Oblate spheroidal coordinates

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci.

(Rotation about the other axis produces prolate spheroidal coordinates.)

Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid of revolution.

For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Prize in Physics to Jean Baptiste Perrin.

These friction factors determine the rotational diffusion of molecules, which affects the feasibility of many techniques such as protein NMR and from which the hydrodynamic volume and shape of molecules can be inferred.

Oblate spheroidal coordinates are also useful in problems of electromagnetism (e.g., dielectric constant of charged oblate molecules), acoustics (e.g., scattering of sound through a circular hole), fluid dynamics (e.g., the flow of water through a firehose nozzle) and the diffusion of materials and heat (e.g., cooling of a red-hot coin in a water bath) The most common definition of oblate spheroidal coordinates

describes a unique point in Cartesian coordinates

The surfaces of constant μ form oblate spheroids, by the trigonometric identity

An ellipse in the x-z plane (Figure 2) has a major semiaxis of length a cosh μ along the x-axis, whereas its minor semiaxis has length a sinh μ along the z-axis.

The foci of all the ellipses in the x-z plane are located on the x-axis at ±a.

Similarly, the surfaces of constant ν form one-sheet half hyperboloids of revolution by the hyperbolic trigonometric identity

For positive ν, the half-hyperboloid is above the x-y plane (i.e., has positive z) whereas for negative ν, the half-hyperboloid is below the x-y plane (i.e., has negative z).

and its distances to the foci in the plane defined by φ is given by

The remaining coordinates μ and ν can be calculated from the equations

The scale factors for the coordinates μ and ν are equal

can be expressed in the coordinates (μ, ν, φ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

+ sinh ⁡ μ cos ⁡ ν sin ⁡ ϕ

− cosh ⁡ μ sin ⁡ ν cos ⁡ ϕ

is the outward normal vector to the oblate spheroidal surface of constant

lies in the tangent plane to the oblate spheroid surface and completes the right-handed basis set.

As is the case with spherical coordinates and spherical harmonics, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate.

Following the technique of separation of variables, a solution to Laplace's equation is written:

This yields three separate differential equations in each of the variables:

An alternative and geometrically intuitive set of oblate spheroidal coordinates (σ, τ, φ) are sometimes used, where σ = cosh μ and τ = cos ν.

[1] Therefore, the coordinate σ must be greater than or equal to one, whereas τ must lie between ±1, inclusive.

Similar to its counterpart μ, the surfaces of constant σ form oblate spheroids

Similarly, the surfaces of constant τ form full one-sheet hyperboloids of revolution

The scale factors for the alternative oblate spheroidal coordinates

by substituting the scale factors into the general formulae found in orthogonal coordinates.

As is the case with spherical coordinates, Laplaces equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate (See Smythe, 1968).

Figure 1: Coordinate isosurfaces for a point P (shown as a black sphere) in oblate spheroidal coordinates ( μ , ν , φ ) . The z -axis is vertical, and the foci are at ±2 . The red oblate spheroid (flattened sphere) corresponds to μ = 1 , whereas the blue half-hyperboloid corresponds to ν = 45° . The azimuth φ = −60° measures the dihedral angle between the green xz half-plane and the yellow half-plane that includes the point P . The Cartesian coordinates of P are roughly (1.09, −1.89, 1.66) .
Figure 2: Plot of the oblate spheroidal coordinates μ and ν in the x - z plane, where φ is zero and a equals one. The curves of constant μ form red ellipses, whereas those of constant ν form cyan half-hyperbolae in this plane. The z -axis runs vertically and separates the foci; the coordinates z and ν always have the same sign. The surfaces of constant μ and ν in three dimensions are obtained by rotation about the z -axis, and are the red and blue surfaces, respectively, in Figure 1.
Figure 3: Coordinate isosurfaces for a point P (shown as a black sphere) in the alternative oblate spheroidal coordinates (σ, τ, φ). As before, the oblate spheroid corresponding to σ is shown in red, and φ measures the azimuthal angle between the green and yellow half-planes. However, the surface of constant τ is a full one-sheet hyperboloid, shown in blue. This produces a two-fold degeneracy, shown by the two black spheres located at ( x , y , ± z ).