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Projective geometry is a non-metrical form of geometry, notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th century, although it did not achieve prominence as a field of mathematics until the early 19th century through the work of Poncelet and others. Projective geometry involves affine, metrical (with similitude), and Euclidean geometries as its special and more restrictive cases. Consequently, the projective geometry is a natural unifying frame for a large class of geometries, a point of view emphasized by Felix Klein in his Erlangen program. The incidence structure and the cross-ratio are fundamental invariants under the projective transformations. Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e. Euclidean - metric (similarity) - affine - projective. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the cross-ratio are preserved. It is a non-Euclidean geometry. In particular, it formalizes one of the central principles of perspective art that parallel lines meet at infinity and therefore are to be 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 will meet on a horizon line in virtue of their possessing 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 plane in this regard — those at infinity are treated just like any others. Because a Euclidean geometry is contained within a Projective geometry, with Projective geometry having a simpler foundation, general results in Euclidean geometry may be arrived at in a more transparent fashion, where separate but similar theorems in 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 - we single out some arbitrary projective plane as the ideal plane and locate it "at infinity" using homogeneous coordinates.
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