Plane (mathematics)

Last updated April 18, 2024

In mathematics, a plane is a two-dimensional space or 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.

Euclidean plane

Bi-dimensional Cartesian coordinate system Cartesian-coordinate-system.svg — Bi-dimensional Cartesian coordinate system

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted ${\textbf {E}}^{2}$ or $\mathbb {E} ^{2}$ . It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of 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 coordinate system is called a Cartesian plane .

The set

\mathbb {R} ^{2}

of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it.

Embedding in three-dimensional space

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space $\mathbb {R} ^{3}$ . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.

While a pair of real numbers

\mathbb {R} ^{2}

suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space

\mathbb {R} ^{3}

.

Elliptic plane

The elliptic plane is the real projective plane provided with a metric. Kepler and 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

In mathematics, a projective plane is a geometric structure that extends the concept of a 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 projective space. Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as 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 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 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 construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with 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.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or 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 complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.

In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (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 one-point compactification of the plane is homeomorphic to a sphere (see stereographic 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 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.

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.

Related Research Articles

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

<span class="mw-page-title-main">Homeomorphism</span> Mapping which preserves all topological properties of a given space

In mathematics and more specifically in topology, a homeomorphism, also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of 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.

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

<span class="mw-page-title-main">Projective space</span> Completion of the usual space with "points at infinity"

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

In the mathematical disciplines of topology and geometry, an orbifold is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry.

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of $with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H 2, which was the first instance studied, is also called the hyperbolic plane.$

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

William Mark Goldman is a professor of mathematics at the University of Maryland, College Park. He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an $-dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.$

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.

In mathematics, a space is a set with a definition (structure) of relationships among the elements of the set. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.

Topological geometry deals with incidence structures consisting of a point set $and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous. As in the case of topological groups, many deeper results require the point space to be (locally) compact and connected. This generalizes the observation that the line joining two distinct points in the Euclidean plane depends continuously on the pair of points and the intersection point of two lines is a continuous function of these lines.$

A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensional spaces are often called planes, or, more generally, surfaces.

References

↑ Janich, P.; Zook, D. (1992). Euclid's Heritage. Is Space Three-Dimensional?. The Western Ontario Series in Philosophy of Science. Springer Netherlands. p. 50. ISBN 978-0-7923-2025-8 . Retrieved 2023-03-11.
↑ H. S. M. Coxeter (1965) Introduction to Geometry, page 92
↑ Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London , 149: 61–90, doi: 10.1098/rstl.1859.0004 , ISSN 0080-4614, JSTOR 108690
↑ 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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[Janich_Zook_1992_p._50-1] Janich, P.; Zook, D. (1992). Euclid's Heritage. Is Space Three-Dimensional?. The Western Ontario Series in Philosophy of Science. Springer Netherlands. p. 50. ISBN 978-0-7923-2025-8 . Retrieved 2023-03-11.

[2] H. S. M. Coxeter (1965) Introduction to Geometry, page 92

[3] Cayley, Arthur (1859), "A sixth memoir upon quantics", Philosophical Transactions of the Royal Society of London , 149: 61–90, doi: 10.1098/rstl.1859.0004 , ISSN 0080-4614, 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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