 | This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(February 2026) (Learn how and when to remove this message) |
Inmathematics, aplane is atwo-dimensional space orflatsurface that extends indefinitely. A plane is the two-dimensional analogue of apoint (zero dimensions), aline (one dimension) andthree-dimensional space. When working exclusively in two-dimensionalEuclidean space, the definite article is used, sotheEuclidean plane refers to the whole space.
Several notions of a plane may be defined. The Euclidean plane followsEuclidean geometry, and in particular theparallel postulate. Aprojective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. Theelliptic plane may be further defined by adding ametric to the real projective plane. One may also conceive of ahyperbolic plane, which obeyshyperbolic geometry and has a negativecurvature.
Abstractly, one may forget all structure except the topology, producing the topological plane, which is homeomorphic to anopen disk. Viewing the plane as anaffine space produces theaffine plane, which lacks a notion of distance but preserves the notion ofcollinearity. Conversely, in adding more structure, one may view the plane as a1-dimensionalcomplex manifold, called thecomplex line.
Many fundamental tasks in mathematics,geometry,trigonometry,graph theory, andgraphing are performed in a two-dimensional orplanar space.[1]
Inmathematics, aEuclidean plane is aEuclidean space ofdimension two, denoted or. It is ageometric space in which tworeal numbers are required to determine theposition of eachpoint. It is anaffine space, which includes in particular the concept ofparallel lines. It has alsometrical properties induced by adistance, which allows to definecircles, andangle measurement.
A Euclidean plane with a chosenCartesian coordinate system is called aCartesian plane.The set of the ordered pairs of real numbers (thereal coordinate plane), equipped with thedot product, is often calledthe Euclidean plane orstandard Euclidean plane, since every Euclidean plane isisomorphic to it.
InEuclidean geometry, aplane is aflat two-dimensionalsurface that extends indefinitely.Euclidean planes often arise assubspaces ofthree-dimensional space.A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimally thin.While a pair of real numbers suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for theirembedding in theambient space.
The elliptic plane is thereal projective plane provided with ametric.Kepler andDesargues used thegnomonic projection to relate a plane σ to points on ahemisphere tangent to it. WithO the center of the hemisphere, a pointP in σ determines a lineOP intersecting the hemisphere, and any lineL ⊂ σ determines a planeOL which intersects the hemisphere in half of agreat circle. The hemisphere is bounded by a plane through O and parallel to σ. No ordinary line ofσ corresponds to this plane; instead aline at infinity is appended toσ. As any line in this extension of σ corresponds to a plane throughO, and since any pair of such planes intersects in a line throughO, 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]
GivenP andQ inσ, the elliptic distance between them is the measure of the anglePOQ, 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 byFelix Klein andBernhard Riemann leading tonon-Euclidean geometry andRiemannian geometry.
Inmathematics, aprojective plane is a geometric structure that extends the concept of aplane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thusany two distinct lines in a projective plane intersect at exactly one point.
Renaissance artists, in developing the techniques of drawing inperspective, laid the groundwork for this mathematical topic. The archetypical example is thereal projective plane, also known as the extended Euclidean plane.[4] This example, in slightly different guises, is important inalgebraic geometry,topology andprojective geometry where it may be denoted variously byPG(2, R), RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as thecomplex projective plane, and finite, such as theFano plane.
A projective plane is a 2-dimensionalprojective space. Not all projective planes can beembedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known asDesargues' theorem, not shared by all projective planes.
In addition to its familiargeometric structure, withisomorphisms that areisometries with respect to the usual inner product, the plane may be viewed at various other levels ofabstraction. Each level of abstraction corresponds to a specificcategory.
At one extreme, all geometrical andmetric concepts may be dropped to leave thetopological plane, which may be thought of as an idealizedhomotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to constructsurfaces (or 2-manifolds) classified inlow-dimensional topology. Isomorphisms of the topological plane are allcontinuousbijections. The topological plane is the natural context for the branch ofgraph theory that deals withplanar graphs, and results such as thefour color theorem.
The plane may also be viewed as anaffine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, butcollinearity and ratios of distances on any line are preserved.
Differential geometry views a plane as a 2-dimensional realmanifold, a topological plane which is provided with adifferential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example adifferentiable orsmooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.
In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to thecomplex plane and the major area ofcomplex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity andconjugation.
In the same way as in the real case, the plane may also be viewed as the simplest,one-dimensional (in terms ofcomplex dimension, over the complex numbers)complex manifold, sometimes called thecomplex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are allconformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation.
In addition, the Euclidean geometry (which has zerocurvature everywhere) is not the only geometry that the plane may have. The plane may be given aspherical geometry by using thestereographic 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 thehyperbolic plane. The latter possibility finds an application in the theory ofspecial relativity in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is atimelikehypersurface in three-dimensionalMinkowski space.)
Theone-point compactification of the plane is homeomorphic to asphere (seestereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is amanifold referred to as theRiemann sphere or thecomplexprojective line. The projection from the Euclidean plane to a sphere without a point is adiffeomorphism and even aconformal map.
The plane itself is homeomorphic (and diffeomorphic) to an opendisk. For thehyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.