19 Facts About Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.

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 .

Topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry and projective differential geometry .

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance.

Projective geometry can be seen as a geometry of constructions with a straight-edge alone.

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Projective geometry can be modeled by the affine plane plus a line "at infinity" and then treating that line as "ordinary".

Projective geometry is not "ordered" and so it is a distinct foundation for geometry.

Projective geometry made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system.

Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning hyperbolic geometry by providing models for the hyperbolic plane: for example, the Poincare disc model where generalised circles perpendicular to the unit circle correspond to "hyperbolic lines", and the "translations" of this model are described by Mobius transformations that map the unit disc to itself.

Towards the end of the century, the Italian school of algebraic Projective geometry broke out of the traditional subject matter into an area demanding deeper techniques.

Projective geometry later proved key to Paul Dirac's invention of quantum mechanics.

In more advanced work, Dirac used extensive drawings in projective geometry to understand the intuitive meaning of his equations, before writing up his work in an exclusively algebraic formalism.

Projective geometry is less restrictive than either Euclidean geometry or affine geometry.

Projective geometry includes a full theory of conic sections, a subject extensively developed in Euclidean geometry.

Only projective geometry of dimension 0 is a single point.

Smallest 2-dimensional projective geometry is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:.

Parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.

Projective geometry geometries are characterised by the "elliptic parallel" axiom, that any two planes always meet in just one line, or in the plane, any two lines always meet in just one point.

The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the complete quadrangle configuration.