Editing Projective geometry (section)

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In [[mathematics]], '''projective geometry''' is the study of geometric properties that are invariant with respect to [[projective transformation]]s. 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 transformation]]s 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 can be expressed by a [[transformation matrix]] and [[translation (geometry)|translation]]s (the [[affine transformation]]s). 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 (geometry)|parallel line]]s 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 [[Algebraic variety#Projective varieties|projective varieties]]) and [[projective differential geometry]] (the study of [[differential geometry|differential invariants]] of the projective transformations).