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Prolate spheroidal coordinates μ and ν for a = 1.The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0.The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.
If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M.
Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions". [9] These are of great utility in disciplines such as geodesy, [10] cosmology, [11] or tomography [12]
If the third axis is shorter, the ellipsoid is a sphere that has been flattened (called an oblate spheroid). If the third axis is longer, it is a sphere that has been lengthened (called a prolate spheroid). If the three axes have the same length, the ellipsoid is a sphere.
The unusual cosmic abundance of alpha nuclides has inspired geometric arrangements of alpha particles as a solution to nuclear shapes, although the atomic nucleus generally assumes a prolate spheroid shape. Nuclides can also be discus-shaped (oblate deformation), triaxial (a combination of oblate and prolate deformation) or pear-shaped. [7] [8]
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.
If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called oblate harmonics and with prolate spheroidal coordinates, prolate harmonics.
This sequence is called a 0 th –order Slepian sequence (also known as a discrete prolate spheroidal sequence, or DPSS), which is a unique taper with maximally suppressed sidelobes. It turns out that the number of dominant eigenvalues of the matrix M that are close to 1, corresponds to N=2WT called the Shannon number.