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 $(r,\theta )$ the standard polar coordinates.

To Cartesian coordinates

From polar coordinates

{\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\[5pt]{\frac {\partial (x,y)}{\partial (r,\theta )}}&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &{\phantom {-}}r\cos \theta \end{bmatrix}}\\[5pt]{\text{Jacobian}}=\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 }

That is, 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)\\[1ex]y&=\pm {\frac {1}{4c}}{\sqrt {16c^{2}r_{1}^{2}-\left(r_{1}^{2}-r_{2}^{2}+4c^{2}\right)^{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 '&=\arctan \left|{\frac {y}{x}}\right|\end{aligned}}

Note: solving for

\theta '

returns the resultant angle in the first quadrant (

{\textstyle 0<\theta <{\frac {\pi }{2}}}

). To find

\theta ,

one must refer to the original Cartesian coordinate, determine the quadrant in which

\theta

lies (for example, (3,−3) [Cartesian] lies in QIV), then use the following to solve for

\theta :

\theta ={\begin{cases}\theta '&{\text{for }}\theta '{\text{ in QI: }}&0<\theta '<{\frac {\pi }{2}}\\[1.2ex]\pi -\theta '&{\text{for }}\theta '{\text{ in QII: }}&{\frac {\pi }{2}}<\theta '<\pi \\[1.2ex]\pi +\theta '&{\text{for }}\theta '{\text{ in QIII: }}&\pi <\theta '<{\frac {3\pi }{2}}\\[1.2ex]2\pi -\theta '&{\text{for }}\theta '{\text{ in QIV: }}&{\frac {3\pi }{2}}<\theta '<2\pi \end{cases}}

The value for $\theta$ must be solved for in this manner because for all values of $\theta$ , $\tan \theta$ is only defined for ${\textstyle -{\frac {\pi }{2}}<\theta <+{\frac {\pi }{2}}}$ , 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 '&=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''}{\left({x'}^{2}+{y'}^{2}\right)^{\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}^{\varphi }{\sqrt {r^{2}+{r'}^{2}}}\,d\varphi \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.