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Created page with "In mathematics, '''projective geometry''' is the study of geometric properties that are invariant with respect to projective transformations (see Homography). == Desc..."

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In [[mathematics]], '''projective geometry''' is the study of geometric properties that are invariant with respect to projective transformations (see [[Homography]]).

== Description ==

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 [[Point at infinity|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 expressible by a [[transformation matrix]] and [[Translation (geometry)|translations]] (the [[Affine transformation|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 (graphical)|perspective]].

Another difference from elementary geometry is the way in which [[Parallel (geometry)|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 group|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 [[Algebraic variety|projective varieties]]) and [[projective differential geometry]] (the study of [[Differential geometry|differential invariants]] of the projective transformations).

== See also ==

* [[Complete quadrangle]]
* [[Complex projective space]]
* [[Cross-ratio]]
* [[Desargues' theorem]]
* [[Duality (projective geometry)]]
* [[Fundamental theorem of projective geometry]]
* [[Grassmann–Cayley algebra]]
* [[Homogeneous coordinates]]
* [[Incidence]]
* [[Joseph Wedderburn]]
* [[Möbius transformation]]
* [[Pappus's hexagon theorem]]
* [[Pascal's theorem]]
* [[Perspective (graphical)]]
* [[Projective configuration]]
* [[Projective line]]
* [[Projective line over a ring]]
* [[Projective plane]]
* [[Projective space]]
* [[Projective transformation]]

== External links ==

* [https://en.wikipedia.org/wiki/Projective_geometry Projective geometry] @ Wikipedia

[[Category:Geometry]]
[{Category:Mathematics]]

Projective geometry - Revision history

Karl Jones: Created page with "In mathematics, '''projective geometry''' is the study of geometric properties that are invariant with respect to projective transformations (see Homography). == Desc..."