Mathematical tables: coordinate conversion formulas
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.

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Polar coordinate system

NameFormulaLegend
Conversion from cartesian to polar system: rShow sourcer=x2+y2 r=\sqrt{{ x}^{2}+{ y}^{2}}
  • rr - radial coordinate in polar system,
  • x, y - coordinates in two-dimensional cartesian system.
Conversion from cartesian to polar system: φShow sourceϕ=arctan(xy) \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)
  • ϕ\phi - angular coordinate in polar system,
  • x, y - coordinates in two-dimensional cartesian system.
Conversion from polar to cartesian system: xShow sourcex=r cos(ϕ) x= r~\cos\left( \phi\right)
  • x - x-coordinate in cartesian system,
  • rr, ϕ\phi - polar coordinates: radial and angular.
Conversion from polar to cartesian system: yShow sourcey=r sin(ϕ) y= r~\sin\left( \phi\right)
  • y - y-coordinate in cartesian system,
  • rr, ϕ\phi - polar coordinates: radial and angular.
Conversion from cartesian to polar systemShow source{r=x2+y2ϕ=arctan(xy)\begin{dcases} r=\sqrt{{ x}^{2}+{ y}^{2}}\\ \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)\end{dcases}
  • rr, ϕ\phi - polar coordinates: radial and angular,
  • x, y - coordinates in two-dimensional cartesian system.
Conversion from polar to cartesian systemShow source{x=r cos(ϕ)y=r sin(ϕ)\begin{dcases} x= r~\cos\left( \phi\right)\\ y= r~\sin\left( \phi\right)\end{dcases}
  • x, y - coordinates in two-dimensional cartesian system,
  • rr, ϕ\phi - polar coordinates: radial and angular.

Cylindrical coordinate system

NameFormulaLegend
Conversion from cartesian to cylindrical system: ρShow sourceρ=x2+y2 \rho=\sqrt{{ x}^{2}+{ y}^{2}}
  • ρ\rho - axial distance in cylindrical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to cylindrical system: φShow sourceϕ=arctan(xy) \phi=\mathrm{arctan}\left(\frac{ x}{ y}\right)
  • ϕ\phi - azimuth in cylindrical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to cylindrical system: zShow sourcez=z z= z
  • zz - height in cylindrical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cylindrical to cartesian system: xShow sourcex=ρ cos(ϕ) x= \rho~\cos\left( \phi\right)
  • x - x-coordinate in cartesian system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to cartesian system: yShow sourcey=ρ sin(ϕ) y= \rho~\sin\left( \phi\right)
  • y - y-coordinate in cartesian system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to cartesian system: zShow sourcez=z z= z
  • z - z-coordinate in cartesian system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: rShow sourcer=ρ2+z2 r=\sqrt{{ \rho}^{2}+{ z}^{2}}
  • rr - radial distance in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: θShow sourceθ=arctan(ρz) \theta=\mathrm{arctan}\left(\frac{ \rho}{ z}\right)
  • θ\theta - polar angle in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: φShow sourceϕ=ϕ \phi= \phi
  • ϕ\phi - azimuthal angle in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from spherical to cylindrical system: ρShow sourceρ=r sin(θ) \rho= r~\sin\left( \theta\right)
  • ρ\rho - axial distance in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: φShow sourceϕ=ϕ \phi= \phi
  • ϕ\phi - azimuth in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: zShow sourcez=r cos(θ) z= r~\cos\left( \theta\right)
  • zz - height in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cartesian to cylindrical systemShow source{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\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cylindrical to cartesian systemShow source{x=ρ cos(ϕ)y=ρ sin(ϕ)z=z\begin{dcases} x= \rho~\cos\left( \phi\right)\\ y= \rho~\sin\left( \phi\right)\\ z= z\end{dcases}
  • x, y, z - coordinates in three-dimensional cartesian system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.

Spherical coordinate system

NameFormulaLegend
Conversion from cartesian to spherical system: rShow sourcer=x2+y2+z2 r=\sqrt{{ x}^{2}+{ y}^{2}+{ z}^{2}}
  • rr - radial distance in spherical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to spherical system: θShow sourceθ=arccos(zx2+y2+z2) \theta=\mathrm{arccos}\left(\frac{ z}{{ x}^{2}+{ y}^{2}+{ z}^{2}}\right)
  • θ\theta - polar angle in spherical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from cartesian to spherical system: φShow sourceϕ=arctan(yx) \phi=\mathrm{arctan}\left(\frac{ y}{ x}\right)
  • ϕ\phi - azimuthal angle in spherical system,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from spherical to cartesian system: xShow sourcex=r sin(θ) cos(ϕ) x= r~\sin\left( \theta\right)~\cos\left( \phi\right)
  • x - x-coordinate in cartesian system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cartesian system: yShow sourcey=r sin(θ) sin(ϕ) y= r~\sin\left( \theta\right)~\sin\left( \phi\right)
  • y - y-coordinate in cartesian system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cartesian system: zShow sourcez=r cos(θ) z= r~\cos\left( \theta\right)
  • z - z-coordinate in cartesian system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cylindrical to spherical system: rShow sourcer=ρ2+z2 r=\sqrt{{ \rho}^{2}+{ z}^{2}}
  • rr - radial distance in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: θShow sourceθ=arctan(ρz) \theta=\mathrm{arctan}\left(\frac{ \rho}{ z}\right)
  • θ\theta - polar angle in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from cylindrical to spherical system: φShow sourceϕ=ϕ \phi= \phi
  • ϕ\phi - azimuthal angle in spherical system,
  • ρ\rho, ϕ\phi, zz - cylindrical coordinates: axial distance, azimuth and height.
Conversion from spherical to cylindrical system: ρShow sourceρ=r sin(θ) \rho= r~\sin\left( \theta\right)
  • ρ\rho - axial distance in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: φShow sourceϕ=ϕ \phi= \phi
  • ϕ\phi - azimuth in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from spherical to cylindrical system: zShow sourcez=r cos(θ) z= r~\cos\left( \theta\right)
  • zz - height in cylindrical system,
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle.
Conversion from cartesian to spherical systemShow source{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}
  • rr, θ\theta, ϕ\phi - spherical coordinates: radial distance, polar angle and azimuthal angle,
  • x, y, z - coordinates in three-dimensional cartesian system.
Conversion from spherical to cartesian systemShow source{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}
  • x, y, z - coordinates in three-dimensional cartesian system,
  • rr, θ\theta, ϕ\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)(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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