Plane (mathematics)

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely.

It is a geometric space in which two real numbers are required to determine the position of each point.

It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely.

A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.

Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere tangent to it.

Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.

Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic.

In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction.

At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances.

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps.

The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.

The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line.

The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.