View source for Plane (geometry)

{{short description|Flat, two-dimensional surface}}
[[File:Intersecting planes.svg|thumb|Two intersecting planes in three-dimensional space]]

In [[mathematics]], a ''plane'' is a flat, two-[[dimension]]al [[surface (topology)|surface]] that extends infinitely far. A plane is the [[two-dimensional space|two-dimensional analogue]] of a [[point (geometry)|point]] (zero dimensions), a [[line (geometry)|line]] (one dimension) and [[three-dimensional space]]. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of [[Euclidean geometry]].

When working exclusively in two-dimensional [[Euclidean space]], the definite article is used, so, ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, [[geometry]], [[trigonometry]], [[graph theory]], and [[graph of a function|graphing]] are performed in a two-dimensional space, or, in other words, in the plane.

==Euclidean geometry==
{{Main|Euclidean geometry}}

[[Euclid]] set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.<ref>{{harvnb|Eves|1963|loc= pg. 19}}</ref> He selected a small core of undefined terms (called ''common notions'') and postulates (or [[axiom]]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.<ref>{{Citation
|last= Joyce
|first= D.E.
|title= Euclid's Elements, Book I, Definition 7
|publisher= Clark University
|year= 1996
|url= http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI7.html
|accessdate= 8 August 2009}}</ref> Euclid never used numbers to measure length, angle, or area. In this way the Euclidean plane is not quite the same as the [[Cartesian coordinate system|Cartesian]] plane.
<!--In higher-dimensional Euclidean space, a plane inside this space can be uniquely determined by any of the following sets of objects:
* three [[non-collinear points]] (i.e. not lying on the same [[line (mathematics)|line]])
* a line and a point not on the line
* two distinct intersecting lines -->

[[File:Planes parallel.svg|thumb|right|150px|Three parallel planes.]]
<!--In 3 dimensional Euclidean space, planes, like [[line (geometry)|lines]], can be [[parallel (geometry)|parallel]] or intersecting. Lines drawn on two parallel planes will either be parallel or [[skew lines|skew]] but will not intersect. Intersecting planes may be [[perpendicular]] or may form any number of other angles. In Euclidean spaces of more than 3 dimensions, it is possible to have two planes that intersect in a single point.-->

A plane is a [[ruled surface]].

==Planes embedded in three-dimensional Euclidean space==
This section is solely concerned with planes embedded in three dimensions: specifically, in [[Cartesian product|'''R'''<sup>3</sup>]].

===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]] points (points not on a single line).
* A line and a point not on that line.
* Two distinct but intersecting lines.
* Two [[parallel (geometry)|parallel]] 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 (geometry)|line]].
* A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
* Two distinct lines [[perpendicular]] 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]]) to indicate its "inclination".

Specifically, let {{math|'''''r'''''<sub>0</sub>}} be the position vector of some point {{math|1=''P''<sub>0</sub> = (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>)}}, and let {{math|1='''''n''''' = (''a'', ''b'', ''c'')}} be a nonzero vector. The plane determined by the point {{math|''P''<sub>0</sub>}} and the vector {{mvar|'''n'''}} consists of those points {{math|''P''}}, with position vector {{mvar|'''r'''}}, such that the vector drawn from {{math|''P''<sub>0</sub>}} to {{math|''P''}} is perpendicular to {{mvar|'''n'''}}. 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 {{mvar|'''r'''}} such that
:<math>\mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0)=0.</math>
(The dot here means a [[dot product|dot (scalar) product]].)
Expanded this becomes
:<math> a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0,</math>
which is the ''point-normal'' form of the equation of a plane.<ref>{{harvnb|Anton|1994|loc=p. 155}}</ref> This is just a [[linear equation]]
:<math> ax + by + cz + d = 0,</math>
where
:<math> d = -(ax_0 + by_0 + cz_0).</math>
Conversely, it is easily shown that if {{math|''a'', ''b'', ''c''}} and {{math|''d''}} are constants and {{math|''a'', ''b''}}, and {{math|''c''}} are not all zero, then the graph of the equation
::<math> ax + by + cz + d = 0,</math>
is a plane having the vector {{math|1='''''n''''' = (''a'', ''b'', ''c'')}} as a normal.<ref>{{harvnb|Anton|1994|loc=p. 156}}</ref> This familiar equation for a plane is called the ''general form'' of the equation of the plane.<ref name=Weisstein2009>{{Citation
|title= Plane
|url= http://mathworld.wolfram.com/Plane.html
|year= 2009
|author= Weisstein, Eric W.
|journal= MathWorld--A Wolfram Web Resource
|accessdate= 2009-08-08
}}</ref>

Thus for example a [[regression equation]] of the form {{math|1=''y'' = ''d'' + ''ax'' + ''cz''}} (with {{math|1=''b'' = −1}}) 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
:<math>\mathbf r = \mathbf {r}_0 + s \mathbf{v} + t \mathbf{w},</math>

[[File:PlaneR.jpg|thumb|Vector description of a plane]]
where ''s'' and ''t'' range over all real numbers, '''{{math|v}}''' and {{math|'''w'''}} are given [[linear independence|linearly independent]] [[vector (geometry)|vectors]] defining the plane, and {{math|'''r'''<sub>0</sub>}} is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors '''{{math|v}}''' and {{math|'''w'''}} can be visualized as vectors starting at {{math|'''r'''<sub>0</sub>}} and pointing in different directions along the plane. The vectors '''{{math|v}}''' and {{math|'''w'''}} can be [[perpendicular]], but cannot be parallel.

===Describing a plane through three points===
Let {{math|'''p'''<sub>1</sub>{{=}}(x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>)}}, {{math|'''p'''<sub>2</sub>{{=}}(x<sub>2</sub>, y<sub>2</sub>, z<sub>2</sub>)}}, and {{math|'''p'''<sub>3</sub>{{=}}(x<sub>3</sub>, y<sub>3</sub>, z<sub>3</sub>)}} be non-collinear points.

====Method 1====
The plane passing through {{math|'''p'''<sub>1</sub>}}, {{math|'''p'''<sub>2</sub>}}, and {{math|'''p'''<sub>3</sub>}} can be described as the set of all points (x,y,z) that satisfy the following [[determinant]] equations:
:<math>\begin{vmatrix}
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{vmatrix} =\begin{vmatrix}
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{vmatrix} = 0. </math>

====Method 2====
To describe the plane by an equation of the form <math> ax + by + cz + d = 0 </math>, solve the following system of equations:
:<math>\, ax_1 + by_1 + cz_1 + d = 0</math>
:<math>\, ax_2 + by_2 + cz_2 + d = 0</math>
:<math>\, ax_3 + by_3 + cz_3 + d = 0.</math>

This system can be solved using [[Cramer's rule]] and basic matrix manipulations. Let
: <math>D = \begin{vmatrix}
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
x_3 & y_3 & z_3
\end{vmatrix}</math>.

If ''D'' is non-zero (so for planes not through the origin) the values for ''a'', ''b'' and ''c'' can be calculated as follows:
:<math>a = \frac{-d}{D} \begin{vmatrix}
1 & y_1 & z_1 \\
1 & y_2 & z_2 \\
1 & y_3 & z_3
\end{vmatrix}</math>
:<math>b = \frac{-d}{D} \begin{vmatrix}
x_1 & 1 & z_1 \\
x_2 & 1 & z_2 \\
x_3 & 1 & z_3
\end{vmatrix}</math>
:<math>c = \frac{-d}{D} \begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{vmatrix}.</math>

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 "[[plane (geometry)#Point-normal form and general form of the equation of a plane|point and a normal vector]]" prescription above. A suitable normal vector is given by the [[cross product]]
:<math>\mathbf n = ( \mathbf p_2 - \mathbf p_1 ) \times ( \mathbf p_3 - \mathbf p_1 ), </math>
and the point {{math|'''r'''<sub>0</sub>}} can be taken to be any of the given points {{math|'''p'''<sub>1</sub>}},{{math|'''p'''<sub>2</sub>}} or {{math|'''p'''<sub>3</sub>}}<ref name=PaulsPlanes>{{citation
|last= Dawkins
|first= Paul
|title= Calculus III
|chapter= Equations of Planes
|chapterurl= http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx
}}</ref> (or any other point in the plane).

===Distance from a point to a plane===
{{Main|Distance from a point to a plane}}
For a plane <math>\Pi : ax + by + cz + d = 0</math> and a point <math>\mathbf p_1 = (x_1,y_1,z_1) </math> not necessarily lying on the plane, the shortest distance from <math>\mathbf p_1</math> to the plane is
:<math> D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}. </math>

It follows that <math>\mathbf p_1</math> lies in the plane [[if and only if]] ''D=0''.

If <math>\sqrt{a^2+b^2+c^2}=1</math> meaning that ''a'', ''b'', and ''c'' are normalized<ref>To normalize arbitrary coefficients, divide each of ''a'', ''b'', ''c'' and ''d'' by <math>\sqrt{a^2+b^2+c^2}</math> (which can not be 0). The "new" coefficients are now normalized and the following formula is valid for the "new" coefficients.</ref> then the equation becomes
:<math> D = \ | a x_1 + b y_1 + c z_1+d |.</math>

Another vector form for the equation of a plane, known as the [[Hesse normal form]] relies on the parameter ''D''. This form is:<ref name=Weisstein2009/>
:<math>\mathbf{n} \cdot \mathbf{r} - D_0 = 0,</math>
where <math>\mathbf{n}</math> is a unit normal vector to the plane, <math>\mathbf{r}</math> a position vector of a point of the plane and ''D''<sub>0</sub> the distance of the plane from the origin.

The general formula for higher dimensions can be quickly arrived at using [[vector notation]]. Let the [[hyperplane]] have equation <math> \mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0 </math>, where the <math>\mathbf{n}</math> is a [[normal vector]] and <math>\mathbf{r}_0 = (x_{10},x_{20},\dots,x_{N0})</math> is a [[position vector]] to a point in the [[hyperplane]]. We desire the perpendicular distance to the point <math>\mathbf{r}_1 = (x_{11},x_{21},\dots,x_{N1})</math>. The [[hyperplane]] may also be represented by the scalar equation <math>\textstyle\sum_{i=1}^N a_i x_i = -a_0</math>, for constants <math>\{a_i\}</math>. Likewise, a corresponding <math>\mathbf{n}</math> may be represented as <math>(a_1,a_2, \dots, a_N)</math>. We desire the [[scalar projection]] of the vector <math>\mathbf{r}_1 - \mathbf{r}_0</math> in the direction of <math>\mathbf{n}</math>. Noting that <math>\mathbf{n} \cdot \mathbf{r}_0 = \mathbf{r}_0 \cdot \mathbf{n} = -a_0</math> (as <math>\mathbf{r}_0</math> satisfies the equation of the [[hyperplane]]) we have
:<math>
\begin{align}
D &= \frac{|(\mathbf{r}_1 - \mathbf{r}_0) \cdot \mathbf{n}|}{|\mathbf{n}|} \\
&= \frac{|\mathbf{r}_1\cdot \mathbf{n} - \mathbf{r}_0 \cdot \mathbf{n}|}{|\mathbf{n}|} \\
&= \frac{|\mathbf{r}_1\cdot \mathbf{n} + a_0|}{|\mathbf{n}|} \\
&= \frac{|a_1x_{11} + a_2x_{21} + \dots + a_Nx_{N1} + a_0|}{\sqrt{a_1^2 + a_2^2 + \dots + a_N^2}}
\end{align}
</math>.

===Line of intersection between two planes===
The line of intersection between two planes <math>\Pi_1 : \mathbf {n}_1 \cdot \mathbf r = h_1</math> and <math>\Pi_2 : \mathbf {n}_2 \cdot \mathbf r = h_2</math> where <math>\mathbf {n}_i</math> are normalized is given by
:<math> \mathbf {r} = (c_1 \mathbf {n}_1 + c_2 \mathbf {n}_2) + \lambda (\mathbf {n}_1 \times \mathbf {n}_2) </math>

where
:<math> c_1 = \frac{ h_1 - h_2(\mathbf {n}_1 \cdot \mathbf {n}_2) }{ 1 - (\mathbf {n}_1 \cdot \mathbf {n}_2)^2 } </math>
:<math> c_2 = \frac{ h_2 - h_1(\mathbf {n}_1 \cdot \mathbf {n}_2) }{ 1 - (\mathbf {n}_1 \cdot \mathbf {n}_2)^2 }.</math>

This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product <math>\mathbf {n}_1 \times \mathbf {n}_2</math> (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 <math>\mathbf r = c_1\mathbf {n}_1 + c_2\mathbf {n}_2 + \lambda(\mathbf {n}_1 \times \mathbf {n}_2)</math>, since <math>\{ \mathbf {n}_1, \mathbf {n}_2, (\mathbf {n}_1 \times \mathbf {n}_2) \}</math> is a [[basis (linear algebra)|basis]]. 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 <math>c_1</math> and <math>c_2</math>.

If we further assume that <math>\mathbf {n}_1</math> and <math>\mathbf {n}_2</math> are [[orthonormal]] then the closest point on the line of intersection to the origin is <math>\mathbf r_0 = h_1\mathbf {n}_1 + h_2\mathbf {n}_2</math>. If that is not the case, then a more complex procedure must be used.<ref>[http://mathworld.wolfram.com/Plane-PlaneIntersection.html Plane-Plane Intersection - from Wolfram MathWorld]. Mathworld.wolfram.com. Retrieved on 2013-08-20.</ref>

===Dihedral angle===
{{See also|Dihedral angle}}
Given two intersecting planes described by <math>\Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0</math> and <math>\Pi_2 : a_2 x + b_2 y + c_2 z + d_2 = 0</math>, the [[dihedral angle]] between them is defined to be the angle <math>\alpha</math> between their normal directions:
:<math>\cos\alpha = \frac{\hat n_1\cdot \hat n_2}{|\hat n_1||\hat n_2|} = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}. </math>

==Planes in various areas of mathematics==
In addition to its familiar [[geometric]] structure, with [[isomorphism]]s that are [[isometry|isometries]] with respect to the usual inner product, the plane may be viewed at various other levels of [[abstraction (mathematics)|abstraction]]. Each level of abstraction corresponds to a specific [[category (mathematics)|category]].

At one extreme, all geometrical and [[metric (mathematics)|metric]] concepts may be dropped to leave the [[topological]] plane, which may be thought of as an idealized [[homotopy|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 [[surface (topology)|surfaces]] (or 2-manifolds) classified in [[low-dimensional topology]]. Isomorphisms of the topological plane are all [[continuous function|continuous]] [[bijection]]s. 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 function|differentiable]] or [[smooth function|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 [[complex conjugation|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 map|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 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 geometry|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 numbers|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 (mathematics)|disk]]. For the [[hyperbolic geometry|hyperbolic plane]] such diffeomorphism is conformal, but for the Euclidean plane it is not.

==See also==
{{Div col|colwidth=20em}}
* [[Face (geometry)]]
* [[Flat (geometry)]]
* [[Half-plane]]
* [[Hyperplane]]
* [[Line–plane intersection]]
* [[Plane coordinates]]
* [[Plane of incidence]]
* [[Plane of rotation]]
* [[Point on plane closest to origin]]
* [[Projective plane]]
{{Div col end}}

==Notes==
<references />

==References==
* {{citation|last=Anton|first=Howard|title=Elementary Linear Algebra|edition=7th|publisher=John Wiley & Sons|year=1994|isbn=0-471-58742-7}}
* {{citation|last=Eves|first=Howard|title=A Survey of Geometry|publisher=Allyn and Bacon, Inc.|place=Boston|year=1963|volume= I}}

==External links==
{{Commons category}}
* {{SpringerEOM|title=Plane|id=p/p072790}}
* {{MathWorld|title=Plane|urlname=Plane}}
* [http://www.wdl.org/en/item/2850/ "Easing the Difficulty of Arithmetic and Planar Geometry"] is an Arabic manuscript, from the 15th century, that serves as a tutorial about plane geometry and arithmetic.
{{Use dmy dates|date=April 2012}}

{{DEFAULTSORT:Plane (Geometry)}}
[[Category:Euclidean geometry]]
[[Category:Mathematical concepts]]
[[Category:Surfaces]]