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coords - coordinate systems supported in Maple

Description

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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.

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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.

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

u,v,wx,y,z

  

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

  

bipolarcylindrical:  (Spiegel)

  

x=asinhvcoshvcosu

  

y=asinucoshvcosu

  

z=w

  

bispherical:

  

x=sinucoswd

  

y=sinusinwd

  

z=sinhvd  where d=coshvcosu

  

cardioidal:

  

x=uvcoswu2+v22

  

y=uvsinwu2+v22

  

z=12u2v2u2+v22

  

cardioidcylindrical:

  

x=12u2v2u2+v22

  

y=uvu2+v22

  

z=w

  

casscylindrical:  (Cassinian-oval cylinder)

  

x=12a2ⅇ2u+2ⅇucosv+1+ⅇucosv+1

  

y=12a2ⅇ2u+2ⅇucosv+1ⅇucosv1

  

z=w

  

confocalellip:  (confocal elliptic)

  

x=a2ua2va2wa2b2a2c2

  

y=b2ub2vb2wa2+b2b2c2

  

z=c2uc2vc2wa2+c2b2+c2

  

confocalparab:  (confocal parabolic)

  

x=a2ua2va2wa2+b2

  

y=b2ub2vb2wa2+b2

  

z=12a2+12b212u12v12w

  

conical:

  

x=uvwab

  

y=ub2+v2b2w2a2b2b

  

z=ua2v2a2w2a2b2a

  

cylindrical:

  

x=ucosv

  

y=usinv

  

z=w

  

ellcylindrical:  (elliptic cylindrical)

  

x=acoshucosv

  

y=asinhusinv

  

z = w

  

ellipsoidal:

  

x=uvwab

  

y=b2+u2b2+v2b2w2a2b2b

  

z=a2+u2a2v2a2w2a2b2a

  

hypercylindrical:  (hyperbolic cylinder)

  

x=u2+v2+u

  

y=u2+v2u

  

z=w

  

invcasscylindrical:  (inverse Cassinian-oval cylinder)

  

x=12a2ⅇ2u+2ⅇucosv+1+ⅇucosv+1ⅇ2u+2ⅇucosv+1

  

y=12a2ⅇ2u+2ⅇucosv+1ⅇucosv1ⅇ2u+2ⅇucosv+1

  

z=w

  

invellcylindrical:  (inverse elliptic cylinder)

  

x=acoshucosvcoshu2sinv2

  

y=asinhusinvcoshu2sinv2

  

z=w

  

invoblspheroidal:  (inverse oblate spheroidal)

  

x=acoshusinvcoswcoshu2cosv2

  

y=acoshusinvsinwcoshu2cosv2

  

z=asinhucosvcoshu2cosv2

  

invprospheroidal:  (inverse prolate spheroidal)

  

x=asinhusinvcoswcoshu2sinv2

  

y=asinhusinvsinwcoshu2sinv2

  

z=acoshucosvcoshu2sinv2

  

logcylindrical:  (logarithmic cylinder)

  

x=alnu2+v2π

  

y=2aarctanvuπ

  

z=w

  

logcoshcylindrical:  (ln cosh cylinder)

  

x=alncoshu2sinv2π

  

y=2aarctantanhutanvπ

  

z=w

  

maxwellcylindrical:

  

x=au+1+ⅇucosvπ

  

y=av+ⅇusinvπ

  

z=w

  

oblatespheroidal:

  

x=acoshusinvcosw

  

y=acoshusinvsinw

  

z=asinhucosv

  

paraboloidal:  (Spiegel)

  

x=uvcosw

  

y=uvsinw

  

z=12u212v2

  

paraboloidal2:  (Moon)

  

x=2uaavawab

  

y=2ubbvbwab

  

z=u+v+wab

  

paracylindrical:

  

x=12u212v2

  

y=uv

  

z=w

  

prolatespheroidal:

  

x=asinhusinvcosw

  

y=asinhusinvsinw

  

z=acoshucosv

  

rectangular:

  

x=u

  

y=v

  

z=w

  

rosecylindrical:

  

x=u2+v2+uu2+v2

  

y=u2+v2uu2+v2

  

z=w

  

sixsphere:  (6-sphere)

  

x=uu2+v2+w2

  

y=vu2+v2+w2

  

z=wu2+v2+w2

  

spherical:

  

x=ucosvsinw

  

y=usinvsinw

  

z=ucosw

  

tangentcylindrical:

  

x=uu2+v2

  

y=vu2+v2

  

z=w

  

tangentsphere:

  

x=ucoswu2+v2

  

y=usinwu2+v2

  

z=vu2+v2

  

toroidal:

  

x=asinhvcoswd

  

y=asinhvsinwd

  

z=asinud  where d=coshvcosu

• 

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

u,vx,y

  

are given by:

  

bipolar:  (Spiegel)

  

x=sinhvcoshvcosu

  

y=sinucoshvcosu

  

cardioid:

  

x=12u2v2u2+v22

  

y=uvu2+v22

  

cartesian:

  

x=u

  

y=v

  

cassinian:  (Cassinian-oval)

  

x=12a2ⅇ2u+2ⅇucosv+1+ⅇucosv+1

  

y=12a2ⅇ2u+2ⅇucosv+1ⅇucosv1

  

elliptic:

  

x=coshucosv

  

y=sinhusinv

  

hyperbolic:

  

x=u2+v2+u

  

y=u2+v2u

  

invcassinian:  (inverse Cassinian-oval)

  

x=12a2ⅇ2u+2ⅇucosv+1+ⅇucosv+1ⅇ2u+2ⅇucosv+1

  

y=12a2ⅇ2u+2ⅇucosv+1ⅇucosv1ⅇ2u+2ⅇucosv+1

  

invelliptic:  (inverse elliptic)

  

x=acoshucosvcoshu2sinv2

  

y=asinhusinvcoshu2sinv2

  

logarithmic:

  

x=alnu2+v2π

  

y=2aarctanvuπ

  

logcosh:  (ln cosh)

  

x=alncoshu2sinv2π

  

y=2aarctantanhutanvπ

  

maxwell:

  

x=au+1+ⅇucosvπ

  

y=av+ⅇusinvπ

  

parabolic:

  

x=12u212v2

  

y=uv

  

polar:

  

x=ucosv

  

y=usinv

  

rose:

  

x=u2+v2+uu2+v2

  

y=u2+v2uu2+v2

  

tangent:

  

x=uu2+v2

  

y=vu2+v2

• 

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.

See Also

addcoords, plot3d[coords], plot[coords], plots[changecoords], plots[coordplot3d], plots[coordplot]

References

  

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.


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