List of common coordinate transformations

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

2-dimensionalEdit

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

To Cartesian coordinatesEdit

From polar coordinatesEdit

 

From log-polar coordinatesEdit

 

By using complex numbers  , the transformation can be written as

 

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

From bipolar coordinatesEdit

 

From 2-center bipolar coordinatesEdit

 

From Cesàro equationEdit

 

To polar coordinatesEdit

From Cartesian coordinatesEdit

 

Note: solving for   returns the resultant angle in the first quadrant ( ). To find  , one must refer to the original Cartesian coordinate, determine the quadrant in which   lies (ex (3,-3) [Cartesian] lies in QIV), then use the following to solve for  :

  • For   in QI:
     
  • For   in QII:
     
  • For   in QIII:
     
  • For   in QIV:
     

The value for   must be solved for in this manner because for all values of  ,   is only defined for  , and is periodic (with period  ). 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

 

From 2-center bipolar coordinatesEdit

 

Where 2c is the distance between the poles.

To log-polar coordinates from Cartesian coordinatesEdit

 

Arc-length and curvatureEdit

In Cartesian coordinatesEdit

 

In polar coordinatesEdit

 

3-dimensionalEdit

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.

To Cartesian coordinatesEdit

From spherical coordinatesEdit

 

So for the volume element:

 

From cylindrical coordinatesEdit

 

So for the volume element:

 

To spherical coordinatesEdit

From Cartesian coordinatesEdit

 

See also the article on atan2 for how to elegantly handle some edge cases.

So for the element:

 

From cylindrical coordinatesEdit

 

To cylindrical coordinatesEdit

From Cartesian coordinatesEdit

 
 

From spherical coordinatesEdit

 

Arc-length, curvature and torsion from Cartesian coordinatesEdit

 

See alsoEdit

ReferencesEdit

  • Arfken, George (2013). Mathematical Methods for Physicists. Academic Press. ISBN 978-0123846549.