TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, a plane is a flat, two-
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
al
surface Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
that extends infinitely far. A plane is the two-dimensional analogue of a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
(zero dimensions), a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Literatu ...
(one dimension) and
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the t ...
. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's method consists in assuming a small set of intuitively appealing axioms, ...
. When working exclusively in two-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics,
geometry Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
,
trigonometry Trigonometry (from Greek ''trigōnon'', "triangle" and ''metron'', "measure") is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century ...
,
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''vertices'' (also called ''nodes'' or ''points'') which are connec ...
, and graphing are performed in a two-dimensional space, or, in other words, in the plane.

# Euclidean geometry

Euclid Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of ... set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined terms (called ''common notions'') and postulates (or
axiom An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that ... s) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the '' Elements'', it may be thought of as part of the common notions. Euclid never used numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane. A plane is a
ruled surface In geometry, a surface ''S'' is ruled (also called a scroll) if through every point of ''S'' there is a straight line that lies on ''S''. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical direc ...

# Representation

This section is solely concerned with planes embedded in three dimensions: specifically, in R3.

## Determination by contained points and lines

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: * Three non-
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned obj ...
points (points not on a single line). * A line and a point not on that line. * Two distinct but intersecting lines. * Two distinct but
parallel Parallel may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM mainframes * Parallel communication * P ...
lines.

## Properties

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: * Two distinct planes are either parallel or they intersect in a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Literatu ...
. * A line is either parallel to a plane, intersects it at a single point, or is contained in the plane. * Two distinct lines
perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
to the same plane must be parallel to each other. * Two distinct planes perpendicular to the same line must be parallel to each other.

## Point–normal form and general form of the equation of a plane

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at t ...
) to indicate its "inclination". Specifically, let be the position vector of some point , and let be a nonzero vector. The plane determined by the point and the vector consists of those points , with position vector , such that the vector drawn from to is perpendicular to . Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points such that :$\boldsymbol \cdot \left(\boldsymbol-\boldsymbol_0\right)=0.$ The dot here means a
dot (scalar) product .
Expanded this becomes :$a \left(x-x_0\right)+ b\left(y-y_0\right)+ c\left(z-z_0\right)=0,$ which is the ''point–normal'' form of the equation of a plane. This is just a
linear equation In mathematics, a linear equation is an equation that may be put in the form :a_1x_1+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variables (or unknowns), and b, a_1, \ldots, a_n are the coefficients, which are often real numbers. The coeffi ...
:$ax + by + cz + d = 0,$ where :$d = -\left(ax_0 + by_0 + cz_0\right)$, which is the expanded form of $- \boldsymbol \cdot \boldsymbol_0.$ In mathematics it is a common convention to express the normal as a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vect ...
, but the above argument holds for a normal vector of any non-zero length. Conversely, it is easily shown that if and are constants and , and are not all zero, then the graph of the equation ::$ax + by + cz + d = 0,$ is a plane having the vector as a normal. This familiar equation for a plane is called the ''general form'' of the equation of the plane. Thus for example a regression equation of the form (with ) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

## Describing a plane with a point and two vectors lying on it

Alternatively, a plane may be described parametrically as the set of all points of the form :$\boldsymbol = \boldsymbol_0 + s \boldsymbol + t \boldsymbol,$ where ''s'' and ''t'' range over all real numbers, and are given
linearly independent In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be ...
vectors defining the plane, and is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors and can be visualized as vectors starting at and pointing in different directions along the plane. The vectors and can be
perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
, but cannot be parallel.

## Describing a plane through three points

Let , , and be non-collinear points.

### Method 1

The plane passing through , , and can be described as the set of all points (x,y,z) that satisfy the following
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero i ...
equations: :$\begin x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1& z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end =\begin x - x_1 & y - y_1 & z - z_1 \\ x - x_2 & y - y_2 & z - z_2 \\ x - x_3 & y - y_3 & z - z_3 \end = 0.$

### Method 2

To describe the plane by an equation of the form $ax + by + cz + d = 0$, solve the following system of equations: :$\, ax_1 + by_1 + cz_1 + d = 0$ :$\, ax_2 + by_2 + cz_2 + d = 0$ :$\, ax_3 + by_3 + cz_3 + d = 0.$ This system can be solved using
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
and basic matrix manipulations. Let : $D = \begin x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end$. If ''D'' is non-zero (so for planes not through the origin) the values for ''a'', ''b'' and ''c'' can be calculated as follows: :$a = \frac \begin 1 & y_1 & z_1 \\ 1 & y_2 & z_2 \\ 1 & y_3 & z_3 \end$ :$b = \frac \begin x_1 & 1 & z_1 \\ x_2 & 1 & z_2 \\ x_3 & 1 & z_3 \end$ :$c = \frac \begin x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end.$ These equations are parametric in ''d''. Setting ''d'' equal to any non-zero number and substituting it into these equations will yield one solution set.

### Method 3

This plane can also be described by the " point and a normal vector" prescription above. A suitable normal vector is given by the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space \mathbb^3, and is denoted by the symbol \times. Given ... :$\boldsymbol n = \left( \boldsymbol p_2 - \boldsymbol p_1 \right) \times \left( \boldsymbol p_3 - \boldsymbol p_1 \right),$ and the point can be taken to be any of the given points , or (or any other point in the plane).

# Operations

## Distance from a point to a plane

For a plane $\Pi : ax + by + cz + d = 0$ and a point $\boldsymbol p_1 = \left(x_1,y_1,z_1\right)$ not necessarily lying on the plane, the shortest distance from $\boldsymbol p_1$ to the plane is :$D = \frac.$ It follows that $\boldsymbol p_1$ lies in the plane
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondit ...
''D=0''. If $\sqrt=1$ meaning that ''a'', ''b'', and ''c'' are normalized then the equation becomes :$D = \ , a x_1 + b y_1 + c z_1+d , .$ Another vector form for the equation of a plane, known as the
Hesse normal form The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. Sp ...
relies on the parameter ''D''. This form is: :$\boldsymbol \cdot \boldsymbol - D_0 = 0,$ where $\boldsymbol$ is a unit normal vector to the plane, $\boldsymbol$ a position vector of a point of the plane and ''D''0 the distance of the plane from the origin. The general formula for higher dimensions can be quickly arrived at using
vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be geometric vectors, or, more generally, members of a vector space. For representing a vector, the common typographic convention is lowe ...
. Let the
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
have equation $\boldsymbol \cdot \left(\boldsymbol - \boldsymbol_0\right) = 0$, where the $\boldsymbol$ is a
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at t ...
and $\boldsymbol_0 = \left(x_,x_,\dots,x_\right)$ is a
position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s, ...
to a point in the
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
. We desire the perpendicular distance to the point $\boldsymbol_1 = \left(x_,x_,\dots,x_\right)$. The
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
may also be represented by the scalar equation $\textstyle\sum_^N a_i x_i = -a_0$, for constants $\$. Likewise, a corresponding $\boldsymbol$ may be represented as $\left(a_1,a_2, \dots, a_N\right)$. We desire the
scalar projection 300px, If 0° ≤ ''θ'' ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection. In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scala ...
of the vector $\boldsymbol_1 - \boldsymbol_0$ in the direction of $\boldsymbol$. Noting that $\boldsymbol \cdot \boldsymbol_0 = \boldsymbol_0 \cdot \boldsymbol = -a_0$ (as $\boldsymbol_0$ satisfies the equation of the
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
) we have :$\begin D &= \frac \\ &= \frac \\ &= \frac \\ &= \frac \end$.

## Line–plane intersection

In analytic geometry, the intersection of a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Literatu ...
and a plane in
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the t ...
can be the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other the ...
, a
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
, or a line.

## Line of intersection between two planes The line of intersection between two planes $\Pi_1 : \boldsymbol _1 \cdot \boldsymbol r = h_1$ and $\Pi_2 : \boldsymbol _2 \cdot \boldsymbol r = h_2$ where $\boldsymbol _i$ are normalized is given by :$\boldsymbol = \left(c_1 \boldsymbol _1 + c_2 \boldsymbol _2\right) + \lambda \left(\boldsymbol _1 \times \boldsymbol _2\right)$ where :$c_1 = \frac$ :$c_2 = \frac.$ This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product $\boldsymbol _1 \times \boldsymbol _2$ (this cross product is zero if and only if the planes are parallel, and are therefore non-intersecting or entirely coincident). The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as $\boldsymbol r = c_1\boldsymbol _1 + c_2\boldsymbol _2 + \lambda\left(\boldsymbol _1 \times \boldsymbol _2\right)$, since $\$ is a
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of t ...
. We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for $c_1$ and $c_2$. If we further assume that $\boldsymbol _1$ and $\boldsymbol _2$ are
orthonormalIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit l ...
then the closest point on the line of intersection to the origin is $\boldsymbol r_0 = h_1\boldsymbol _1 + h_2\boldsymbol _2$. If that is not the case, then a more complex procedure must be used.Plane-Plane Intersection - from Wolfram MathWorld
Mathworld.wolfram.com. Retrieved on 2013-08-20.

### Dihedral angle

Given two intersecting planes described by $\Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0$ and $\Pi_2 : a_2 x + b_2 y + c_2 z + d_2 = 0$, the
dihedral angle#REDIRECT Dihedral angle#REDIRECT Dihedral angle {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
between them is defined to be the angle $\alpha$ between their normal directions: :$\cos\alpha = \frac = \frac.$

# Planes in various areas of mathematics

geometric Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
structure, with
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomo ...
s that are
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. "We shall find it convenient to use the word ''transformation'' in the specia ...
with respect to the usual inner product, the plane may be viewed at various other levels of
abstraction Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstract ...
. Each level of abstraction corresponds to a specific
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *Cat ...
. At one extreme, all geometrical and
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model developed by the University of Idaho, that uses Landsat satellite data to compute and map evapotranspiration (ET). METRIC calculates ET as a re ...
concepts may be dropped to leave the
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformation ...
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 Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
(or 2-manifolds) classified in
low-dimensional topologyA three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot. Knot theory">trivial_knot.html" style="text-decoration: none;"class="mw-redirect" title="trefoil knot, the simplest non-trivial knot">trefoil knot, the si ...
. Isomorphisms of the topological plane are all
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous gam ...
bijection In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ... s. The topological plane is the natural context for the branch of
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''vertices'' (also called ''nodes'' or ''points'') which are connec ...
that deals with
planar graphs In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross e ...
, and results such as the
four color theorem In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a ''map'', no more than four colors are required to color the regions of the map ...
. The plane may also be viewed as an
affine 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 ...
, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but
collinearity In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned obj ...
and ratios of distances on any line are preserved.
Differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
views a plane as a 2-dimensional real
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics, a manifold is a topological space that locally resembles Euclidean space ...
, a topological plane which is provided with a
differential structureIn mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for diffe ...
. 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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
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 Image:Complex conjugate picture.svg, Geometric representation of ''z'' and its conjugate ''z̅'' in the complex plane. The distance along the light blue line from the origin to the point ''z'' is the ''modulus'' or ''absolute value'' of ''z''. The ...
and the major area of
complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex n ...
. The complex field has only two isomorphisms that leave the real line fixed, the identity and
conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
. 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 In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in C''n'', such that the transition maps are holomorphic. The term complex manifold is variously used to mean a complex m ...
, 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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonica ...
everywhere) is not the only geometry that the plane may have. The plane may be given a
spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...
by using the
stereographic projection In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smoot ...
. This can be thought of as placing a sphere on 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' no ...
. The latter possibility finds an application in the theory of
special relativity#REDIRECT Special relativity#REDIRECT Special relativity {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a
timelike In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. The fabric of space-time is a conceptual model combining the three dimensions of space ...
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean s ...
in three-dimensional
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial ...
.)

# Topological and differential geometric notions

The
one-point compactificationIn the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexa ...
of the plane is homeomorphic to a
sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...
(see
stereographic projection In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smoot ...
); 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics, a manifold is a topological space that locally resembles Euclidean space ...
referred to as the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus ...
or the
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London (UCL). The Faculty, the UCL Faculty of Engineering Sciences and the UCL Faculty of the Built Envirornment (The Bartlett) toget ...
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; f ...
. The projection from the Euclidean plane to a sphere without a point is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Definition Given two manifolds M a ...
and even a
conformal map Conformal may refer to: * Conformal (software), in ASIC Software * Conformal coating in electronics * Conformal cooling channel, in injection or blow moulding * Conformal field theory in physics, such as: ** Boundary conformal field theory ** Co ... . The plane itself is homeomorphic (and diffeomorphic) to an open
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), a Briti ...
. For the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' no ...
such diffeomorphism is conformal, but for the Euclidean plane it is not.

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Face (geometry)In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dime ...
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Flat (geometry) In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines ...
* Half-plane *
Hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
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Line–plane intersection In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane ...
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Plane coordinates 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set ...
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Plane of incidence In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave optics, ... *
Plane of rotationIn geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two ...
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Point on plane closest to originIn Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane or the nearest point on the plane. It can be found starting with a change of variables that moves the origin to ...
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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 in a single point, but there are some pairs of lines (namely, parallel lines) that do n ...

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