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). In mathematics, it is commonly represented
by the symbol ℝ2. For a generalization of
the concept, see dimension.
Two-dimensional space can be seen as a projection
of the physical universe onto a plane. Usually,
it is thought of as a Euclidean space and
the two dimensions are called length and width.
== 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 areas, 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 uniquely
in a plane by a pair of numerical coordinates,
which are the signed distances from the point
to two fixed perpendicular directed lines,
measured in the same 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, usually at
ordered pair (0, 0). The coordinates can
also be defined as the positions of the 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. Both authors used a single
axis in their treatments and have a variable
length measured in reference to this axis.
The concept of using a pair of 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.Later, the plane was thought of as a
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).
Argand diagrams are frequently used to plot
the positions of the poles and zeroes of a
function in the complex plane.
== In geometry ==
=== Coordinate systems ===
In mathematics, analytic geometry (also called
Cartesian geometry) describes every point
in two-dimensional space by means of two coordinates.
Two perpendicular coordinate axes are given
which cross each other at the 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 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.
=== Polytopes ===
In two dimensions, there are infinitely many
polytopes: the polygons. The first few regular
ones are shown below:
==== Convex ====
The Schläfli symbol {p} represents a regular
p-gon.
==== Degenerate (spherical) ====
The regular henagon {1} and regular digon
{2} can be considered degenerate regular polygons.
They can exist nondegenerately in non-Euclidean
spaces like on a 2-sphere or a 2-torus.
==== 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 polygons and share the
same vertex arrangements 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.
=== Circle ===
The hypersphere in 2 dimensions is a circle,
sometimes called a 1-sphere (S1) because it
is a one-dimensional manifold. In a Euclidean
plane, it has the length 2πr and the area
of its interior is
A
=
π
r
2
{\displaystyle A=\pi r^{2}}
where
r
{\displaystyle r}
is the radius.
=== Other shapes ===
There are an infinitude of other curved shapes
in two dimensions, notably including the conic
sections: 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 vectors.
=== Dot product, angle, and length ===
The dot product of two vectors A = [A1, A2]
and B = [B1, B2] is defined as:
A
⋅
B
=
A
1
B
1
+
A
2
B
2
{\displaystyle \mathbf {A} \cdot \mathbf {B}
=A_{1}B_{1}+A_{2}B_{2}}
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
‖
A
‖
{\displaystyle \|\mathbf {A} \|}
. In this viewpoint, the dot product of two
Euclidean vectors A and B is defined by
A
⋅
B
=
‖
A
‖
‖
B
‖
cos
⁡
θ
,
{\displaystyle \mathbf {A} \cdot \mathbf {B}
=\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta
,}
where θ is the angle between A and B.
The dot product of a vector A by itself is
A
⋅
A
=
‖
A
‖
2
,
{\displaystyle \mathbf {A} \cdot \mathbf {A}
=\|\mathbf {A} \|^{2},}
which gives
‖
A
‖
=
A
⋅
A
,
{\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf
{A} \cdot \mathbf {A} }},}
the formula for the Euclidean length of the
vector.
== In calculus ==
=== Gradient ===
In a rectangular coordinate system, the gradient
is given by
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
{\displaystyle \nabla f={\frac {\partial f}{\partial
x}}\mathbf {i} +{\frac {\partial f}{\partial
y}}\mathbf {j} }
=== Line integrals and double integrals ===
For some scalar field f : U ⊆ R2 → R,
the line integral along a piecewise smooth
curve C ⊂ U is defined as
∫
C
f
d
s
=
∫
a
b
f
(
r
(
t
)
)
|
r
′
(
t
)
|
d
t
.
{\displaystyle \int \limits _{C}f\,ds=\int
_{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.}
where r: [a, b] → C is an arbitrary bijective
parametrization of the curve C such that r(a)
and r(b) give the endpoints of C and
a
<
b
{\displaystyle a<b}
.
For a vector field F : U ⊆ R2 → R2, the
line integral along a piecewise smooth curve
C ⊂ U, in the direction of r, is defined
as
∫
C
F
(
r
)
⋅
d
r
=
∫
a
b
F
(
r
(
t
)
)
⋅
r
′
(
t
)
d
t
.
{\displaystyle \int \limits _{C}\mathbf {F}
(\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf
{F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.}
where · is the dot product and r: [a, b]
→ C is a bijective 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 R2 of a function
f
(
x
,
y
)
,
{\displaystyle f(x,y),}
and is usually written as:
∬
D
f
(
x
,
y
)
d
x
d
y
.
{\displaystyle \iint \limits _{D}f(x,y)\,dx\,dy.}
=== 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
φ
:
U
⊆
R
2
→
R
{\displaystyle \varphi :U\subseteq \mathbb
{R} ^{2}\to \mathbb {R} }
. Then
φ
(
q
)
−
φ
(
p
)
=
∫
γ
[
p
,
q
]
∇
φ
(
r
)
⋅
d
r
.
{\displaystyle \varphi \left(\mathbf {q} \right)-\varphi
\left(\mathbf {p} \right)=\int _{\gamma [\mathbf
{p} ,\,\mathbf {q} ]}\nabla \varphi (\mathbf
{r} )\cdot d\mathbf {r} .}
=== Green's theorem ===
Let C be a positively oriented, piecewise
smooth, simple closed curve in a plane, and
let D be the region bounded by C. If L and
M are functions of (x, y) defined on an open
region containing D and have continuous partial
derivatives there, then
∮
C
⁡
(
L
d
x
+
M
d
y
)
=
∬
D
(
∂
M
∂
x
−
∂
L
∂
y
)
d
x
d
y
{\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint
_{D}\left({\frac {\partial M}{\partial x}}-{\frac
{\partial L}{\partial y}}\right)\,dx\,dy}
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
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 each other. 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
