3-D Coordinate Systems - Maple Programming Help

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3-D Coordinate Systems

Main Concept

The Cartesian coordinate system is the default 3-D coordinate system used by Maple.

Additionally, Maple supports the following 3-D coordinate systems:

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

toroidal

 

 

 

 

 

Conversions

The conversions from the various coordinate systems to cartesian coordinates in three dimensions

u,v,wx,y,z

  

are given as follows:

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=u2v22u2+v22

cardioidcylindrical

  

x=u2v22u2+v22

  

y=uvu2+v22

  

z=w

casscylindrical (Cassinian-oval cylinder)

  

x=a2ⅇ2u+2ⅇucosv+1+ⅇucosv+12

  

y=a2ⅇ2u+2ⅇucosv+1ⅇucosv12

  

z=w

confocalellip (confocal elliptic)

  

x=a2ua2va2wa2b2a2c2

  

y=b2ub2vb2wa2+b2b2c2

  

z=c2uc2vc2wa2+c2b2+c2

confocalparab (confocal parabolic)

  

x=a2ua2va2wa2+b2

  

y=b2ub2vb2wa2+b2

  

z=a22+b22u2v2w2

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

  

y=a2ⅇ2u+2ⅇucosv+1ⅇucosv12ⅇ2u+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=u22v22

paraboloidal2 (Moon)

  

x=2uaavawab

  

y=2ubbvbwab

  

z=u+v+wab

paracylindrical

  

x=u22v22

  

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

 

Instructions: Adjust the sliders to see how the surface depends on each parameter.

Coordinate System:

 

 

 

 

 

 

 

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