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.

• 
  Cartesian coordinate system
• 
  Polar coordinate system

Embedding in three-dimensional space

This section is an excerpt from Euclidean planes in three-dimensional space.[edit]

Plane equation in normal form

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 infinitesimal 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.

Name	Triangle (2-simplex)	Square (2-orthoplex) (2-cube)	Pentagon	Hexagon	Heptagon	Octagon
Schläfli symbol	{3}	{4}	{5}	{6}	{7}	{8}
Image
Name	Nonagon	Decagon	Hendecagon	Dodecagon	Tridecagon	Tetradecagon
Schläfli	{9}	{10}	{11}	{12}	{13}	{14}
Image
Name	Pentadecagon	Hexadecagon	Heptadecagon	Octadecagon	Enneadecagon	Icosagon	...n-gon
Schläfli	{15}	{16}	{17}	{18}	{19}	{20}	{n}
Image

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.

Name	Monogon	Digon
Schläfli	{1}	{2}
Image

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.

Name	Pentagram	Heptagrams		Octagram	Enneagrams		Decagram	...n-agrams
Schläfli	{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}	{10/3}	{n/m}
Image

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.

Dot product, angle, and length

The dot product of two vectors A = [A₁, A₂] and B = [B₁, B₂] is defined as:^[4]

${\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{1}B_{1}+A_{2}B_{2}}$

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 ${\displaystyle \|\mathbf {A} \|}$. In this viewpoint, the dot product of two Euclidean vectors A and B is defined by^[5]

${\displaystyle \mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}$

where θ is the angle between A and B.

The dot product of a vector A by itself is

${\displaystyle \mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2},}$

which gives

${\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }},}$

the formula for the Euclidean length of the vector.

Gradient

In a rectangular coordinate system, the gradient is given by

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

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

${\displaystyle \int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt,}$

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 ${\displaystyle a<b}$.

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

${\displaystyle \int \limits _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt,}$

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

${\displaystyle \iint \limits _{D}f(x,y)\,dx\,dy.}$

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

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

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^[6][7]

${\displaystyle \oint _{C}(L\,dx+M\,dy)=\iint _{D}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)\,dx\,dy}$

where the path of integration along C is counterclockwise.

Works cited

• Burton, David M. (2011), The History of Mathematics / An Introduction (7th ed.), McGraw Hill, ISBN 978-0-07-338315-6