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Tangent-Sphere Coordinates

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Tangent-SphereCoordinates

A coordinate system (mu,nu,psi) given by the coordinate transformation

x=(mucospsi)/(mu^2+nu^2)
(1)
y=(musinpsi)/(mu^2+nu^2)
(2)
z=nu/(mu^2+nu^2)
(3)

and defined for mu>0, nu in (-infty,infty), and psi in [0,2pi). Surfaces of constant mu are given by the toroids

x^2+y^2+z^2=1/musqrt(x^2+y^2),
(4)

surface of constant nu by the spheres tangent to the xy-plane

x^2+y^2+(z-1/(2nu))^2=1/(4nu^2),
(5)

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(6)

The metric coefficients are

g_(xx)=1/((mu^2+nu^2)^2)
(7)
g_(yy)=1/((mu^2+nu^2)^2)
(8)
g_(zz)=(mu^2)/((mu^2+nu^2)^2).
(9)

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References

Moon, P. and Spencer, D. E. "Tangent-Sphere Coordinate (mu,nu,psi)." Fig. 4.01 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 104-106, 1988.

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Tangent-Sphere Coordinates

Cite this as:

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

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