Editing Geodesy (section)

== Geodetic problems {{anchor|Problems}} ==
{{further|Geodesics on an ellipsoid#Solution of the direct and inverse problems}}
{{see also|Bearing (navigation)#Arcs}}
{{unsourced section|date=February 2024}}

In geometrical geodesy, there are two main problems:

* '''First geodetic problem''' (also known as ''direct'' or ''forward geodetic problem''): given the coordinates of a point and the directional ([[azimuth]]) and [[distance]] to a second point, determine the coordinates of that second point.
* '''Second geodetic problem''' (also known as ''inverse'' or ''reverse geodetic problem''): given the coordinates of two points, determine the azimuth and length of the (straight, curved, or [[geodesic]]) line connecting those points.

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]].

The general solution is called the [[geodesic]] for the surface considered, and the [[differential equation]]s for the [[geodesic]] are solvable numerically. On the ellipsoid of revolution, geodesics are expressible in terms of elliptic integrals, which are usually evaluated in terms of a series expansion — see, for example, [[Vincenty's formulae]].