However, the inverse problem is intrinsically more complex. The direct or forward geodesic problem is much simpler than the inverse or backward problem because the equatorial azimuth \(\alpha0\) can be determined directly from the given quantities \(\phi1\) and \(\alpha1\). The input data includes the latitudes for the two points \(\phi1\) and \(\phi2\), and their longitudinal difference \(\lambda12\). On the contrary, the inverse geodesic problem deals with finding the azimuths for the two points \(\alpha1\) and \(\alpha2\), and the distance between them \(s12\). The information we are provided for solving this is the latitude \(\phi1\) and azimuth \(\alpha1\) at the first point, and the distance between the two points \(s12\). Our goal in the direct geodesic problem is to find the longitudinal difference \(\lambda12\), the latitude \(\phi2\), and the azimuth \(\alpha2\) for the second point. Geodesics algorithms are employed for solving the direct and inverse geodetic problems. The algorithms described below model the Earth as an ellipsoid, which is obtained by deforming a sphere by means of directional scalings, or more generally, of an affine transformation. A solution is then proposed following the paper Algorithm for geodesics, by Charles Karney, the implementation for which is provided in the GeographicLib library. This post provides a general description of geodesic algorithms and shows their inaccuracy for nearly antipodal points in the Boost Geometry library. The software can handle this either by doing an error analysis check and providing specific values through which the inaccuracy can be identified, or by choosing a different method altogether. This can have major implications in applications which rely on accurate results, such as flight navigation systems. In case of Vincenty's formulae, the solution fails to converge, or provides inaccurate results. If a line is drawn between these two points, it passes through the center of the sphere and forms its diameter.Ĭomputing the great circle distance between these two points is often a corner case for most geodesic computations, and the distance is either overestimated or underestimated. Nearly antipodal points or antipodes refer to the most geographically distant points on a sphere, that is, the points are diametrically opposite to each other.
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