Coordinate systems converter
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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This online calculator is currently under heavy development. It may or it may NOT work correctly.
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However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
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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
X-coordinate (xx)
Y-coordinate (yy)
Z-coordinate (zz)
Polar coordinates
Radial coordinate (rr)
Angular coordinate (ϕ\phi)
Cylindrical coordinates
Radial coordinate (ρ\rho)
Angular coordinate (ϕ\phi)
Z-coordinate (zz)
Spherical coordinates
Radial distance (rr)
Polar angle (θ\theta)
Azimuthal angle (ϕ\phi)

Results#

Cartesian coordinates
X-coordinate (xx)Show source-
Y-coordinate (yy)Show source-
Z-coordinate (zz)Show source-
Polar coordinates
Radial coordinate (rr)Show source-
Angular coordinate (ϕ\phi)Show source-
Cylindrical coordinates
Radial coordinate (ρ\rho)Show source-
Angular coordinate (ϕ\phi)Show source-
Z-coordinate (zz)Show source-
Spherical coordinates
Radial distance (rr)Show source-
Polar angle (θ\theta)Show source-
Azimuthal angle (ϕ\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)(x, y), which determine the position of the point on two perpendicular axes.
    • polar coordinate system - pair of numbers (r,ϕ)(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:
      {r=x2+y2ϕ=arctan(xy)\begin{dcases} r=\sqrt{{ x}^{2}+{ y}^{2}}\\ \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)\end{dcases}
      {x=r cos(ϕ)y=r sin(ϕ)\begin{dcases} x= r~\cos\left( \phi\right)\\ y= r~\sin\left( \phi\right)\end{dcases}
      where:
      • rr, ϕ\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)(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 (ρ,ϕ,z)(\rho, \phi, z):
      {ρ=x2+y2ϕ=arctan(xy)z=z\begin{dcases} \rho=\sqrt{{ x}^{2}+{ y}^{2}}\\ \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)\\ z= z\end{dcases}
      {ρ=x2+y2ϕ=arctan(xy)z=z\begin{dcases} \rho=\sqrt{{ x}^{2}+{ y}^{2}}\\ \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)\\ z= z\end{dcases}
      where:
      • ρ\rho, ϕ\phi, zz - 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,θ,ϕ)(r, \theta, \phi):
      {r=x2+y2+z2θ=arccos(zx2+y2+z2)ϕ=arctan(yx)\begin{dcases} r=\sqrt{{ x}^{2}+{ y}^{2}+{ z}^{2}}\\ \theta=\mathrm{arccos}\left(\frac{ z}{{ x}^{2}+{ y}^{2}+{ z}^{2}}\right)\\ \phi=\mathrm{arctan}\left(\frac{ y}{ x}\right)\end{dcases}
      {x=r sin(θ) cos(ϕ)y=r sin(θ) sin(ϕ)z=r cos(θ)\begin{dcases} x= r~\sin\left( \theta\right)~\cos\left( \phi\right)\\ y= r~\sin\left( \theta\right)~\sin\left( \phi\right)\\ z= r~\cos\left( \theta\right)\end{dcases}
      gdzie:
      • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle,
      • x, y, z - coordinates in three-dimensional cartesian system.

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