Plane (mathematics)

Source: Wikipedia, the free encyclopedia.
For other uses, see Plane (disambiguation).

In

flat surface
that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.

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

hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature
.

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

1-dimensional complex manifold, called the complex line
.

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

Euclidean plane

This section is an excerpt from Euclidean plane.[edit]
Bi-dimensional Cartesian coordinate system

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

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

A Euclidean plane with a chosen

Cartesian plane
.

The set of the ordered pairs of real numbers (the
isomorphic
to it.

Embedding in three-dimensional space

This section is an excerpt from Euclidean planes in three-dimensional space.[edit]
Plane equation in normal form

In

surface
that extends indefinitely.
subspaces of three-dimensional space
. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal 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 their
ambient space
.

Elliptic plane

This section is an excerpt from Elliptic geometry § Elliptic plane.[edit]

The elliptic plane is the

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

Given 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

This section is an excerpt from Projective plane.[edit]
Drawings of the finite projective planes of orders 2 (the Fano plane) and 3, in grid layout, showing a method of creating such drawings for prime orders
These parallel lines appear to intersect in the vanishing point "at infinity". In a projective plane this is actually true.

In

plane
. 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. Thus 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), RP2, or P2(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.

A projective plane is a 2-dimensional
Desargues' theorem
, not shared by all projective planes.

Further generalizations

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. Each level of abstraction corresponds to a specific category
.

At one extreme, all geometrical and

planar graphs, and results such as the four color theorem
.

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.

smooth
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 the

conjugation
.

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

one-dimensional (in terms of complex dimension, over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal
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 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

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

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

References

  1. ISBN 978-0-7923-2025-8
    . Retrieved 2023-03-11.
  2. H. S. M. Coxeter
    (1965) Introduction to Geometry, page 92
  3. JSTOR 108690
  4. ^ 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.
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