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Bispherical coordinates are a three-dimensional orthogonalcoordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci and in bipolar coordinates remain points (on the -axis, the axis of rotation) in the bispherical coordinate system.
The scale factors for the bispherical coordinates and are equal
whereas the azimuthal scale factor equals
Thus, the infinitesimal volume element equals
and the Laplacian is given by
Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Parts I and II. New York: McGraw-Hill. pp. 665–666, 1298–1301.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 182. LCCN59014456.
Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 113. ISBN0-86720-293-9.
Moon PH, Spencer DE (1988). "Bispherical Coordinates (η, θ, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 110–112 (Section IV, E4Rx). ISBN0-387-02732-7.