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", "θ") the standard polar coordinates.
To Cartesian coordinates.
From log-polar coordinates.
By using complex numbers formula_3, the transformation can be written as
That is, it is given by the complex exponential function.
To polar coordinates.
From Cartesian coordinates.
Note: solving for formula_9 returns the resultant angle in the first quadrant (formula_10). To find formula_11, one must refer to the original Cartesian coordinate, determine the quadrant in which formula_11 lies (for example, (3,−3) [Cartesian] lies in QIV), then use the following to solve for formula_11:
The value for formula_11 must be solved for in this manner because for all values of formula_11, formula_24 is only defined for formula_25, and is periodic (with period formula_26). 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 coordinates.
Where 2"c" is the distance between the poles.
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 , 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 coordinates.
From spherical coordinates.
So for the volume element:
From cylindrical coordinates.
So for the volume element:
To spherical coordinates.
From Cartesian coordinates.
See also the article on atan2 for how to elegantly handle some edge cases.
So for the element:
Arc-length, curvature and torsion from Cartesian coordinates.
 s &= \int_0^t \sqrt{({x'}^2 + {y'}^2 + {z'}^2)^\frac{3}{2}} \\[3pt]