Projective Geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under 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, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa.
Properties meaningful in projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometric language is adequate to the novel situation? It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly 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 nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.

The field of projective geometry is itself now divided into many research subfields, 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).

Read more about Projective Geometry:  Overview, History, Description, Duality, Axioms of Projective Geometry

Other articles related to "projective geometry, geometry, projective":

Line (geometry) - Projective Geometry
... In many models of projective geometry, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry ... In Elliptic geometry we see a typical example of this ... In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified ...
Axioms of Projective Geometry - Axioms For Projective Planes
... In incidence geometry, most authors give a treatment that embraces the Fano plane PG(2, 2) as the minimal finite projective plane ... Coxeter's Introduction to Geometry gives a list of five axioms for a more restrictive concept of a projective plane attributed to Bachmann, adding Pappus ... in this manner more closely resemble the real projective plane ...
Luis Santaló - Works - Geometria Proyectiva (1966)
... A curious feature of this book on projective geometry is the opening on abstract algebra including laws of composition, group theory, ring theory, fields, finite fields, vector spaces and ... provide an enhanced vocabulary for the treatment of 15 classical topics of projective geometry ... are covered such as (4) Fundamental theorem of projective geometry, (11) projective plane, (12) cross-ratio, (13) harmonic quadruples, (18) pole and polar, (21) Klein model of non-Euclidean geometry, (22–4 ...
List Of Algebraic Geometry Topics - Classical Topics in Projective Geometry
... Affine space Projective space Projective line, cross-ratio Projective plane Line at infinity Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame ...
Scale Invariance - Scale-invariant Curves and Self-similarity - Projective Geometry
... Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry ... Projective geometry is a particularly rich field of mathematics in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory ...

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