# Coordinate systems

From this article you will learn how to determine the space, what are the coordinate systems

## Setting the space

To determine the location of a point in space, you can use any coordinate system, depending on the task. For example, if you design a lamp in the shape of a ball, then you use spherical coordinates if your task needs to describe spiral movement - You choose cylindrical coordinates. So, ahead often the coordinate systems used.

#### Cartesian coordinate system *x, y, z*

Cartesian or rectangular coordinate system. In the Cartesian coordinate system, the position of a point is determined using coordinates along each of the axes, in a two-dimensional coordinate system it is a pair of numbers (x,y), in three-dimensional space it is a group of three numbers (x,y,z). The coordinates of the Cartesian system belong to the set of real numbers, i.e. x,y and z are any real number (-∞;+∞)

#### Polar coordinate system *ρ, θ*

A polar coordinate system is a plane coordinate system in which the position of any point is determined using the distance **r** from the center of the coordinate system
and the angle between the radius vector to the x axis. The polar coordinate system is used when it is more convenient to determine the distances between points by angle and distance.
The polar coordinate system is also used to represent complex numbers. In the polar coordinate system r ≥ 0, angle φ ∈ [0;2π),
however, for convenience, the angle φ can be written both as a negative value and as values greater than 2π. Polar coordinates are related to Cartesian coordinates
with the following expressions:

Converting polar coordinates to Cartesian coordinates:

x = r cosφ

y = r sinφ

#### Cylindrical coordinates *r, φ, z*

Cylindrical coordinates were introduced to work with cylindrical bodies and surfaces, cylindrical coordinates are convenient to use, for example, for a spiral, spiral equations in a cylindrical coordinate system will look much simpler than in Cartesian coordinates. In cylindrical coordinates, the XY plane is defined in the same way as in polar coordinates: using the distance and angle between the radius vector and the axis The x, z coordinate is the same as in Cartesian coordinates. Relation of cylindrical and Cartesian coordinates:

Converting cylindrical coordinates to Cartesian coordinates:

x = r cosφ

y = r sinφ

z = z

#### Spherical coordinates *ρ, φ, θ*

Spherical coordinates, as the name implies, were introduced to work with spherical bodies, the position of any point in spherical coordinates is determined by using two angles φ and θ and a radius vector ρ. Spherical coordinates are related to Cartesian coordinates as follows:

Converting spherical coordinates to Cartesian coordinates:

x = ρ sinφ cosθ

y = ρ sinφ sinθ

z = ρ cosφ