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 a {displaystyle a} in the x-y plane. (Rotation about the other axis produces prolate spheroidal coordinates.) 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 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 a {displaystyle a} in the x-y plane. (Rotation about the other axis produces prolate spheroidal coordinates.) 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 (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 ( μ , ν , φ ) {displaystyle (mu , u ,varphi )} is where μ {displaystyle mu } is a nonnegative real number and the angle ν ∈ [ − π / 2 , π / 2 ] {displaystyle u in left} . The azimuthal angle φ {displaystyle varphi } can fall anywhere on a full circle, between ± π {displaystyle pm pi } . These coordinates are favored over the alternatives below because they are not degenerate; the set of coordinates ( μ , ν , φ ) {displaystyle (mu , u ,varphi )} describes a unique point in Cartesian coordinates ( x , y , z ) {displaystyle (x,y,z)} . The reverse is also true, except on the z {displaystyle z} -axis and the disk in the x y {displaystyle xy} plane inside the focal ring. 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 (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). 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. The (μ, ν, φ) coordinates may be calculated from the Cartesian coordinates (x, y, z) as follows. The azimuthal angle φ is given by the formula