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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.
Tables of numerical values of oblate spheroidal wave functions are given in Flammer, [4] Hanish et al., [16] [17] [18] and Van Buren et al. [19] Asymptotic expansions of angular oblate spheroidal wave functions for large values of have been derived by Müller., [20] also similarly for prolate spheroidal wave functions. [21]
The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis separating the foci.
The planet Jupiter is a slight oblate spheroid with a flattening of 0.06487. The oblate spheroid is the approximate shape of rotating planets and other celestial bodies, including Earth, Saturn, Jupiter, and the quickly spinning star Altair. Saturn is the most oblate planet in the Solar System, with a flattening of 0.09796. [5]
They are based on the assumption that the figure of the Earth is an oblate spheroid, and hence are more accurate than methods that assume a spherical Earth, such as great-circle distance. The first (direct) method computes the location of a point that is a given distance and azimuth (direction) from another point.
Better approximations can be made by modeling the entire surface as an oblate spheroid, using spherical harmonics to approximate the geoid, or modeling a region with a best-fit reference ellipsoid. For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature.
The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry .