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.
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