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VectorCalculus Coordinate Systems

 

Description

References

Description

• 

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

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

  

 

  

logarithmic:

  

x=alnu2+v2π

  

y=2aarctanvuπ

  

 

  

logcosh:  (ln cosh)

  

x=alncoshu2sinv2π

  

y=2aarctantanhutanvπ

  

 

  

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

u,v,wx,y,z

  

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

  

 

  

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

  

12u2v2u2+v22

  

 

  

cardioidcylindrical:

  

x=12u2v2u2+v22

  

y=uvu2+v22

  

z=w

  

 

  

cartesian:

  

x=u

  

y=v

  

z=w

  

 

  

casscylindrical:  (Cassinian-oval cylinder)

  

x=12a2ⅇ2u+2ⅇucosv+1+ⅇucosv+1

  

y=12a2ⅇ2u+2ⅇucosv+1ⅇucosv1

  

z=w

  

 

  

conical:

  

x=uvwab

  

y=ub2+v2b2w2a2b2b

  

z=ua2v2a2w2a2b2a

  

 

  

cylindrical:

  

x=ucosv

  

y=usinv

  

z=w

  

 

  

ellcylindrical:  (elliptic cylindrical)

  

x=acoshucosv

  

y=asinhusinv

  

z=w

  

 

  

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

  

 

  

logcylindrical:  (logarithmic cylinder)

  

x=alnu2+v2π

  

y=2aarctanvuπ

  

z=w

  

 

  

logcoshcylindrical:  (ln cosh cylinder)

  

x=alncoshu2sinv2π

  

y=2aarctantanhutanvπ

  

z=w

  

 

  

oblatespheroidal:

  

x=acoshusinvcosw

  

y=acoshusinvsinw

  

z=asinhucosv

  

 

  

paraboloidal:  (Spiegel)

  

x=uvcosw

  

y=uvsinw

  

z=12u212v2

  

 

  

paracylindrical:

  

x=12u212v2

  

y=uv

  

z=w

  

 

  

prolatespheroidal:

  

x=asinhusinvcosw

  

y=asinhusinvsinw

  

z=acoshucosv

  

 

  

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=ucoswsinv

  

y=usinwsinv

  

z=ucosv

  

 

  

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 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=12, and c=13.

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

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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.

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 Book Company, 1968, pp. 126-130.

See Also

VectorCalculus

VectorCalculus[AddCoordinates]

VectorCalculus[GetCoordinateParameters]

VectorCalculus[GetCoordinates]

VectorCalculus[SetCoordinateParameters]

VectorCalculus[SetCoordinates]

 


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