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{{short description|Geometric model of the planar projection of the physical universe}}
{{redirect|Plane (geometry)|generalizations|Plane (mathematics)|its applications|Plane (physics)}}
[[Image:Cartesian-coordinate-system.svg|right|thumb|300px|Bi-dimensional [[Cartesian coordinate system]]]]
{{General geometry}}

In [[mathematics]], a '''Euclidean plane''' is a [[Euclidean space]] of [[Two-dimensional space|dimension two]], denoted <math>\textbf{E}^2</math> or <math>\mathbb{E}^2</math>. It is a [[geometric space]] in which two [[real number]]s are required to determine the [[position (geometry)|position]] of each [[point (mathematics)|point]]. It is an [[affine space]], which includes in particular the concept of [[parallel lines]]. It has also [[measurement|metrical properties]] induced by a [[Euclidean distance|distance]], which allows to define [[circle]]s, and [[angle|angle measurement]].

A Euclidean plane with a chosen [[Cartesian coordinate system]] is called a ''[[Cartesian plane]]''.
The set <math>\mathbb{R}^2</math> of the ordered pairs of real numbers (the [[real coordinate plane]]), equipped with the [[dot product]], is often called ''the'' Euclidean plane or ''standard Euclidean plane'', since every Euclidean plane is [[isomorphic]] to it.

==History==
{{see also|Euclidean geometry#History}}

Books I through IV and VI of [[Euclid's Elements]] dealt with two-dimensional geometry, developing such notions as similarity of shapes, the [[Pythagorean theorem]] (Proposition 47), equality of angles and [[area]]s, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics.

Later, the plane was described in a so-called ''[[Cartesian coordinate system]]'', a [[coordinate system]] that specifies each [[point (geometry)|point]] uniquely in a [[plane (geometry)|plane]] by a pair of [[number|numerical]] ''coordinates'', which are the [[positive and negative numbers|signed]] distances from the point to two fixed [[perpendicular]] directed lines, measured in the same [[unit length|unit of length]]. Each reference line is called a ''coordinate axis'' or just ''axis'' of the system, and the point where they meet is its ''[[Origin (mathematics)|origin]]'', usually at ordered pair (0,&nbsp;0). The coordinates can also be defined as the positions of the [[orthogonal projection|perpendicular projections]] of the point onto the two axes, expressed as signed distances from the origin.

The idea of this system was developed in 1637 in writings by Descartes and independently by [[Pierre de Fermat]], although Fermat also worked in three dimensions, and did not publish the discovery.<ref>{{cite encyclopedia |title=Analytic geometry |encyclopedia=Encyclopædia Britannica |year=2008 |edition=Online |url=https://www.britannica.com/science/analytic-geometry }}</ref> Both authors used a single ([[abscissa]]) axis in their treatments, with the lengths of [[ordinate]]s measured along lines not-necessarily-perpendicular to that axis.<ref>{{Cite book |last=Katz |first=Victor J. |title=A History of Mathematics |year=2009 |orig-year=1993 |publisher=Addison-Wesley |isbn=978-0-321-38700-4 |edition=3rd |location=Boston |page=484}}</ref> The concept of using a pair of fixed axes was introduced later, after Descartes' ''[[La Géométrie]]'' was translated into Latin in 1649 by [[Frans van Schooten]] and his students. These commentators  introduced several concepts while trying to clarify the ideas contained in Descartes' work.<ref>{{harvnb|Burton|2011|loc=p. 374}}</ref>

Later, the plane was thought of as a [[field (mathematics)|field]], where any two points could be multiplied and, except for 0, divided. This was known as the [[complex plane]]. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after [[Jean-Robert Argand]] (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician [[Caspar Wessel]] (1745–1818).<ref>Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9)</ref> Argand diagrams are frequently used to plot the positions of the [[pole (complex analysis)|poles]] and [[Zero (complex analysis)|zeroes]] of a [[mathematical function|function]] in the complex plane.

==In geometry==
{{See also|Euclidean geometry}}

===Coordinate systems===
<!-- now correct: if a sphere included, then there is no unique 2d space -->
{{main|Rectangular coordinate system|Polar coordinate system}}
{{redirect-distinguish|Plane coordinates|Coordinate plane}}

In mathematics, [[analytic geometry]] (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates.  Two perpendicular [[coordinate axis|coordinate axes]] are given which cross each other at the [[Origin (mathematics)|origin]].  They are usually labeled ''x'' and ''y''. Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the [[Origin (mathematics)|origin]] measured along the given axis, which is equal to the distance of that point from the other axis.

Another widely used coordinate system is the [[polar coordinate system]], which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray.

<gallery>
Image:Coord XY.svg|[[Cartesian coordinate system]]
Image:Coord Circular.svg|[[Polar coordinate system]]
</gallery>

===Embedding in three-dimensional space===
{{excerpt|Euclidean planes in three-dimensional space}}

===Polytopes===
{{main article|Polygon}}
In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below:

====Convex====
The [[Schläfli symbol]] <math>\{n\}</math> represents a [[Regular polygon|regular {{mvar|n}}-gon]].

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
![[Equilateral triangle|Triangle]]<br />([[Simplex|2-simplex]])
![[Square (geometry)|Square]]<br />([[Cross-polytope|2-orthoplex]])<br />([[Hypercube|2-cube]])
![[Pentagon]]
![[Hexagon]]
![[Heptagon]]
![[Octagon]]
|- bgcolor="#ffd0d0"
![[Schläfli symbol]]
|{3}
|{4}
|{5}
|{6}
|{7}
|{8}
|-
!Image
|[[Image:Regular triangle.svg|75px]]
|[[Image:Regular quadrilateral.svg|75px]]
|[[Image:Regular pentagon.svg|75px]]
|[[Image:Regular hexagon.svg|75px]]
|[[Image:Regular heptagon.svg|75px]]
|[[Image:Regular octagon.svg|75px]]
|-
!Name
![[Nonagon]]
![[Decagon]]
![[Hendecagon]]
![[Dodecagon]]
![[Tridecagon]]
![[Tetradecagon]]
|- bgcolor="#ffd0d0"
!Schläfli
|{9}
|{10}
|{11}
|{12}
|{13}
|{14}
|-
!Image
|[[Image:Regular nonagon.svg|75px]]
|[[Image:Regular decagon.svg|75px]]
|[[Image:Regular hendecagon.svg|75px]]
|[[Image:Regular dodecagon.svg|75px]]
|[[Image:Regular tridecagon.svg|75px]]
|[[Image:Regular tetradecagon.svg|75px]]
|-
!Name
![[Pentadecagon]]
![[Hexadecagon]]
![[Heptadecagon]]
![[Octadecagon]]
![[Enneadecagon]]
![[Icosagon]]
|...[[Regular polygon|n-gon]]
|- bgcolor="#ffd0d0"
!Schläfli
|{15}
|{16}
|{17}
|{18}
|{19}
|{20}
|{''n''}
|-
!Image
|[[Image:Regular pentadecagon.svg|75px]]
|[[Image:Regular hexadecagon.svg|75px]]
|[[Image:Regular heptadecagon.svg|75px]]
|[[Image:Regular octadecagon.svg|75px]]
|[[Image:Regular enneadecagon.svg|75px]]
|[[Image:Regular icosagon.svg|75px]]
|}

====Degenerate (spherical)====
The regular [[monogon]] (or henagon) {1} and regular [[digon]] {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a [[sphere|2-sphere]], [[torus|2-torus]], or [[right circular cylinder]].

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
|[[Monogon]]
|[[Digon]]
|- bgcolor="#eeeedd"
![[Schläfli]]
|{1}
|{2}
|-
!Image
|[[Image:Monogon.svg|75px]]
|[[Image:Digon.svg|75px]]
|}

====Non-convex====
There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called [[star polygon]]s and share the same [[vertex arrangement]]s of the convex regular polygons.

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {''n''/''m''} for all ''m'' such that ''m'' < ''n''/2 (strictly speaking {''n''/''m''} = {''n''/(''n'' − ''m'')}) and ''m'' and ''n'' are [[coprime]].

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0"
!Name
|[[Pentagram]]
| colspan="2" | [[Heptagram]]s
|[[Octagram]]
| colspan="2" | [[Enneagram (geometry)|Enneagrams]]
|[[Decagram (geometry)|Decagram]]
|...[[star polygon|n-agrams]]
|- bgcolor="#ffd0d0"
![[Schläfli]]
|{5/2}
|{7/2}
|{7/3}
|{8/3}
|{9/2}
|{9/4}
|{10/3}
|{''n/m''}
|-
!Image
|[[Image:Star polygon 5-2.svg|75px]]
|[[Image:Star polygon 7-2.svg|75px]]
|[[Image:Star polygon 7-3.svg|75px]]
|[[Image:Star polygon 8-3.svg|75px]]
|[[Image:Star polygon 9-2.svg|75px]]
|[[Image:Star polygon 9-4.svg|75px]]
|[[Image:Star polygon 10-3.svg|75px]]
|&nbsp;
|}

===Circle===
{{main article|Circle}}
[[File:CIRCLE 1.svg|200px|right]]
The [[hypersphere]] in 2 dimensions is a [[circle]], sometimes called a 1-sphere (''S''<sup>1</sup>) because it is a one-dimensional [[manifold]]. In a Euclidean plane, it has the length 2&pi;''r'' and the [[area]] of its [[interior (topology)|interior]] is
:<math>A = \pi r^{2}</math>
where <math>r</math> is the radius.

===Other shapes===
{{Main article| List of two-dimensional geometric shapes}}

There are an infinitude of other curved shapes in two dimensions, notably including the [[conic section]]s: the [[ellipse]], the [[parabola]], and the [[hyperbola]].

==In linear algebra==
Another mathematical way of viewing two-dimensional space is found in [[linear algebra]], where the idea of independence is crucial.  The plane has two dimensions because the length of a [[rectangle]] is independent of its width.  In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent [[coordinate vector|vector]]s.

===Dot product, angle, and length===
{{main article|Dot product}}
The dot product of two vectors {{nowrap|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>]}} and {{nowrap|1='''B''' = [''B''<sub>1</sub>, ''B''<sub>2</sub>]}} is defined as:<ref name="Lipschutz2009">{{cite book |author1=S. Lipschutz |author2=M. Lipson |title= Linear Algebra (Schaum's Outlines)|edition= 4th |year= 2009|publisher= McGraw Hill|isbn=978-0-07-154352-1}}</ref>

:<math>\mathbf{A}\cdot \mathbf{B} = A_1B_1 + A_2B_2</math>

A vector can be pictured as an arrow.  Its magnitude is its length, and its direction is the direction the arrow points.  The magnitude of a vector '''A''' is denoted by <math>\|\mathbf{A}\|</math>.  In this viewpoint, the dot product of two Euclidean vectors '''A''' and '''B''' is defined by<ref name="Spiegel2009">{{cite book |author1=M.R. Spiegel |author2=S. Lipschutz |author3=D. Spellman |title= Vector Analysis (Schaum's Outlines)|edition= 2nd |year= 2009|publisher= McGraw Hill|isbn=978-0-07-161545-7}}</ref>
:<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,</math>
where θ is the [[angle]] between '''A''' and '''B'''.

The dot product of a vector '''A''' by itself is
:<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2,</math>
which gives
: <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A},</math>
the formula for the [[Euclidean length]] of the vector.

==In calculus==

===Gradient===
In a rectangular coordinate system, the gradient is given by

:<math>\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y}  \mathbf{j} \,.</math>

===Line integrals and double integrals===
For some [[scalar field]] ''f'' : ''U'' ⊆ '''R'''<sup>''2''</sup> → '''R''', the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂  ''U'' is defined as
:<math>\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\,dt,</math>
where '''r''': [a, b] → ''C'' is an arbitrary [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C'' and <math>a < b</math>.

For a [[vector field]] '''F''' : ''U'' ⊆ '''R'''<sup>''2''</sup> → '''R'''<sup>''2''</sup>, the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂ ''U'', in the direction of '''r''', is defined as

:<math>\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt,</math>

where · is the [[dot product]] and '''r''': [a, b] → ''C'' is a [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C''.

A [[double integral]] refers to an [[integral]] within a region ''D'' in '''R'''<sup>2</sup> of a [[function (mathematics)|function]] <math>f(x,y),</math> and is usually written as:

:<math>\iint\limits_D f(x,y)\,dx\,dy.</math>

===Fundamental theorem of line integrals===
{{main article|Fundamental theorem of line integrals}}
The [[fundamental theorem of line integrals]] says that a [[line integral]] through a [[gradient]] field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let <math> \varphi : U \subseteq \mathbb{R}^2 \to \mathbb{R}</math>. Then

:<math> \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\mathbf{q}]} \nabla\varphi(\mathbf{r})\cdot d\mathbf{r} , </math>

with '''p''', '''q''' the endpoints of the curve γ.

===Green's theorem===
{{main article|Green's theorem}}
Let ''C'' be a positively [[Curve orientation|oriented]], [[piecewise smooth]], [[simple closed curve]] in a [[plane (mathematics)|plane]], and let ''D'' be the region bounded by ''C''. If ''L'' and ''M'' are functions of (''x'', ''y'') defined on an [[Open set|open region]] containing ''D'' and have [[Continuous function|continuous]] [[partial derivatives]] there, then<ref>Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, {{isbn|978-0-521-86153-3}}</ref><ref>Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum's Outlines, McGraw Hill (USA), 2009, {{isbn|978-0-07-161545-7}}</ref>

:<math>\oint_{C} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dx\, dy</math>

where the path of integration along C is [[counterclockwise]].

==In topology==
In [[topology]], the plane is characterized as being the unique [[contractible]] [[2-manifold]].

Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not [[simply connected]].

==In graph theory==
In [[graph theory]], a [[planar graph]] is a [[graph (discrete mathematics)|graph]] that can be [[graph embedding|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 each other.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=64|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|access-date=8 August 2012|quote=Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.}}</ref>  Such a drawing is called a ''plane graph'' or ''planar embedding of the graph''. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a [[plane curve]] on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

==See also==
* [[Geometric space]]
* [[Planimetrics]]

== References ==
{{Reflist}}

===Works cited===
* {{citation |first=David M. |last=Burton |title=The History of Mathematics / An Introduction |year=2011 |edition=7th |publisher=McGraw Hill |isbn=978-0-07-338315-6}}

{{Dimension topics}}
{{Authority control}}

[[Category:Dimension]]
[[Category:Multi-dimensional geometry]]
[[Category:2 (number)]]
[[Category:Euclidean plane geometry|*]]