stylistic:style_transfer, information:extraction
History

The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see Perspective (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) and Girard Desargues (1591–1661) independently developed the 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 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-old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard 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 until Michel 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 concrete pole and polar relation with respect to a circle, established a relationship between metric and projective properties. The non-Euclidean geometries discovered soon thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.

In 1855 A. F. Möbius wrote an article about permutations, now called Möbius transformations, of generalised circles in the complex plane. These transformations represent projectivities of the complex projective line. In the study of lines in space, Julius Plücker used homogeneous coordinates in his description, and the set of lines was viewed on the Klein quadric, one of the early contributions of projective geometry to a new field called algebraic geometry, an offshoot of analytic geometry with projective ideas.

Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane: 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. The distance between points is given by a Cayley–Klein metric, known to be invariant under the translations since it depends on cross-ratio, a key projective invariant. The translations are described variously as isometries in metric space theory, as linear fractional transformations formally, and as projective linear transformations of the projective linear group, in this case SU(1, 1).

The work of Poncelet, Jakob Steiner and others was not intended to extend analytic geometry. 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 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.