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the inverse geodesic problem or second geodesic problem, given A and B, determine s 12, α 1, and α 2. As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α 1 for the direct problem and λ 12 = λ 2 − λ 1 for the inverse problem, and its two adjacent sides.
Given the coordinates of the two points (Φ 1, L 1) and (Φ 2, L 2), the inverse problem finds the azimuths α 1, α 2 and the ellipsoidal distance s. Calculate U 1, U 2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges:
The solutions to both problems in plane geometry reduce to simple trigonometry and are valid for small areas on Earth's surface; on a sphere, solutions become significantly more complex as, for example, in the inverse problem, the azimuths differ going between the two end points along the arc of the connecting great circle.
Finding the geodesic between two points on the Earth, the so-called inverse geodetic problem, was the focus of many mathematicians and geodesists over the course of the 18th and 19th centuries with major contributions by Clairaut, [5] Legendre, [6] Bessel, [7] and Helmert English translation of Astron. Nachr. 4, 241–254 (1825). Errata. [8]
The inverse problem for earth sections is: given two points, and on the surface of the reference ellipsoid, find the length, , of the short arc of a spheroid section from to and also find the departure and arrival azimuths (angle from true north) of that curve, and . The figure to the right illustrates the notation used here.
Figure 1. The great circle path between (φ 1, λ 1) and (φ 2, λ 2).. The great circle path may be found using spherical trigonometry; this is the spherical version of the inverse geodetic problem.
Beltrami's theorem, named for Italian mathematician Eugenio Beltrami, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics. It is nontrivial to see that, on any Riemannian manifold of constant curvature, there are smooth coordinates relative to which all nonconstant geodesics appear as straight lines.
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because ...