The VectorCalculus package supports the following coordinate systems:

In two dimensions - bipolar, cardioid, cassinian, cartesian, elliptic, hyperbolic, invcassinian, logarithmic, logcosh, parabolic, polar, rose, and tangent.

In three dimensions - bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, cartesian, casscylindrical, conical, cylindrical, ellcylindrical, hypercylindrical, invcasscylindrical, logcoshcylindrical, logcylindrical, oblatespheroidal, paraboloidal, paracylindrical, prolatespheroidal, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.

Note: Only the positive roots have been used for the following transformations:

In two dimensions - cassinian, hyperbolic, invcassinian, and rose

In three dimensions - casscylindrical, conical, hypercylindrical, invcasscylindrical, and rosecylindrical

The conversions from the various coordinate systems to cartesian (rectangular) coordinates in 2-space

are given by:

bipolar: (Spiegel)

$x=\frac{\sinh \left(v\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

$y=\frac{\sin \left(u\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

cardioid:

$x=\frac{u^2-v^2}{2{\left(u^2+v^2\right)}^2}$

$y=\frac{uv}{{\left(u^2+v^2\right)}^2}$

cartesian:

$x=u$

$y=v$

cassinian:  (Cassinian-oval)

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2}$

elliptic:

$x=\cosh \left(u\right)\cos \left(v\right)$

$y=\sinh \left(u\right)\sin \left(v\right)$

hyperbolic:

$x=\sqrt{\sqrt{u^2+v^2}+u}$

$y=\sqrt{\sqrt{u^2+v^2}-u}$

invcassinian:  (inverse Cassinian-oval)

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

logarithmic:

$x=\frac{a\ln \left(u^2+v^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\frac{v}{u}\right)}{\mathrm{\pi }}$

logcosh:  (ln cosh)

$x=\frac{a\ln \left({\cosh \left(u\right)}^2-{\sin \left(v\right)}^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\tanh \left(u\right)\tan \left(v\right)\right)}{\mathrm{\pi }}$

parabolic:

$x=\frac{u^2}{2}-\frac{v^2}{2}$

$y=uv$

polar:

$x=u\cos \left(v\right)$

$y=u\sin \left(v\right)$

rose:

$x=\frac{\sqrt{\sqrt{u^2+v^2}+u}}{\sqrt{u^2+v^2}}$

$y=\frac{\sqrt{\sqrt{u^2+v^2}-u}}{\sqrt{u^2+v^2}}$

tangent:

$x=\frac{u}{u^2+v^2}$

$y=\frac{v}{u^2+v^2}$

The conversions from the various coordinate systems to cartesian coordinates in 3-space

are given as follows (the author is indicated where applicable):

bipolarcylindrical:  (Spiegel)

$x=\frac{a\sinh \left(v\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

$y=\frac{a\sin \left(u\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

$z=w$

bispherical:

$x=\frac{\sin \left(u\right)\cos \left(w\right)}{d}$

$y=\frac{\sin \left(u\right)\sin \left(w\right)}{d}$

$z=\frac{\sinh \left(v\right)}{d}$ where $d=\cosh \left(v\right)-\cos \left(u\right)$

cardioidal:

$x=\frac{uv\cos \left(w\right)}{{\left(u^2+v^2\right)}^2}$

$y=\frac{uv\sin \left(w\right)}{{\left(u^2+v^2\right)}^2}$

$\frac{u^2-v^2}{2{\left(u^2+v^2\right)}^2}$

cardioidcylindrical:

$x=\frac{u^2-v^2}{2{\left(u^2+v^2\right)}^2}$

$y=\frac{uv}{{\left(u^2+v^2\right)}^2}$

$z=w$

cartesian:

$x=u$

$y=v$

$z=w$

casscylindrical:  (Cassinian-oval cylinder)

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2}$

$z=w$

conical:

$x=\frac{uvw}{ab}$

$y=\frac{u\sqrt{\frac{\left(-b^2+v^2\right)\left(b^2-w^2\right)}{a^2-b^2}}}{b}$

$z=\frac{u\sqrt{\frac{\left(a^2-v^2\right)\left(a^2-w^2\right)}{a^2-b^2}}}{a}$

cylindrical:

$x=u\cos \left(v\right)$

$y=u\sin \left(v\right)$

$z=w$

ellcylindrical:  (elliptic cylindrical)

$x=a\cosh \left(u\right)\cos \left(v\right)$

$y=a\sinh \left(u\right)\sin \left(v\right)$

$z=w$

hypercylindrical:  (hyperbolic cylinder)

$x=\sqrt{\sqrt{u^2+v^2}+u}$

$y=\sqrt{\sqrt{u^2+v^2}-u}$

$z=w$

invcasscylindrical:  (inverse Cassinian-oval cylinder)

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

$z=w$

logcylindrical:  (logarithmic cylinder)

$x=\frac{a\ln \left(u^2+v^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\frac{v}{u}\right)}{\mathrm{\pi }}$

$z=w$

logcoshcylindrical:  (ln cosh cylinder)

$x=\frac{a\ln \left({\cosh \left(u\right)}^2-{\sin \left(v\right)}^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\tanh \left(u\right)\tan \left(v\right)\right)}{\mathrm{\pi }}$

$z=w$

oblatespheroidal:

$x=a\cosh \left(u\right)\sin \left(v\right)\cos \left(w\right)$

$y=a\cosh \left(u\right)\sin \left(v\right)\sin \left(w\right)$

$z=a\sinh \left(u\right)\cos \left(v\right)$

paraboloidal:  (Spiegel)

$x=uv\cos \left(w\right)$

$y=uv\sin \left(w\right)$

$z=\frac{u^2}{2}-\frac{v^2}{2}$

paracylindrical:

$x=\frac{u^2}{2}-\frac{v^2}{2}$

$y=uv$

$z=w$

prolatespheroidal:

$x=a\sinh \left(u\right)\sin \left(v\right)\cos \left(w\right)$

$y=a\sinh \left(u\right)\sin \left(v\right)\sin \left(w\right)$

$z=a\cosh \left(u\right)\cos \left(v\right)$

rosecylindrical:

$x=\frac{\sqrt{\sqrt{u^2+v^2}+u}}{\sqrt{u^2+v^2}}$

$y=\frac{\sqrt{\sqrt{u^2+v^2}-u}}{\sqrt{u^2+v^2}}$

$z=w$

sixsphere:  (6-sphere)

$x=\frac{u}{u^2+v^2+w^2}$

$y=\frac{v}{u^2+v^2+w^2}$

$z=\frac{w}{u^2+v^2+w^2}$

spherical:

$x=u\cos \left(w\right)\sin \left(v\right)$

$y=u\sin \left(w\right)\sin \left(v\right)$

$z=u\cos \left(v\right)$

tangentcylindrical:

$x=\frac{u}{u^2+v^2}$

$y=\frac{v}{u^2+v^2}$

$z=w$

tangentsphere:

$x=\frac{u\cos \left(w\right)}{u^2+v^2}$

$y=\frac{u\sin \left(w\right)}{u^2+v^2}$

$z=\frac{v}{u^2+v^2}$

toroidal:

$x=\frac{a\sinh \left(v\right)\cos \left(w\right)}{d}$

$y=\frac{a\sinh \left(v\right)\sin \left(w\right)}{d}$

$z=\frac{a\sin \left(u\right)}{d}$ where $d=\cosh \left(v\right)-\cos \left(u\right)$

The a, b, and c values in the above coordinate transformations can be queried and set by using the GetCoordinateParameters and SetCoordinateParameters commands from the VectorCalculus package.  The default values are $a=1$, $b=\frac{1}{2}$, and $c=\frac{1}{3}$.

The GetCoordinateParameters command returns an expression sequence containing the current values of a, b, and c.

The SetCoordinateParameters command takes either 1, 2, or 3 arguments, and sets the values of a, a and b, or a, b, and c respectively.

Moon, P., and Spencer, D.E. Field Theory Handbook. 2d ed. Berlin: Springer-Verlag, 1971.

Spiegel, Murray R.  Mathematical Handbook of Formulas and Tables. New York:  McGraw Hill Book Company, 1968, pp. 126-130.