Conical coordinates

Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius r) and by two families of perpendicular elliptic cones, aligned along the z- and x-axes, respectively.The intersection between one of the cones and the sphere forms a spherical conic.are defined by with the following limitations on the coordinates Surfaces of constant r are spheres of that radius centered on the origin whereas surfaces of constantare mutually perpendicular cones and In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.The scale factor for the radius r is one (hr = 1), as in spherical coordinates.
Coordinate surfaces of the conical coordinates. The constants b and c were chosen as 1 and 2, respectively. The red sphere represents r = 2 , the blue elliptic cone aligned with the vertical z -axis represents μ=cosh(1) and the yellow elliptic cone aligned with the (green) x -axis corresponds to ν 2 = 2/3 . The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (1.26, -0.78, 1.34). The elliptic cones intersect the sphere in spherical conics .
Cartesian coordinatesspherical conicsorthogonalcoordinate systemspherical conicLaplace's equationHelmholtz equationspherical coordinatesMorse PMFeshbach HMargenau HKorn TMOrthogonal coordinate systemsCartesianLog-polarParabolicBipolarEllipticCylindricalSphericalParaboloidalOblate spheroidalProlate spheroidalEllipsoidalElliptic cylindricalToroidalBisphericalBipolar cylindrical6-sphere