Coordinate system

Author: Leandro Alegsa Creation date: November 13, 2022

A coordinate system is a system of numbers used to uniquely determine the position of a point. For two-dimensional systems, the numbers (scalars) are in ordered pairs. More dimensions call for more numbers.

Number line (top), plane Cartesian coordinates (bottom)

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

The squares of the chessboard are designated with a pair of numbers and letters.

Affine coordinate system

→ Main article: Affine coordinates

Selecting three points $P_{0},P_{1},P_{2}$ not on a straight line in the Euclidean plane, the two vectors are ${\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}$ linearly independent. With the point $P_{0}$ as origin, the location vector ${\overrightarrow {P_{0}P}}$ of any point written like this

${\overrightarrow {P_{0}P}}={\color {green}u}{\overrightarrow {P_{0}P_{1}}}+{\color {green}v}{\overrightarrow {P_{0}P_{2}}}\$

and $P_{0},P_{1},P_{2}$ assign to the point number pair $\ ({\color {green}u},{\color {green}v})_{a}\$ as affine coordinates with respect to the base points

If the vectors ${\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}$ form an orthonormal basis, Cartesian coordinates result. In this case, for a point $(u_{0},v_{0})_{a}$ the sets of points $u=u_{0}$ and are $v=v_{0}$ Straight lines that intersect orthogonally.
If the basis vectors are not orthogonal (see picture), we speak of oblique coordinates.

Accordingly, affine coordinates are explained for higher dimensions.

Defining coordinates in this way is possible for any n-dimensional affine space over a body, so is not limited to a Euclidean space.

Since the 3-dimensional space plays an important role in practice, there are certain agreements for the introduction of coordinates, e.g. the legal system.

Left- and right-handed (right) three-dimensional coordinate system

Non-affine coordinate systems

In the plane

Since for calculations of lengths and angles it is advantageous if the curves $u=u_{0}$ or $v=v_{0}$ intersect orthogonally, one is interested in curve systems which intersect orthogonally. Because, then it is easy to specify a local Cartesian coordinate system at each point (see below).

The simplest non-linear orthogonal curve system consists of the rays in a point $P_{0}$ and the associated concentric circles. Thus one obtains the polar coordinates. Note: They are $P\neq P_{0}$ defined only for points For the radius coordinate, The angular range consists only of the half-open interval $[0,2\pi [$ .

Elliptical coordinates use perpendicularly intersecting systems of confocal ellipses and hyperbolas. These coordinates are not defined for the focal points and the points between them.

In space

The spatial coordinates corresponding to the polar coordinates are the

Cylinder coordinates. They are defined only for points outside the cylinder axis. and the

Spherical coordinates. Restrictions of the definition ranges and points, to which spherical coordinates are assigned, apply also here.

The (plane) elliptical coordinates correspond to the ellipsoid coordinates. The orthogonal surface system used here consists of confocal ellipsoids, single-shell and double-shell hyperboloids.

Furthermore there are the ellipsoidal coordinates, which are used for the description of points of a rotation ellipsoid (earth).

On surfaces

Parameter representations of surfaces can be seen as coordinate systems of these surfaces. E.g. the parameter representation of a plane, the usual parameter representation of a spherical surface with geographical longitude and latitude or the paramer representation of an ellipsoid.

Other curvilinear coordinate systems

Further examples and applications of coordinate systems can be found in the article curvilinear coordinates.

Local coordinate systems

If are $(u,v)_{c}$ plane curvilinear orthogonal coordinates (polar coordinates, elliptic coordinates,...) and if one determines in a point $P_{0}=(u_{0},v_{0})_{c}$ the tangent directions of the curves $v=v_{0}$ and $u=u_{0}$ and normalizes these, then one receives local basis vectors, which one can use for a local Cartesian coordinate system.

In polar coordinates, one vector points in the direction of the radius and the other in the direction of the tangent of the circle through $P_{0}$ . Here, the local system can be thought to have arisen from the global system by displacement and suitable rotation.

In space, one determines the tangent vectors to the curves passing through a point $P_{0}=(u_{0},v_{0},w_{0})_{c}$ $(u,v_{0},w_{0})_{c}$ , $(u_{0},v,w_{0})_{c}$ and $(u_{0},v_{0},w)_{c}$ and normalizes them. See (in the picture) the example spherical coordinates.