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Cardioid Coordinates

CardioidCoordinates

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

x=(munu)/((mu^2+nu^2)^2)cospsi
(1)
y=(munu)/((mu^2+nu^2)^2)sinpsi
(2)
z=1/2(nu^2-nu^2)/((mu^2+nu^2)^2)
(3)

with mu,nu>=0 and psi in [0,2pi). Surfaces of constant mu are given by the cardioids of revolution intersecting the positive half of the z-axis

x^2+y^2+z^2=1/(4mu^2)[sqrt(x^2+y^2+z^2)+z],
(4)

surfaces of constant nu by the cardioids of revolution intersecting the negative half of the z-axis

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

and surfaces of constant psi by the half-planes

tanpsi=y/x.
(6)

The metric coefficients are

g_(mumu)=1/((mu^2+nu^2)^3)
(7)
g_(nunu)=1/((mu^2+nu^2)^3)
(8)
g_(psipsi)=(mu^2nu^2)/((mu^2+nu^2)^4).
(9)

See also

Cardioid

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References

Moon, P. and Spencer, D. E. "Cardioid Coordinate (mu,nu,psi)." Fig. 4.02 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 107-109, 1988.

Referenced on Wolfram|Alpha

Cardioid Coordinates

Cite this as:

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

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