Projective geometry

That there is indeed some geometric interest in this sparse setting was first established by Desargues and others in their exploration of the principles of perspective art.

[1] In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the principle of duality.

During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics.

[3] Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century.

After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood.

[6][7] On the other hand, axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.

Projective geometry formalizes one of the central principles of perspective art: that parallel lines meet at infinity, and therefore are drawn that way.

In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem.

Using Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically.

The resulting operations satisfy the axioms of a field – except that the commutativity of multiplication requires Pappus's hexagon theorem.

As a result, the points of each line are in one-to-one correspondence with a given field, F, supplemented by an additional element, ∞, such that r ⋅ ∞ = ∞, −∞ = ∞, r + ∞ = ∞, r / 0 = ∞, r / ∞ = 0, ∞ − r = r − ∞ = ∞, except that 0 / 0, ∞ / ∞, ∞ + ∞, ∞ − ∞, 0 ⋅ ∞ and ∞ ⋅ 0 remain undefined.

The whole family of circles can be considered as conics passing through two given points on the line at infinity — at the cost of requiring complex coordinates.

[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.

The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry.

Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane:[12] for example, the Poincaré disc model where generalised circles perpendicular to the unit circle correspond to "hyperbolic lines" (geodesics), and the "translations" of this model are described by Möbius transformations that map the unit disc to itself.

Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically.

As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult.

Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra.

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.

Towards the end of the century, the Italian school of algebraic geometry (Enriques, Segre, Severi) broke out of the traditional subject matter into an area demanding deeper techniques.

During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous.

Some important work was done in enumerative geometry in particular, by Schubert, that is now considered as anticipating the theory of Chern classes, taken as representing the algebraic topology of Grassmannians.

Projective geometry later proved key to Paul Dirac's invention of quantum mechanics.

At a foundational level, the discovery that quantum measurements could fail to commute had disturbed and dissuaded Heisenberg, but past study of projective planes over noncommutative rings had likely desensitized Dirac.

In 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 practice, the principle of duality allows us to set up a dual correspondence between two geometric constructions.

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 of axioms.

Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).

One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear.

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.