Projective Geometry - Overview

Overview

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions it begins with the study of configurations of points and lines. That there is indeed some geometric interest in this sparse setting was seen as projective geometry was developed by Desargues and others in their exploration of the principles of perspective art. In higher dimensional spaces there are considered hyperplanes (that always meet), and other linear subspaces, which exhibit the 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 a straight-edge alone. Since projective geometry excludes compass constructions, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy. It was realised that the theorems that do hold in projective geometry are simpler statements. For example the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be seen as special cases of these general theorems.

In the early 19th century the work of Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics . Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano late in the 19th century. Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants under transformations of the projective group.

After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. 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.

In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. Projective geometry is not "ordered" and so it is a distinct foundation for geometry.