Euclidean 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.

Embedding in three-dimensional space

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space ${\displaystyle {\mathbb{R}}^3}$. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimally thin. While a pair of real numbers ${\displaystyle {\mathbb{R}}^2}$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space ${\displaystyle {\mathbb{R}}^3}$.

Polytopes

In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below:

Convex

The Schläfli symbol ${\displaystyle \{n\}}$ represents a regular n-gon.

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 2-sphere, 2-torus, or right circular cylinder.

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 $${\displaystyle A=\pi r^2}$$ where ${\displaystyle r}$ 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:

$${\displaystyle \mathbf{A}\cdot \mathbf{B}=A_1{B}_1+A_2{B}_2}$$

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 ${\displaystyle \Vert \mathbf{A}\Vert }$. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by $${\displaystyle \mathbf{A}\cdot \mathbf{B}=\Vert \mathbf{A}\Vert \Vert \mathbf{B}\Vert \cos \theta ,}$$ where θ is the angle between A and B.

The dot product of a vector A by itself is $${\displaystyle \mathbf{A}\cdot \mathbf{A}=\Vert \mathbf{A}{\Vert }^2,}$$ which gives $${\displaystyle \Vert \mathbf{A}\Vert =\sqrt{\mathbf{A}\cdot \mathbf{A}},}$$ the formula for the Euclidean length of the vector.

In calculus

Gradient

In a rectangular coordinate system, the gradient is given by

$${\displaystyle \mathrm{\nabla }f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\mathbf{i}+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\mathbf{j}.}$$

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 $${\displaystyle \underset{C}{\int }fds={\int }_a^{b}f(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|dt,}$$ 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 ${\displaystyle a<b}$.

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

$${\displaystyle \underset{C}{\int }\mathbf{F}(\mathbf{r})\cdot d\mathbf{r}={\int }_a^b\mathbf{F}(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt,}$$

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 ${\displaystyle f(x,y),}$ and is usually written as:

$${\displaystyle \underset{D}{\iint }f(x,y)dxdy.}$$

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 ${\displaystyle \phi :U\subseteq {\mathbb{R}}^2\to \mathbb{R}}$. Then

$${\displaystyle \phi \left(\mathbf{q}\right)-\phi \left(\mathbf{p}\right)={\int }_{\gamma [\mathbf{p},\mathbf{q}]}\mathrm{\nabla }\phi (\mathbf{r})\cdot d\mathbf{r},}$$

with p, q the endpoints of the curve γ.

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

$${\displaystyle {\oint }_C(Ldx+Mdy)={\iint }_D\left(\frac{\mathrm{\partial }M}{\mathrm{\partial }x}-\frac{\mathrm{\partial }L}{\mathrm{\partial }y}\right)dxdy}$$

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.