At present, Maple supports the following coordinate systems:

In three dimensions - bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, casscylindrical, confocalellip, confocalparab, conical, cylindrical, ellcylindrical, ellipsoidal, hypercylindrical, invcasscylindrical, invellcylindrical, invoblspheroidal, invprospheroidal, logcoshcylindrical, logcylindrical, maxwellcylindrical, oblatespheroidal, paraboloidal, paraboloidal2, paracylindrical, prolatespheroidal, rectangular, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.

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

NOTE that only the positive roots have been used for the following transformations: (in three dimensions) casscylindrical, confocalellip, confocalparab, conical, ellipsoidal, hypercylindrical, invcasscylindrical, paraboloidal2, rosecylindrical; (in two dimensions) cassinian, hyperbolic, invcassinian, and rose.

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

are given as follows (note that the author is indicated where necessary):

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:

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

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

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

cardioidcylindrical:

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

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

$z=w$

casscylindrical:  (Cassinian-oval cylinder)

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

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

$z=w$

confocalellip:  (confocal elliptic)

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

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

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

confocalparab:  (confocal parabolic)

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

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

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

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

ellipsoidal:

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

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

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

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)

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

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

$z=w$

invellcylindrical:  (inverse elliptic cylinder)

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

$y=\frac{a\sinh \left(u\right)\sin \left(v\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

$z=w$

invoblspheroidal:  (inverse oblate spheroidal)

$x=\frac{a\cosh \left(u\right)\sin \left(v\right)\cos \left(w\right)}{{\cosh \left(u\right)}^2-{\cos \left(v\right)}^2}$

$y=\frac{a\cosh \left(u\right)\sin \left(v\right)\sin \left(w\right)}{{\cosh \left(u\right)}^2-{\cos \left(v\right)}^2}$

$z=\frac{a\sinh \left(u\right)\cos \left(v\right)}{{\cosh \left(u\right)}^2-{\cos \left(v\right)}^2}$

invprospheroidal:  (inverse prolate spheroidal)

$x=\frac{a\sinh \left(u\right)\sin \left(v\right)\cos \left(w\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

$y=\frac{a\sinh \left(u\right)\sin \left(v\right)\sin \left(w\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

$z=\frac{a\cosh \left(u\right)\cos \left(v\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

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)

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

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

$z=w$

maxwellcylindrical:

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

$y=\frac{a\left(v+{\ⅇ}^u\sin \left(v\right)\right)}{\mathrm{\pi }}$

$z=w$

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)

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

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

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

paraboloidal2:  (Moon)

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

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

$z=u+v+w-a-b$

paracylindrical:

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

$y=uv$

$z=w$

prolatespheroidal:

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

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

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

rectangular:

$x=u$

$y=v$

$z=w$

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:

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

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

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

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 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)

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

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

invelliptic:  (inverse elliptic)

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

$y=\frac{a\sinh \left(u\right)\sin \left(v\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

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)

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

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

maxwell:

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

$y=\frac{a\left(v+{\ⅇ}^u\sin \left(v\right)\right)}{\mathrm{\pi }}$

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 a, b, and c values in the above coordinate transformations can be given using the coordinate specification as a function, e.g., conical(a,b) or ellcylindrical(2). The values a, b, and c if necessary, should be specified.  If not specified, the default values used are a = 1, b = 1/2, and c = 1/3.

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, 1968.