Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with <a href="/facts/Geodesy/EJH0Te0m">geodesy</a> specifically with the solution of <a href="/facts/Triangulation_network/XDQvlPzl">triangulation networks</a>. The <a href="/facts/Figure_of_the_Earth/z1LvpVgd">figure of the Earth</a> is well approximated by an <a href="/facts/Oblate_ellipsoid/GJyyLxQ7">oblate ellipsoid</a>, a slightly flattened sphere. A <a href="/facts/Geodesic/fbWR6l80">geodesic</a> is the shortest path between two points on a curved surface, analogous to a <a href="/facts/Straight_line/gJqsr4Gj">straight line</a> on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry (Euler 1755).
If the Earth is treated as a <a href="/facts/Sphere/jywYLNM2">sphere</a>, the geodesics are <a href="/facts/Great_circles/x4fq6eNm">great circles</a> (all of which are closed) and the problems reduce to ones in <a href="/facts/Spherical_trigonometry/3G3Fhk2h">spherical trigonometry</a>. However, Newton (1687) showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the <a href="/facts/Equator/CjaxbVaN">equator</a> and the <a href="/facts/Meridian_(geography)/zdbk1YPt">meridians</a> are the only simple closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a <a href="/facts/Triaxial_ellipsoid/pe9vm0Sh">triaxial ellipsoid</a> (with three distinct semi-axes), only three geodesics are closed.

Geodesics on an ellipsoid open-in-new

Geodesics on an ellipsoid