coords - Maple Help

coords

coordinate systems supported in Maple

Description

 • 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.
 • 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.
 • 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 (note that the author is indicated where necessary):
 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 $d=\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)$
 • 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)
 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 invelliptic:  (inverse elliptic)
 $x=\frac{a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}$
 $y=\frac{a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}$
 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)
 $x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}$
 maxwell:
 $x=\frac{a\left(u+1+{ⅇ}^{u}\mathrm{cos}\left(v\right)\right)}{\mathrm{\pi }}$
 $y=\frac{a\left(v+{ⅇ}^{u}\mathrm{sin}\left(v\right)\right)}{\mathrm{\pi }}$
 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 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.

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.