3-D Coordinate Systems - Maple 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:

 $\mathrm{bipolarcylindrical}$ $\mathrm{bispherical}$ $\mathrm{cardioidal}$ $\mathrm{cardioidcylindrical}$ $\mathrm{casscylindrical}$ $\mathrm{confocalellip}$ $\mathrm{confocalparab}$ $\mathrm{conical}$ $\mathrm{cylindrical}$ $\mathrm{ellcylindrical}$ $\mathrm{ellipsoidal}$ $\mathrm{hypercylindrical}$ $\mathrm{invcasscylindrical}$ $\mathrm{invellcylindrical}$ $\mathrm{invoblspheroidal}$ $\mathrm{invprospheroidal}$ $\mathrm{logcoshcylindrical}$ $\mathrm{logcylindrical}$ $\mathrm{maxwellcylindrical}$ $\mathrm{oblatespheroidal}$ $\mathrm{paraboloidal}$ $\mathrm{paraboloidal2}$ $\mathrm{paracylindrical}$ $\mathrm{prolatespheroidal}$ $\mathrm{rectangular}$ $\mathrm{rosecylindrical}$ $\mathrm{sixsphere}$ $\mathrm{spherical}$ $\mathrm{tangentcylindrical}$ $\mathrm{tangentsphere}$ $\mathrm{toroidal}$

Conversions

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

$\left(u,v,w\right)\to \left(x,y,z\right)$

 are given as follows:



bipolarcylindrical (Spiegel)

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

bispherical

 $x=\frac{\mathrm{sin}\left(u\right)\mathrm{cos}\left(w\right)}{d}$
 $y=\frac{\mathrm{sin}\left(u\right)\mathrm{sin}\left(w\right)}{d}$
 $z=\frac{\mathrm{sinh}\left(v\right)}{d}$  where $d=\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)$

cardioidal

 $x=\frac{uv\mathrm{cos}\left(w\right)}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $y=\frac{uv\mathrm{sin}\left(w\right)}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $z=\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}$

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)

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

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

 $x=u\mathrm{cos}\left(v\right)$
 $y=u\mathrm{sin}\left(v\right)$
 $z=w$

ellcylindrical (elliptic cylindrical)

 $x=a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$
 $y=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)$
 $z=w$

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)

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

invellcylindrical (inverse elliptic cylinder)

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

invoblspheroidal (inverse oblate spheroidal)

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

invprospheroidal (inverse prolate spheroidal)

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

logcylindrical (logarithmic cylinder)

 $x=\frac{a\mathrm{ln}\left({u}^{2}+{v}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\frac{v}{u}\right)}{\mathrm{\pi }}$
 $z=w$

logcoshcylindrical (ln cosh cylinder)

 $x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}$
 $z=w$

maxwellcylindrical

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

oblatespheroidal

 $x=a\mathrm{cosh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{cos}\left(w\right)$
 $y=a\mathrm{cosh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{sin}\left(w\right)$
 $z=a\mathrm{sinh}\left(u\right)\mathrm{cos}\left(v\right)$

paraboloidal (Spiegel)

 $x=uv\mathrm{cos}\left(w\right)$
 $y=uv\mathrm{sin}\left(w\right)$
 $z=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}$

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

 $x=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{cos}\left(w\right)$
 $y=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{sin}\left(w\right)$
 $z=a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$

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

 $x=u\mathrm{cos}\left(v\right)\mathrm{sin}\left(w\right)$
 $y=u\mathrm{sin}\left(v\right)\mathrm{sin}\left(w\right)$
 $z=u\mathrm{cos}\left(w\right)$

tangentcylindrical

 $x=\frac{u}{{u}^{2}+{v}^{2}}$
 $y=\frac{v}{{u}^{2}+{v}^{2}}$
 $z=w$

tangentsphere

 $x=\frac{u\mathrm{cos}\left(w\right)}{{u}^{2}+{v}^{2}}$
 $y=\frac{u\mathrm{sin}\left(w\right)}{{u}^{2}+{v}^{2}}$
 $z=\frac{v}{{u}^{2}+{v}^{2}}$

toroidal

 $x=\frac{a\mathrm{sinh}\left(v\right)\mathrm{cos}\left(w\right)}{d}$
 $y=\frac{a\mathrm{sinh}\left(v\right)\mathrm{sin}\left(w\right)}{d}$
 $z=\frac{a\mathrm{sin}\left(u\right)}{d}$  where $d=\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)$



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

  Coordinate System:bipolarcylindricalbisphericalcardioidalcardioidcylindricalcasscylindricalconfocalellipconfocalparabconicalcylindricalellcylindricalellipsoidalhypercylindricalinvcasscylindricalinvellcylindricalinvoblspheroidalinvprospheroidallogcoshcylindricallogcylindricalmaxwellcylindricaloblatespheroidalparaboloidalparaboloidal2paracylindricalprolatespheroidalrectangularrosecylindricalsixspheresphericaltangentcylindricaltangentspheretoroidal 



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