VectorCalculus パッケージは、以下の座標系をサポートしています:

2 次元 - bipolar, cardioid, cassinian, cartesian, elliptic, hyperbolic, invcassinian, logarithmic, logcosh, parabolic, polar, rose, tangent 

3 次元 - bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, cartesian, casscylindrical, conical, cylindrical, ellcylindrical, hypercylindrical, invcasscylindrical, invprospheroidal, logcoshcylindrical, logcylindrical, oblatespheroidal, paraboloidal, paracylindrical, prolatespheroidal, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, toroidal 

注意: 以下の変換では、正の根だけが使用されています:

2 次元 - cassinian, hyperbolic, invcassinian, rose

3 次元 - casscylindrical, conical, hypercylindrical, invcasscylindrical, rosecylindrical

2 空間における、様々な座標系からデカルト(直交)座標系への変換

    (u, v) --> (x, y)

は、以下のように与えられます:

 bipolar:                    (Spiegel)
   x = sinh(v)/(cosh(v)-cos(u))
   y = sin(u)/(cosh(v)-cos(u))

 cardioid:
   x = 1/2*(u^2-v^2)/(u^2+v^2)^2
   y = u*v/(u^2+v^2)^2

 cartesian:
   x = u
   y = v

 cassinian:                  (Cassinian-oval)
   x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +
          exp(u)*cos(v)+1)^(1/2)
   y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) -
          exp(u)*cos(v)-1)^(1/2)]

 elliptic:
   x = cosh(u)*cos(v)
   y = sinh(u)*sin(v)

 hyperbolic:
   x = ((u^2+v^2)^(1/2)+u)^(1/2)
   y = ((u^2+v^2)^(1/2)-u)^(1/2)

 invcassinian:               (inverse Cassinian-oval)
   x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +
          exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)
   y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) -
          exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)

 logarithmic:
   x = a/Pi*ln(u^2+v^2)
   y = 2*a/Pi*arctan(v/u)

 logcosh:                    (ln cosh)
   x = a/Pi*ln(cosh(u)^2-sin(v)^2)
   y = 2*a/Pi*arctan(tanh(u)*tan(v))

 parabolic:
   x = (u^2-v^2)/2
   y = u*v

 polar:
   x = u*cos(v)
   y = u*sin(v)

 rose:
   x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2)
   y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2)

 tangent:
   x = u/(u^2+v^2)
   y = v/(u^2+v^2)

3空間における、様々な座標系からデカルト座標系への変換

    (u, v, w) --> (x, y, z)

は、以下のように与えられます（どこで応用可能か、作者が示されています):

 bipolarcylindrical:         (Spiegel)
   x = a*sinh(v)/(cosh(v)-cos(u))
   y = a*sin(u)/(cosh(v)-cos(u))
   z = w

 bispherical:
   x = sin(u)*cos(w)/d
   y = sin(u)*sin(w)/d
   z = sinh(v)/d             where d = cosh(v) - cos(u)

 cardioidal:
   x = u*v*cos(w)/(u^2+v^2)^2
   y = u*v*sin(w)/(u^2+v^2)^2
   z = (u^2-v^2)/2/(u^2+v^2)^2

 cardioidcylindrical:
   x = (u^2-v^2)/2/(u^2+v^2)^2
   y = u*v/(u^2+v^2)^2
   z = w

 cartesian:
   x = u
   y = v
   z = w

 casscylindrical:            (Cassinian-oval cylinder)
   x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)+exp(u)*cos(v)+1)^(1/2)
   y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)-exp(u)*cos(v)-1)^(1/2)
   z = w

 conical:
   x = u*v*w/(a*b)
   y = u/b*((v^2 - b^2)*(b^2-w^2)/(a^2-b^2))^(1/2)
   z = u/a*((a^2 - v^2)*(a^2 - w^2)/(a^2-b^2))^(1/2)

 cylindrical:
   x = u*cos(v)
   y = u*sin(v)
   z = w

 ellcylindrical:             (elliptic cylindrical)
   x = a*cosh(u)*cos(v)
   y = a*sinh(u)*sin(v)
   z = w

 hypercylindrical:           (hyperbolic cylinder)
   x = ((u^2+v^2)^(1/2)+u)^(1/2)
   y = ((u^2+v^2)^(1/2)-u)^(1/2)
   z = w

 invcasscylindrical:         (inverse Cassinian-oval cylinder)
   x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +
          exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)
   y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) -
          exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)
   z = w

 invprospheroidal:           (inverse prolate spheroidal)
   x = a*sinh(u)*sin(v)*cos(w)/(cosh(u)^2-sin(v)^2)
   y = a*sinh(u)*sin(v)*sin(w)/(cosh(u)^2-sin(v)^2)
   z = a*cosh(u)*cos(v)/(cosh(u)^2-sin(v)^2)

 logcylindrical:             (logarithmic cylinder)
   x = a/Pi*ln(u^2+v^2)
   y = 2*a/Pi*arctan(v/u)
   z = w

 logcoshcylindrical:         (ln cosh cylinder)
   x = a/Pi*ln(cosh(u)^2-sin(v)^2)
   y = 2*a/Pi*arctan(tanh(u)*tan(v))
   z = w

 oblatespheroidal:
   x = a*cosh(u)*sin(v)*cos(w)
   y = a*cosh(u)*sin(v)*sin(w)
   z = a*sinh(u)*cos(v)

 paraboloidal:               (Spiegel)
   x = u*v*cos(w)
   y = u*v*sin(w)
   z = (u^2 - v^2)/2

 paracylindrical:
   x = (u^2 - v^2)/2
   y = u*v
   z = w

 prolatespheroidal:
   x = a*sinh(u)*sin(v)*cos(w)
   y=a*sinh(u)*sin(v)*sin(w)
   z=a*cosh(u)*cos(v)

 rosecylindrical:
   x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2)
   y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2)
   z = w

 sixsphere:                  (6-sphere)
   x = u/(u^2+v^2+w^2)
   y = v/(u^2+v^2+w^2)
   z = w/(u^2+v^2+w^2)

 spherical:
   x = u*cos(w)*sin(v)
   y = u*sin(w)*sin(v)
   z = u*cos(v)

 tangentcylindrical:
   x = u/(u^2+v^2)
   y = v/(u^2+v^2)
   z = w

 tangentsphere:
   x = u*cos(w)/(u^2+v^2)
   y = u*sin(w)/(u^2+v^2)
   z = v/(u^2+v^2)

 toroidal:
   x = a*sinh(v)*cos(w)/d
   y = a*sinh(v)*sin(w)/d
   z = a*sin(u)/d            where d = cosh(v) - cos(u)

VectorCalculus パッケージの GetCoordinateParameters および SetCoordinateParameters コマンドを用いることで、上記の座標変換における a, b, c の値を質問し、そして設定することが可能です。デフォルトの値は、a = 1, b = 1/2, c = 1/3 です。

GetCoordinateParameters コマンドは、現在の a, b, c の値を含む式列を返します。

SetCoordinateParameters コマンドは、1, 2, 3 個の引数いずれかをとり、a の値、a と b の値、a, b および c の値をそれぞれ設定します。

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.