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. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.
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, acoustics, fluid dynamics and the diffusion of materials and heat

Definition (µ,ν,φ)

The most common definition of oblate spheroidal coordinates is
where is a nonnegative real number and the angle. The azimuthal angle can fall anywhere on a full circle, between. These coordinates are favored over the alternatives below because they are not degenerate; the set of coordinates describes a unique point in Cartesian coordinates. The reverse is also true, except on the -axis and the disk in the plane inside the focal ring.

Coordinate surfaces

The surfaces of constant μ form oblate spheroids, by the trigonometric identity
since they are ellipses rotated about the z-axis, which separates their foci. An ellipse in the x-z plane 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 whereas for negative ν, the half-hyperboloid is below the x-y plane. Geometrically, the angle ν corresponds to the angle of the asymptotes of the hyperbola. The foci of all the hyperbolae are likewise located on the x-axis at ±a.

Inverse transformation

The coordinates may be calculated from the Cartesian coordinates as follows. The azimuthal angle φ is given by the formula
The cylindrical radius ρ of the point P is given by
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
where the sign of μ is always non-negative, and the sign of ν is the same as that of z.
Another method to compute the inverse transform is
where

Scale factors

The scale factors for the coordinates μ and ν are equal
whereas the azimuthal scale factor equals
Consequently, an infinitesimal volume element equals
and the Laplacian can be written
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.

Basis Vectors

The orthonormal basis vectors for the coordinate system can be expressed in Cartesian coordinates as
where are the Cartesian unit vectors. Here, is the outward normal vector to the oblate spheroidal surface of constant, is the same azimuthal unit vector from spherical coordinates, and lies in the tangent plane to the oblate spheroid surface and completes the right-handed basis set.

Definition (ζ, ξ, φ)

Another set of oblate spheroidal coordinates are sometimes used where and . The curves of constant are oblate spheroids and the curves of constant are the hyperboloids of revolution. The coordinate is restricted by and is restricted by.
The relationship to Cartesian coordinates is

Scale factors

The scale factors for are:
Knowing the scale factors, various functions of the coordinates can be calculated by the general method outlined in the orthogonal coordinates article. The infinitesimal volume element is:
The gradient is:
The divergence is:
and the Laplacian equals

Oblate spheroidal harmonics

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:
where m is a constant which is an integer because the φ variable is periodic with period 2π. n will then be an integer. The solution to these equations are:
where the are constants and and are associated Legendre polynomials of the first and second kind respectively. The product of the three solutions is called an oblate spheroidal harmonic and the general solution to Laplace's equation is written:
The constants will combine to yield only four independent constants for each harmonic.

Definition (σ, τ, φ)

An alternative and geometrically intuitive set of oblate spheroidal coordinates are sometimes used, where σ = cosh μ and τ = cos ν. Therefore, the coordinate σ must be greater than or equal to one, whereas τ must lie between ±1, inclusive. The surfaces of constant σ are oblate spheroids, as were those of constant μ, whereas the curves of constant τ are full hyperboloids of revolution, including the half-hyperboloids corresponding to ±ν. Thus, these coordinates are degenerate; two points in Cartesian coordinates map to one set of coordinates. This two-fold degeneracy in the sign of z is evident from the equations transforming from oblate spheroidal coordinates to the Cartesian coordinates
The coordinates and have a simple relation to the distances to the focal ring. For any point, the sum of its distances to the focal ring equals, whereas their difference equals. Thus, the "far" distance to the focal ring is, whereas the "near" distance is.

Coordinate surfaces

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

Scale factors

The scale factors for the alternative oblate spheroidal coordinates are
whereas the azimuthal scale factor is.
Hence, the infinitesimal volume element can be written
and the Laplacian equals
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.
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.

No angles convention