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, concept of angle does not apply in projective geometry, because no measure of angle 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).
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respect to projective transformations, as is seen in perspective drawing from a changing perspective. One source for projectivegeometry was indeed the...
concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus...
the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can...
In projectivegeometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces...
complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...
Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the...
geometry that are related to symmetry. In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective...
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space P n {\displaystyle \mathbb {P}...
In mathematics, the complex projective plane, usually denoted P2(C) or CP2, is the two-dimensional complex projective space. It is a complex manifold...
theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet...
geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative, the simplest projective differential...
outline of an egg. The term is not very specific, but in some areas (projectivegeometry, technical drawing, etc.) it is given a more precise definition,...
especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action...
geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin. The real projective plane...
that are disregarded—projectivegeometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept...
form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. Nowadays, the projective space Pn of...
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically...
absolute geometry, while negating it yields hyperbolic geometry. Other consistent axiom sets can yield other geometries, such as projective, elliptic...
In projectivegeometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following...
affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal form of projective quadrics, below. As the dimension of a Euclidean plane...
are a system of coordinates used in projectivegeometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the...
identified. RP1 is called the real projective line, which is topologically equivalent to a circle. RP2 is called the real projective plane. This space cannot be...
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