Table shows formulas for conversion between various coordinate systems for example from cartesian to cylindrical or vice versa. Formulas for both two and three-dimensional coordinate systems are presented.

Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

⌛ Loading...

Polar coordinate system#

Name	Formula	Legend
Conversion from cartesian to polar system: r	Show source $r=\sqrt{x^{2}+y^{2}}$	$r$ - radial coordinate in polar system, x, y - coordinates in two-dimensional cartesian system.
Conversion from cartesian to polar system: φ	Show source $\phi=arctan\left(\frac{x}{y}\right)$	$\phi$ - angular coordinate in polar system, x, y - coordinates in two-dimensional cartesian system.
Conversion from polar to cartesian system: x	Show source $x=r \cdot cos\left(\phi\right)$	x - x-coordinate in cartesian system, $r$ , $\phi$ - polar coordinates: radial and angular.
Conversion from polar to cartesian system: y	Show source $y=r \cdot sin\left(\phi\right)$	y - y-coordinate in cartesian system, $r$ , $\phi$ - polar coordinates: radial and angular.
Conversion from cartesian to polar system	Show source $\begin{dcases}r=\sqrt{x^{2}+y^{2}}\\\phi=arctan\left(\frac{x}{y}\right)\end{dcases}$	$r$ , $\phi$ - polar coordinates: radial and angular, x, y - coordinates in two-dimensional cartesian system.
Conversion from polar to cartesian system	Show source $\begin{dcases}x=r \cdot cos\left(\phi\right)\\y=r \cdot sin\left(\phi\right)\end{dcases}$	x, y - coordinates in two-dimensional cartesian system, $r$ , $\phi$ - polar coordinates: radial and angular.

Cylindrical coordinate system#

Name	Formula	Legend
Conversion from cartesian to cylindrical system: ρ	Show source $\rho=\sqrt{x^{2}+y^{2}}$	$\rho$ - axial distance in cylindrical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to cylindrical system: φ	Show source $\phi=arctan\left(\frac{x}{y}\right)$	$\phi$ - azimuth in cylindrical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to cylindrical system: z	Show source $z=z$	$z$ - height in cylindrical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cylindrical to cartesian system: x	Show source $x=\rho \cdot cos\left(\phi\right)$	x - x-coordinate in cartesian system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to cartesian system: y	Show source $y=\rho \cdot sin\left(\phi\right)$	y - y-coordinate in cartesian system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to cartesian system: z	Show source $z=z$	z - z-coordinate in cartesian system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: r	Show source $r=\sqrt{\rho^{2}+z^{2}}$	$r$ - radial distance in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: θ	Show source $\theta=arctan\left(\frac{\rho}{z}\right)$	$\theta$ - polar angle in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: φ	Show source $\phi=\phi$	$\phi$ - azimuthal angle in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from spherical to cylindrical system: ρ	Show source $\rho=r \cdot sin\left(\theta\right)$	$\rho$ - axial distance in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: φ	Show source $\phi=\phi$	$\phi$ - azimuth in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: z	Show source $z=r \cdot cos\left(\theta\right)$	$z$ - height in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cartesian to cylindrical system	Show source $\begin{dcases}\rho=\sqrt{x^{2}+y^{2}}\\\phi=arctan\left(\frac{x}{y}\right)\\z=z\end{dcases}$	$\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cylindrical to cartesian system	Show source $\begin{dcases}x=\rho \cdot cos\left(\phi\right)\\y=\rho \cdot sin\left(\phi\right)\\z=z\end{dcases}$	x, y, z - coordinates in three-dimensional cartesian system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.

Spherical coordinate system#

Name	Formula	Legend
Conversion from cartesian to spherical system: r	Show source $r=\sqrt{x^{2}+y^{2}+z^{2}}$	$r$ - radial distance in spherical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to spherical system: θ	Show source $\theta=arccos\left(\frac{z}{x^{2}+y^{2}+z^{2}}\right)$	$\theta$ - polar angle in spherical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to spherical system: φ	Show source $\phi=arctan\left(\frac{y}{x}\right)$	$\phi$ - azimuthal angle in spherical system, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from spherical to cartesian system: x	Show source $x=r \cdot sin\left(\theta\right) \cdot cos\left(\phi\right)$	x - x-coordinate in cartesian system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cartesian system: y	Show source $y=r \cdot sin\left(\theta\right) \cdot sin\left(\phi\right)$	y - y-coordinate in cartesian system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cartesian system: z	Show source $z=r \cdot cos\left(\theta\right)$	z - z-coordinate in cartesian system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cylindrical to spherical system: r	Show source $r=\sqrt{\rho^{2}+z^{2}}$	$r$ - radial distance in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: θ	Show source $\theta=arctan\left(\frac{\rho}{z}\right)$	$\theta$ - polar angle in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: φ	Show source $\phi=\phi$	$\phi$ - azimuthal angle in spherical system, $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height.
Conversion from spherical to cylindrical system: ρ	Show source $\rho=r \cdot sin\left(\theta\right)$	$\rho$ - axial distance in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: φ	Show source $\phi=\phi$	$\phi$ - azimuth in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: z	Show source $z=r \cdot cos\left(\theta\right)$	$z$ - height in cylindrical system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cartesian to spherical system	Show source $\begin{dcases}r=\sqrt{x^{2}+y^{2}+z^{2}}\\\theta=arccos\left(\frac{z}{x^{2}+y^{2}+z^{2}}\right)\\\phi=arctan\left(\frac{y}{x}\right)\end{dcases}$	$r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle, x, y, z - coordinates in three-dimensional cartesian system.
Conversion from spherical to cartesian system	Show source $\begin{dcases}x=r \cdot sin\left(\theta\right) \cdot cos\left(\phi\right)\\y=r \cdot sin\left(\theta\right) \cdot sin\left(\phi\right)\\z=r \cdot cos\left(\theta\right)\end{dcases}$	x, y, z - coordinates in three-dimensional cartesian system, $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle.

Some facts#

To describe the location of a point in space, we need the coordinates.
The number of coordinates needed for an unambiguous description must be equal to the number of dimensions. For example:
- to describe the location of the point on a straight line (one-dimensional space), it is enough to provide only one number,
- to describe the location of the point on the plane it is necessary to provide two numbers e.g. Cartesian coordinates (x, y),
- to describe the location of the point in three-dimensional space it is necessary to provide three numbers e.g. Cartesian coordinates (x, y, z),
- etc.
The amount of numbers is related to number of dimensions, but the meaning of the coordinates may be differently defined. Examples below.
Examples of two-dimensional coordinate systems are:
- Cartesian coordinate system (rectangular) - pair of numbers $(x, y)$ , which determine the position of the point on two perpendicular axes.
- polar coordinate system - pair of numbers $(r, \phi)$ , the first means distance from the origin of the coordinate system and the second one is the angle. The relationship between the polar and Cartesian systems is as follows:
  
  $\begin{dcases}r=\sqrt{x^{2}+y^{2}}\\\phi=arctan\left(\frac{x}{y}\right)\end{dcases}$
  
  $\begin{dcases}x=r \cdot cos\left(\phi\right)\\y=r \cdot sin\left(\phi\right)\end{dcases}$
  where:
  - $r$ , $\phi$ - polar coordinates: radial and angular,
  - x, y - coordinates in two-dimensional cartesian system.
Examples of three-dimensional coordinate systems are:
- three-dimensional Cartesian coordinate system (rectangular) - generalization of the two-dimensional system by adding the third axis perpendicular to the others, the location of the point is described by three numbers usually denoted by $(x, y, z)$ ,
- cylindrical coordinate system - generalization of the polar system by adding the third coordinate z, which plays the same role as in the Cartesian system, thus we get three numbers $(\rho, \phi, z)$ :
  
  $\begin{dcases}\rho=\sqrt{x^{2}+y^{2}}\\\phi=arctan\left(\frac{x}{y}\right)\\z=z\end{dcases}$
  
  $\begin{dcases}\rho=\sqrt{x^{2}+y^{2}}\\\phi=arctan\left(\frac{x}{y}\right)\\z=z\end{dcases}$
  where:
  - $\rho$ , $\phi$ , $z$ - cylindrical coordinates: axial distance, azimuth and height,
  - x, y, z - coordinates in three-dimensional cartesian system.
- spherical coordinate system - another generalization of the polar system, but instead of the z coordinate a second angle is added, in this way we get three coordinates $(r, \theta, \phi)$ :
  
  $\begin{dcases}r=\sqrt{x^{2}+y^{2}+z^{2}}\\\theta=arccos\left(\frac{z}{x^{2}+y^{2}+z^{2}}\right)\\\phi=arctan\left(\frac{y}{x}\right)\end{dcases}$
  
  $\begin{dcases}x=r \cdot sin\left(\theta\right) \cdot cos\left(\phi\right)\\y=r \cdot sin\left(\theta\right) \cdot sin\left(\phi\right)\\z=r \cdot cos\left(\theta\right)\end{dcases}$
  gdzie:
  - $r$ , $\theta$ , $\phi$ - spherical coordinates: radial distance, polar angle and azimuthal angle,
  - x, y, z - coordinates in three-dimensional cartesian system.

Tags and links to this website#

Tags:

mathematical_tables_coordinate_systems · table_of_formulas_for_coordinate_conversion · coordinate_systems_formulas · spherical_system_formulas · polar_system_formulas · cylindrical_system_formulas · spherical_coordinates_formulas · polar_coordinates_formulas · cylindrical_coordinates_formulas · coordinate_conversion_formulas

Tags to Polish version:

tablice_matematyczne_uklady_wspolrzednych · wzory_na_zamiane_wspolrzednych · wzory_na_zamiane_ukladu_wspolrzednych · uklad_sferyczny_wzory · uklad_polarny_wzory · uklad_cylindryczny_wzory

What tags this calculator has#

TABLE · MATH · PHYSICS · A -> Z (all)

Permalink#

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

https://calculla.com/mathematical_tables_coordinate_systems

Links to external sites (leaving Calculla?)#