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Toroidal Coordinates

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A system of curvilinear coordinates for which several different notations are commonly used. In this work (u,v,phi) is used, whereas Arfken (1970) uses (xi,eta,phi) and Moon and Spencer (1988) use (eta,theta,psi). The toroidal coordinates are defined by

x=(asinhvcosphi)/(coshv-cosu)
(1)
y=(asinhvsinphi)/(coshv-cosu)
(2)
z=(asinu)/(coshv-cosu),
(3)

where sinhz is the hyperbolic sine and coshz is the hyperbolic cosine. The coordinates satisfy u in [0,2pi), v in [0,infty), and phi in [0,2pi).

Surfaces of constant v are given by the toroids

x^2+y^2+z^2+a^2=2asqrt(x^2+y^2)cothv,
(4)

surfaces of constant u by the spherical bowls

x^2+y^2+(z-acotu)^2=(a^2)/(sin^2u),
(5)

spheres centered at (0,0,acotu) with radii a|cscu|

2azcotu=x^2+y^2+z^2-a^2,
(6)

and surfaces of constant phi by

tanphi=y/x.
(7)

The scale factors are

h_u=a/(coshv-cosu)
(8)
h_v=a/(coshv-cosu)
(9)
h_phi=(asinhv)/(coshv-cosu).
(10)

The Laplacian is

del ^2=(cschv(coshv-cosu)^3)/(a^2)[partial/(partialu)((sinhv)/(coshv-cosu)partial/(partialu))+partial/(partialv)((sinhv)/(coshv-cosu)partial/(partialv))+partial/(partialphi)((cschv)/(coshv-cosu)partial/(partialphi))].
(11)

The Helmholtz differential equation is not separable in toroidal coordinates, but Laplace's equation is.

See also

Bispherical Coordinates, Flat-Ring Cyclide Coordinates, Laplace's Equation--Toroidal Coordinates

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References

Arfken, G. "Toroidal Coordinates (xi, eta, phi)." §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115, 1970.Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 264, 1959.Moon, P. and Spencer, D. E. "Toroidal Coordinates (eta,theta,psi)." Fig. 4.04 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 112-115, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.

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Toroidal Coordinates

Cite this as:

Weisstein, Eric W. "Toroidal Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ToroidalCoordinates.html

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