Theme: Geodesy

Geodetic problems

Main article: Geodesics on an ellipsoid

In geometric geodesy, two standard problems exist—the first (direct or forward) and the second (inverse or reverse).

First (direct or forward) geodetic problem

Given a point (in terms of its coordinates) and the direction (azimuth) and distance from that point to a second point, determine (the coordinates of) that second point.

Second (inverse or reverse) geodetic problem

Given two points, determine the azimuth and length of the line (straight line, arc or geodesic) that connects them.

In plane geometry (valid for small areas on Earth's surface), the solutions to both problems reduce to simple trigonometry. On a sphere, however, the solution is significantly more complex, because in the inverse problem the azimuths will differ between the two end points of the connecting great circle, arc.

On the ellipsoid of revolution, geodesics may be written in terms of elliptic integrals, which are usually evaluated in terms of a series expansion—see, for example,Vincenty's formulae. In the general case, the solution is called the geodesic for the surface considered. The differential equations for the geodesic can be solved numerically.