Coordinate surfaces of the three-dimensional parabolic coordinates. The red paraboloid corresponds to τ=2, the blue paraboloid corresponds to σ=1, and the yellow half-plane corresponds to φ=-60°. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.0, -1.732, 1.5).
The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the -direction.
Rotation about the symmetry axis of the parabolae produces a set of
confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:
where the parabolae are now aligned with the -axis,
about which the rotation was carried out. Hence, the azimuthal angle is defined
The surfaces of constant form confocal paraboloids
that open upwards (i.e., towards ) whereas the surfaces of constant form confocal paraboloids
that open downwards (i.e., towards ). The foci of all these paraboloids are located at the origin.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 180. LCCN59014456. ASIN B0000CKZX7.
Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 96. LCCN67025285.
Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk.
Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 34–36 (Table 1.08). ISBN978-0-387-18430-2.