Wikipedia / List of common coordinate transformations
Wikipedia / List of common coordinate transformations
This is a list of some of the most commonly used coordinate transformations.
2-dimensional
be the standard Cartesian coordinates the standard polar coordinates.
To Cartesian coordinates
From polar coordinates
From log-polar coordinates
That is, it is given by the complex exponential function.
From bipolar coordinates
From 2-center bipolar coordinates
From Cesàro equation
To polar coordinates
From Cartesian coordinates
). 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.
From 2-center bipolar coordinates
Where 2c is the distance between the poles.
To log-polar coordinates from Cartesian coordinates
Arc-length and curvature
In Cartesian coordinates
In polar coordinates
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 illustrated). 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
To spherical coordinates
From Cartesian coordinates
See also the article on atan2 for how to elegantly handle some edge cases.
From cylindrical coordinates
To cylindrical coordinates
From Cartesian coordinates
From spherical coordinates
Arc-length, curvature and torsion from Cartesian coordinates
See also
References