In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean 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. Unlike in Euclidean geometry, the concept of an angle does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in perspective drawing from a changing perspective. 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 motivated by projective geometry. It was also a subject with many 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).
Overview Projective geometry is an elementary non-metrical form of geometry, meaning that it does not support any 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, excluding compass constructions, common in straightedge and compass constructions. As such, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy (or "betweenness"). 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. Axiomatic studies revealed the existence of non-Desarguesian planes, which shows 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 each involve a minimal set 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.
Description
Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved. A projective range is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that parallel lines meet at infinity, and therefore are drawn that way. 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 meet on a horizon line by virtue of their incorporating 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 lines 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 planes in this regard—those at infinity are treated just like any others.
Because Euclidean geometry is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using homogeneous coordinates.
Additional properties of fundamental importance include Desargues' Theorem and the Pappus's hexagon theorem. In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. But for dimension 2, it must be separately postulated.
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, , supplemented by an additional element, ∞, such that , , , , , , except that , , , , and remain undefined.
Projective geometry also includes a full theory of conic sections, a subject also extensively developed in Euclidean geometry. There are advantages to 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. 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. 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 method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by H. F. Baker.
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 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 Beltrami–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.