Projective geometry

Projective geometry is a topic in mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called 'points at infinity') to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex projective space, the coordinates used (homogeneous coordinates) being complex numbers. Several major types of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme resulting in the study of the classical groups) were based on projective geometry. It was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations). 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 first established 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 apply to 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 considered as special cases of these general theorems. During the early 19th century the work of Jean-Victor 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 during the late 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.