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)

bispherical:

  where

cardioidal:

cardioidcylindrical:

casscylindrical:  (Cassinian-oval cylinder)

confocalellip:  (confocal elliptic)

confocalparab:  (confocal parabolic)

conical:

cylindrical:

ellcylindrical:  (elliptic cylindrical)

z = w

ellipsoidal:

hypercylindrical:  (hyperbolic cylinder)

invcasscylindrical:  (inverse Cassinian-oval cylinder)

invellcylindrical:  (inverse elliptic cylinder)

invoblspheroidal:  (inverse oblate spheroidal)

invprospheroidal:  (inverse prolate spheroidal)

logcylindrical:  (logarithmic cylinder)

logcoshcylindrical:  (ln cosh cylinder)

maxwellcylindrical:

oblatespheroidal:

paraboloidal:  (Spiegel)

paraboloidal2:  (Moon)

paracylindrical:

prolatespheroidal:

rectangular:

rosecylindrical:

sixsphere:  (6-sphere)

spherical:

tangentcylindrical:

tangentsphere:

toroidal:

  where

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

are given by:

bipolar:  (Spiegel)

cardioid:

cartesian:

cassinian:  (Cassinian-oval)

elliptic:

hyperbolic:

invcassinian:  (inverse Cassinian-oval)

invelliptic:  (inverse elliptic)

logarithmic:

logcosh:  (ln cosh)

maxwell:

parabolic:

polar:

rose:

tangent:

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.