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

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