Coordinates - Maple Help

VectorCalculus Coordinate Systems

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

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

 are given by:
 bipolar: (Spiegel)
 $x=\frac{\mathrm{sinh}\left(v\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}$
 $y=\frac{\mathrm{sin}\left(u\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}$
 cardioid:
 $x=\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $y=\frac{uv}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 cartesian:
 $x=u$
 $y=v$
 cassinian:  (Cassinian-oval)
 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2}$
 elliptic:
 $x=\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$
 $y=\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)$
 hyperbolic:
 $x=\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}$
 $y=\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}$
 invcassinian:  (inverse Cassinian-oval)
 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 logarithmic:
 $x=\frac{a\mathrm{ln}\left({u}^{2}+{v}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\frac{v}{u}\right)}{\mathrm{\pi }}$
 logcosh:  (ln cosh)
 $x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}$
 parabolic:
 $x=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}$
 $y=uv$
 polar:
 $x=u\mathrm{cos}\left(v\right)$
 $y=u\mathrm{sin}\left(v\right)$
 rose:
 $x=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}}{\sqrt{{u}^{2}+{v}^{2}}}$
 $y=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}}{\sqrt{{u}^{2}+{v}^{2}}}$
 tangent:
 $x=\frac{u}{{u}^{2}+{v}^{2}}$
 $y=\frac{v}{{u}^{2}+{v}^{2}}$
 • The conversions from the various coordinate systems to cartesian coordinates in 3-space

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

 are given as follows (the author is indicated where applicable):
 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:
 $x=\frac{uv\mathrm{cos}\left(w\right)}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $y=\frac{uv\mathrm{sin}\left(w\right)}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 cardioidcylindrical:
 $x=\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $y=\frac{uv}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $z=w$
 cartesian:
 $x=u$
 $y=v$
 $z=w$
 casscylindrical:  (Cassinian-oval cylinder)
 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2}$
 $z=w$
 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)
 $x=a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$
 $y=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)$
 $z=w$
 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)
 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 $z=w$
 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)
 $x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}$
 $z=w$
 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)
 $x=uv\mathrm{cos}\left(w\right)$
 $y=uv\mathrm{sin}\left(w\right)$
 $z=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}$
 paracylindrical:
 $x=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}$
 $y=uv$
 $z=w$
 prolatespheroidal:
 $x=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{cos}\left(w\right)$
 $y=a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)\mathrm{sin}\left(w\right)$
 $z=a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$
 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:
 $x=u\mathrm{cos}\left(w\right)\mathrm{sin}\left(v\right)$
 $y=u\mathrm{sin}\left(w\right)\mathrm{sin}\left(v\right)$
 $z=u\mathrm{cos}\left(v\right)$
 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 $d=\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)$
 • 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 $a=1$, $b=\frac{1}{2}$, and $c=\frac{1}{3}$.
 • The GetCoordinateParameters command returns an expression sequence containing the current values of a, b, and c.
 • 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.