TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a plane is a flat, two-
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ... al
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
that extends infinitely far. A plane is the two-dimensional analogue of a
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
(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 Lite ... (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 ...
. 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 Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
. 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 (mathematics), dimens ...
, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics,
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ... ,
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ... ,
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, and
graphing are performed in a two-dimensional space, often in the plane.

# Euclidean geometry

Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ... 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 truth, 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' o ... 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

# 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
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 In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
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 Lite ... . * 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ... 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ... ) 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 variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 In statistical modeling, regression analysis is a set of statistical processes for Estimation theory, estimating the relationships between a dependent variable (often called the 'outcome variable') and one or more independent variables (often cal ...
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 space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ... , 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
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 , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ... :$\boldsymbol n = \left( \boldsymbol p_2 - \boldsymbol p_1 \right) \times \left( \boldsymbol p_3 - \boldsymbol p_1 \right),$ 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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
''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 (geometry), line in \mathbb^2 or a plane (geometry), plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: '' ... 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Let the
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
have equation $\boldsymbol \cdot \left(\boldsymbol - \boldsymbol_0\right) = 0$, where the $\boldsymbol$ is a
normal vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ... and $\boldsymbol_0 = \left(x_,x_,\dots,x_\right)$ is a
position vector In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ... to a point in the
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
. We desire the perpendicular distance to the point $\boldsymbol_1 = \left(x_,x_,\dots,x_\right)$. The
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
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 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
) 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 Lite ...
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 ...
can be the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... , a
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
, 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 basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
. 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 vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
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 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 A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ... between them is defined to be the angle $\alpha$ between their normal directions: :$\cos\alpha = \frac = \frac.$

# Planes in various areas of mathematics

In addition to its familiar
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, ... structure, with
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... s that are
isometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...
. Each level of abstraction corresponds to a specific
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
. At one extreme, all geometrical and
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...
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" ;"title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most gener ...
(or 2-manifolds) classified in
low-dimensional topology A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot. Knot theory is an important part of low-dimensional topology. In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or mo ...
. 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 ga ...
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... s. The topological plane is the natural context for the branch of
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
that deals with
planar graphs In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In ...
, and results such as the
four color theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The plane may also be viewed as an
affine space In mathematics, an affine space is a geometric Structure (mathematics), 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 on ...
, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but
collinearity In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
and ratios of distances on any line are preserved.
Differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of su ... , a topological plane which is provided with a
differential structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... or
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ... 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
and the major area of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
. 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 Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ... 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ... . 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. The latter possibility finds an application in the theory of
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a
timelike In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
hypersurface In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
in three-dimensional
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
.)

# Topological and differential geometric notions

The
one-point compactificationIn the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematic ...
of the plane is homeomorphic to a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ... (see
stereographic projection In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ... ); 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 Mathematics (from Ancient Greek, Greek: ) includes the study of su ... referred to as the
Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... or the
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...
projective line In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. The projection from the Euclidean plane to a sphere without a point is a
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
and even a
conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-pres ... . 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), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * Disc (magazin ...
. For the
hyperbolic plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
such diffeomorphism is conformal, but for the Euclidean plane it is not.

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Face (geometry) In solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of ...
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Flat (geometry) In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
* Half-plane *
Hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
* Line–plane intersection * Plane coordinates *
Plane of incidence In describing reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirro ... *
Plane of rotationIn geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
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Point on plane closest to originIn 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 (math ...
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Polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... *
Projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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