Converts coordinates from one system to another for example from cartesian to spherical or vice versa. Both two and three-dimensional coordinate systems are available.

Beta version#

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Symbolic algebra

ⓘ Hint: This calculator supports symbolic math. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations

Input data#

Type of system
Choose the type of coordinates, which you know	Cartesian coordinates (2D)Cartesian coordinates (3D)Polar coordinatesCylindrical coordinatesSpherical coordinates
Cartesian coordinates
X-coordinate ( $x$ )
Y-coordinate ( $y$ )
Z-coordinate ( $z$ )
Polar coordinates
Radial coordinate ( $r$ )
Angular coordinate ( $\phi$ )
Cylindrical coordinates
Radial coordinate ( $\rho$ )
Angular coordinate ( $\phi$ )
Z-coordinate ( $z$ )
Spherical coordinates
Radial distance ( $r$ )
Polar angle ( $\theta$ )
Azimuthal angle ( $\phi$ )

Results#

Cartesian coordinates
X-coordinate ( $x$ )	Show source $-$
Y-coordinate ( $y$ )	Show source $-$
Z-coordinate ( $z$ )	Show source $-$
Polar coordinates
Radial coordinate ( $r$ )	Show source $-$
Angular coordinate ( $\phi$ )	Show source $-$
Cylindrical coordinates
Radial coordinate ( $\rho$ )	Show source $-$
Angular coordinate ( $\phi$ )	Show source $-$
Z-coordinate ( $z$ )	Show source $-$
Spherical coordinates
Radial distance ( $r$ )	Show source $-$
Polar angle ( $\theta$ )	Show source $-$
Azimuthal angle ( $\phi$ )	Show source $-$

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:

coordinate_systems · coordinate_systems_converter · coords_converter

Tags to Polish version:

uklady_wspolrzednych · rodzaje_ukladow_wspolrzednych · konwerter_ukladow_wspolrzednych · zamiana_wspolrzednych

What tags this calculator has#

CONVERTER · 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/coordinate_systems?inSystemType=cartesian3d&inCartesian_x=$symbolic(2)&inCartesian_y=$symbolic(2)&inCartesian_z=$symbolic(3)&inPolar_r=$symbolic(0)&inPolar_fi=$symbolic(0)&inCylindrical_r=$symbolic(0)&inCylindrical_fi=$symbolic(0)&inCylindrical_z=$symbolic(0)&inSpherical_r=$symbolic(0)&inSpherical_theta=$symbolic(0)&inSpherical_fi=$symbolic(0)

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