List of canonical coordinate transformations

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

:$x=r,cos\; heta\; quad$:$y=r,sin\; heta\; quad$:

:$det\{frac\{partial(x,\; y)\}\{partial(r,\; heta)\; =r$

To polar coordinates from Cartesian coordinates

:$r=sqrt\{x^2\; +\; y^2\}$:$heta^prime\; =\; arctanleft|frac\{y\}\{x\}\; ight|$Note: solving for $heta^prime$ returns the resultant angle in the first quadrant (

:For $heta^prime$ in QI::$heta\; =\; heta^prime$

:For $heta^prime$ in QII::$heta=\; pi\; -\; heta^prime$

:For $heta^prime$ in QIII::$heta\; =\; pi\; +\; heta^prime$

:For $heta^prime$ in QIV::$heta\; =\; 2pi\; -\; heta^prime$

The value for $heta$ must be solved for in this manner because for all values of $heta$, $arctan\; heta$ is only defined for

Note that one can also use:$r=sqrt\{x^2\; +\; y^2\}$:$heta\; =\; 2\; arctan\; frac\{y\}\{x+r\}$

To Cartesian coordinates from bipolar coordinates

:$x\; =\; a\; frac\{sinh\; au\}\{cosh\; au\; -\; cos\; sigma\}$

:$y\; =\; a\; frac\{sin\; sigma\}\{cosh\; au\; -\; cos\; sigma\}$

To Cartesian coordinates from two-center bipolar coordinates [Weisstein, Eric W.. "Bipolar Coordinates." "Treasure Troves". 26 May 1999. Sociology and Anthropology China. 14 Feb 2007 [http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/b/b233.htm] ]

:$x\; =\; frac\{r\_1^2-r\_2^2\}\{4c\}$

:$y\; =\; pm\; frac\{1\}\{4c\}sqrt\{16c^2r\_1^2-(r\_1^2-r\_2^2+4c^2)^2\}$

To polar coordinates from two-center bipolar coordinates

:$r\; =\; sqrt\{frac\{r\_1^2+r\_2^2-2c^2\}\{2$

:$heta\; =\; arctan\; left\; [\; sqrt\{frac\{8c^2(r\_1^2+r\_2^2-2c^2)\}\{r\_1^2-r\_2^2\}-1\}\; ight]$

Where 2"c" is the distance between the poles.

To Cartesian coordinates from Cesàro equation

:$x\; =\; int\; cos\; left\; [int\; kappa(s)\; ,ds\; ight]\; ds$

:$y\; =\; int\; sin\; left\; [int\; kappa(s)\; ,ds\; ight]\; ds$

Arc length and curvature from Cartesian coordinates

$kappa\; =\; frac\{x\text{'}y"-y\text{'}x"\}\{(x\text{'}^2+y\text{'}^2)^\{3/2$

$s\; =\; int\_a^t\; sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \},\; dt$

Arc length and curvature from polar coordinates

$kappa=frac\{r^2+2r\text{'}^2-rr"\}\{(r^2+r\text{'}^2)^\{3/2$

$s\; =\; int\_a^phi\; sqrt\; \{\; 1\; +\; y\text{'}^2\; \},\; dphi$

3-Dimensional

Let (x, y, z) be the standard Cartesian coordinates, and (ρ, 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 definition, 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 main axes and are in practice most easily solved by observation.

To Cartesian coordinates

From spherical coordinates

:$\{x\}=\; ho\; ,\; sin\; heta\; ,\; cosphi\; quad$:$\{y\}=\; ho\; ,\; sin\; heta\; ,\; sinphi\; quad$:$\{z\}=\; ho\; ,\; cos\; heta\; quad$

:So for the volume element::$dx;dy;dz=det\{frac\{partial(x,\; y,\; z)\}\{partial(\; ho,\; heta,\; phi)\; d\; ho;d\; heta;dphi\; =\; ho^2\; sin\; heta\; ;\; d\; ho\; ;\; d\; heta\; ;\; dphi\; ;$

From cylindrical coordinates

:$\{x\}=\{r\}\; ,cos\; heta$:$\{y\}=\{r\}\; ,\; sin\; heta$:$\{z\}=\{h\}\; ,$

:So for the volume element::$dx;dy;dz=det\{frac\{partial(x,\; y,\; z)\}\{partial(r,\; heta,\; h)\; dr;d\; heta;dh\; =\{r\};\; dr\; ;\; d\; heta\; ;\; dh\; ;$

To Spherical coordinates

From Cartesian coordinates

:$\{\; ho\}=sqrt\{x^2\; +\; y^2\; +\; z^2\}$

:$\{\; heta\}=arctan\; left(\; frac\{sqrt\{x^2\; +\; y^2\{z\}\; ight)=arccos\; left(\; \{frac\{z\}\{sqrt\{x^2\; +\; y^2\; +\; z^2\}\; ight)$

:$\{phi\}=arctan\; left(\; \{frac\{y\}\{x\; ight)=\; arccos\; left(\; frac\{x\}\{sqrt\{x^2+y^2\; ight)\; =\; arcsin\; left(\; frac\{y\}\{sqrt\{x^2+y^2\; ight)$

:

From cylindrical coordinates

:$\{\; ho\}=sqrt\{r^2+h^2\}$:$\{\; heta\}=\; heta\; quad$:$\{phi\}=arctanfrac\{r\}\{h\}$

:

:$det\; frac\{partial(\; ho,\; heta,\; phi)\}\{partial(r,\; heta,\; h)\}\; =\; frac\{1\}\{sqrt\{r^2+h^2$

To cylindrical coordinates

From Cartesian coordinates

:$r=sqrt\{x^2\; +\; y^2\}$:$heta=arctanfrac\{y\}\{x\}\; +\; pi\; u\_0(-x)\; ,\; operatorname\{sgn\}\; y$ :$h=z\; quad$

:

From spherical coordinates

:$r\; =\; ho\; sin\; phi\; ,$:$heta\; =\; heta\; ,$:$h\; =\; ho\; cos\; phi\; ,$

:

:$detfrac\{partial(r,\; heta,\; h)\}\{partial(\; ho,\; heta,\; phi)\}\; =\; -\; ho$

Arc length, curvature and torsion from cartesian coordinates

:$s\; =\; int\_0^t\; sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; +\; z\text{'}^2\; \},\; dt$

:$kappa=frac\{sqrt\{(z"y\text{'}-z\text{'}y")^2+(x"z\text{'}-z"x\text{'})^2+(y"x\text{'}-x"y\text{'})^2\{(x\text{'}^2+y\text{'}^2+z\text{'}^2)^\{3/2$

:$au=frac\{z$(x'y"-y'x")+z"(xy'-x'y)+z'(x"y-x"'y")}{(x'^2+y'^2+z'^2)(x"^2+y"^2+z"^2)}

References