List of common coordinate transformations

This is a list of some of the most commonly used coordinate transformations.

2-dimensional

Let (x, y) be the standard Cartesian coordinates, and r and θ the standard polar coordinates.

To Cartesian coordinates

From polar coordinates

{\begin{aligned}x&=r\,\cos \theta \\y&=r\,\sin \theta \\{\frac {\partial (x,y)}{\partial (r,\theta )}}&={\begin{pmatrix}\cos \theta &-r\,\sin \theta \\\sin \theta &r\,\cos \theta \end{pmatrix}}\\\det {\frac {\partial (x,y)}{\partial (r,\theta )}}&=r\end{aligned}}

From log-polar coordinates

{\begin{aligned}x&=e^{\rho }\cos \theta ,\\y&=e^{\rho }\sin \theta .\end{aligned}}

By using complex numbers $(x,y)=x+iy'$ , the transformation can be written as

x+iy=e^{\rho +i\theta }

I.e., it is given by the complex exponential function.

From bipolar coordinates

{\begin{aligned}x&=a{\frac {\sinh \tau }{\cosh \tau -\cos \sigma }}\\y&=a{\frac {\sin \sigma }{\cosh \tau -\cos \sigma }}\end{aligned}}

From 2-center bipolar coordinates

{\begin{aligned}x&={\frac {1}{4c}}\left(r_{1}^{2}-r_{2}^{2}\right)\\y&=\pm {\frac {1}{4c}}{\sqrt {16c^{2}r_{1}^{2}-(r_{1}^{2}-r_{2}^{2}+4c^{2})^{2}}}\end{aligned}}

From Cesàro equation

{\begin{aligned}x&=\int \cos \left[\int \kappa (s)\,ds\right]ds\\y&=\int \sin \left[\int \kappa (s)\,ds\right]ds\end{aligned}}

To polar coordinates

From Cartesian coordinates

{\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}\\\theta ^{\prime }&=\arctan \left|{\frac {y}{x}}\right|\end{aligned}}

Note: solving for $\theta ^{\prime }$ returns the resultant angle in the first quadrant ( $0<\theta <{\frac {\pi }{2}}$ ). To find $\theta$ , one must refer to the original Cartesian coordinate, determine the quadrant in which $\theta$ lies (ex (3,-3) [Cartesian] lies in QIV), then use the following to solve for $\theta$ :

For $\theta ^{\prime }$ in QI:
$\theta =\theta ^{\prime }$
For $\theta ^{\prime }$ in QII:
$\theta =\pi -\theta ^{\prime }$
For $\theta ^{\prime }$ in QIII:
$\theta =\pi +\theta ^{\prime }$
For $\theta ^{\prime }$ in QIV:
$\theta =2\pi -\theta ^{\prime }$

The value for $\theta$ must be solved for in this manner because for all values of $\theta$ , $\tan \theta$ is only defined for $-{\frac {\pi }{2}}<\theta <+{\frac {\pi }{2}}$ , and is periodic (with period $\pi$ ). This means that the inverse function will only give values in the domain of the function, but restricted to a single period. Hence, the range of the inverse function is only half a full circle.

Note that one can also use

{\begin{aligned}r&={\sqrt {x^{2}+y^{2}}}\\\theta ^{\prime }&=2\arctan {\frac {y}{x+r}}\end{aligned}}

From 2-center bipolar coordinates

{\begin{aligned}r&={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2c^{2}}{2}}}\\\theta &=\arctan \left[{\sqrt {{\frac {8c^{2}(r_{1}^{2}+r_{2}^{2}-2c^{2})}{r_{1}^{2}-r_{2}^{2}}}-1}}\right]\end{aligned}}

Where 2c is the distance between the poles.

To log-polar coordinates from Cartesian coordinates

{\begin{aligned}\rho &=\log {\sqrt {x^{2}+y^{2}}},\\\theta &=\arctan {\frac {y}{x}}.\end{aligned}}

Arc-length and curvature

In Cartesian coordinates

{\begin{aligned}\kappa &={\frac {x'y''-y'x''}{({x'}^{2}+{y'}^{2})^{\frac {3}{2}}}}\\s&=\int _{a}^{t}{\sqrt {{x'}^{2}+{y'}^{2}}}\,dt\end{aligned}}

In polar coordinates

{\begin{aligned}\kappa &={\frac {r^{2}+2{r'}^{2}-rr''}{(r^{2}+{r'}^{2})^{\frac {3}{2}}}}\\s&=\int _{a}^{\phi }{\sqrt {r^{2}+{r'}^{2}}}\,d\phi \end{aligned}}

3-dimensional

Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as [1], see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent.

If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged.

All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation.