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. (Rotation about the other axis produces
prolate spheroidal coordinates
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are locat ...
.) 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
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
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)
Definition (µ,ν,φ)
The most common definition of oblate spheroidal coordinates
is
where
is a nonnegative real number and the angle