# Plane (mathematics)

In

*the*Euclidean plane refers to the whole space.

Several notions of a plane may be defined. The Euclidean plane follows

Abstractly, one may forget all structure except the topology, producing the topological plane, which is homeomorphic to an

.Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional or *planar* space.^{[1]}

## Euclidean plane

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a

A Euclidean plane with a chosen

### Embedding in three-dimensional space

In

## Elliptic plane

The elliptic plane is the

*O*the center of the hemisphere, a point

*P*in σ determines a line

*OP*intersecting the hemisphere, and any line

*L*⊂ σ determines a plane

*OL*which intersects the hemisphere in half of a great circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line of

*σ*corresponds to this plane; instead a line at infinity is appended to

*σ*. As any line in this extension of σ corresponds to a plane through

*O*, and since any pair of such planes intersects in a line through

*O*, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring all pairs of lines in a plane to intersect, is confirmed.

^{[2]}

*P*and

*Q*in

*σ*, the elliptic distance between them is the measure of the angle

*POQ*, usually taken in radians. Arthur Cayley initiated the study of elliptic geometry when he wrote "On the definition of distance".

^{[3]}

^{: 82 }This venture into abstraction in geometry was followed by Felix Klein and Bernhard Riemann leading to non-Euclidean geometry and Riemannian geometry.

## Projective plane

In

*any*two distinct lines in a projective plane intersect at exactly one point.

Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane.^{[4]} This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP^{2}, or P_{2}(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.

## Further generalizations

In addition to its familiar

At one extreme, all geometrical and

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.

In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the

In the same way as in the real case, the plane may also be viewed as the simplest,

In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere tangent to the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point. This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature.

Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of special relativity in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)

## Topological and differential geometric notions

The

The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.

## See also

- Affine plane
- Half-plane
- Hyperbolic geometry

## References

- ISBN 978-0-7923-2025-8. Retrieved 2023-03-11.
- H. S. M. Coxeter(1965) Introduction to Geometry, page 92
**^**The phrases "projective plane", "extended affine plane" and "extended Euclidean plane" may be distinguished according to whether the line at infinity is regarded as special (in the so-called "projective" plane it is not, in the "extended" planes it is) and to whether Euclidean metric is regarded as meaningful (in the projective and affine planes it is not). Similarly for projective or extended spaces of other dimensions.