literary scholar educator artist playwright Jakob Steiner ( 18 March, 1796 &ndash April 1, 1863) was a Swiss Mathematician.

In 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 of Klein path curves to organic development. In the spirit of D'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 path curves. Sir D'Arcy Wentworth Thompson ( May 2, 1860, Edinburgh &ndash June 21, 1948 St 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. V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a

See also

References

External links

Dictionary

projective geometry

-noun

  1. (mathematics) a branch of mathematics that investigates those properties of figures that are invariant when projected from a point to a line or plane
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