List of canonical coordinate transformations
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This is a list of canonical 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
- [x=r\,\cos\theta \quad]
- [y=r\,\sin\theta \quad]
To polar coordinates from Cartesian coordinates
- [r=\sqrt]
- [\theta = \arctan\frac]
Anyhow these special exceptions, although easily allowed for when calculated by hand, make the writing of a general computer program quite a task. Luckily most computer languages provide in addition to the normal arctangent, also an arctangent with 2 arguments with exactly the wanted behaviour. On electronic pocket calculators that function is usually called R->P (rectangular to polar).
3-Dimensional
Let (x, y, z) be the standard Cartesian coordinates, and (r, θ, φ) the spherical coordinates, with φ the angle measured away from the +Z axis. As θ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. φ 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, φ is chosen to run from −90° to +90°, in opposite direction of the earlier defintion, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in φ 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 mainaxes and are in practice most easily solved by observation.
To Cartesian coordinates
From spherical coordinates
- [=\rho \, \sin\phi \, \cos\theta \quad ]
- [=\rho \, \sin\phi \, \sin\theta \quad ]
- [=\rho \, \cos\phi \quad ]
- [\frac =\begin\sin\phi\cos\theta & -\rho\sin\phi\sin\theta & \rho\cos\phi\cos\theta \\\sin\phi\sin\theta & \rho\sin\phi\cos\theta & \rho\cos\phi\sin\theta \\\cos\phi & 0 & -\rho\sin\phi\end]
- [\det} =\rho^2 \sin\phi \; d\rho \; d\theta \; d\phi \;]
From cylindrical coordinates
- [= \,\cos\theta]
- [= \, \sin\theta]
- [= \,]
- [\frac =\begin\cos\theta & -r\sin\theta & 0 \\\sin\theta & r\cos\theta & 0 \\ 0 & 0 & 1\end]
- [\det} =\; dr \; d\theta \; dh \;]
To Spherical coordinates
From Cartesian coordinates
- [ = \sqrt]
- [ = \arccos \frac} = \arcsin \frac} = \arctan\frac]
- [ = \arccos\frac} = \arctan\frac}]
- [\frac =\begin \frac & \frac & \frac \\ \frac & \frac & 0 \\\frac} & \frac} & \frac}\end]
From cylindrical coordinates
- [=\sqrt]
- [=\theta \quad]
- [=\arctan\frac]
- [\frac =\begin\frac} & 0 & \frac} \\0 & 1 & 0 \\\frac & 0 & \frac \end]
- [ \det \frac = \frac}]
To cylindrical coordinates
From Cartesian coordinates
- [r=\sqrt]
- [\theta=\arctan\frac + \pi u_0(-x) \, \operatorname y ]
- [h=z \quad]
- [\frac =\begin\frac}&\frac}&0\\\frac&\frac&0\\0&0&1\end]
From spherical coordinates
- [ r = \rho \sin \phi ]
- [ \theta = \theta ]
- [ h = \rho \cos \phi ]
- [\frac =\begin\sin\phi & 0 & \rho\cos\phi \\0 & 1 & 0 \\\cos\phi & 0 & -\rho\sin\phi\end]
- [ \det\frac = - \rho ]
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