Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.
If the Earth is treated as a sphere, the geodesics are great circles and the problems reduce to ones in spherical trigonometry. However, showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the equator and the meridians 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 triaxial ellipsoid, only three geodesics are closed.

Geodesics on an ellipsoid of revolution

There are several ways of defining geodesics. A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface that they start to return toward the starting point, so that other routes are more direct, and includes paths that intersect or re-trace themselves. Short enough segments of a geodesics are still the shortest route between their endpoints, but geodesics are not necessarily globally minimal. Every globally-shortest path is a geodesic, but not vice versa.
By the end of the 18th century, an ellipsoid of revolution was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry .
It is possible to reduce the various geodesic problems into one of two types. Consider two points: at latitude and longitude and at latitude and longitude . The connecting geodesic is, of length, which has azimuths and at the two endpoints. The two geodesic problems usually considered are:

the direct geodesic problem or first geodesic problem, given,, and, determine and ;
the inverse geodesic problem or second geodesic problem, given and, determine,, and.

As can be seen from Fig. 1, these problems involve solving the triangle given one angle, for the direct problem and for the inverse problem, and its two adjacent sides.
For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle.
For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by. A systematic solution for the paths of geodesics was given by and .
The full solution for the direct problem is given by.
During the 18th century geodesics were typically referred to as "shortest lines".
The term "geodesic line" was coined by :

Nous désignerons cette ligne sous le nom de ligne géodésique .

This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example,

A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the Earth; and it is therefore the proper itinerary measure of the distance between those two points.

In its adoption by other fields geodesic line, frequently shortened to geodesic, was preferred.
This section treats the problem on an ellipsoid of revolution. The problem on a triaxial ellipsoid is covered in the next section.

Equations for a geodesic

Here the equations for a geodesic are developed; the derivation closely follows that of.,,,,,, and also provide derivations of these equations.
Consider an ellipsoid of revolution with equatorial radius and polar semi-axis. Define the flattening, the eccentricity, and the second eccentricity :
Let an elementary segment of a path on the ellipsoid have length. From Figs. 2 and 3, we see that if its azimuth is, then is related to and by
where is the meridional radius of curvature, is the radius of the circle of latitude, and is the normal radius of curvature.
The elementary segment is therefore given by
or
where and the Lagrangian function depends on through and. The length of an arbitrary path between and is given by
where is a function of satisfying and. The shortest path or geodesic entails finding that function which minimizes. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity,
Substituting for and using Eqs. gives
found this relation, using a geometrical construction; a similar derivation is presented by. Differentiating this relation gives
This, together with Eqs., leads to a system of ordinary differential equations for a geodesic
We can express in terms of the parametric latitude,, using
and Clairaut's relation then becomes
This is the sine rule of spherical trigonometry relating two sides of the triangle ,, and and their opposite angles and.
In order to find the relation for the third side, the spherical arc length, and included angle, the spherical longitude, it is useful to consider the triangle representing a geodesic starting at the equator; see Fig. 5. In this figure, the variables referred to the auxiliary sphere are shown with the corresponding quantities for the ellipsoid shown in parentheses.
Quantities without subscripts refer to the arbitrary point ;, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for, and.
If the side is extended by moving infinitesimally, we obtain
Combining Eqs. and gives differential equations for and
The relation between and is
which gives
so that the differential equations for the geodesic become
The last step is to use as the independent parameter in both of these differential equations and thereby to express and as integrals. Applying the sine rule to the vertices and in the spherical triangle in Fig. 5 gives
where is the azimuth at.
Substituting this into the equation for and integrating the result gives
where
and the limits on the integral are chosen so that. pointed out that the equation for is the same as the equation for the arc on an ellipse with semi-axes and. In order to express the equation for in terms of, we write
which follows from and Clairaut's relation.
This yields
and the limits on the integrals are chosen so that at the equator crossing,.
This completes the solution of the path of a geodesic using the auxiliary sphere. By this device a great circle can be mapped exactly to a geodesic on an ellipsoid of revolution.
There are also several ways of approximating geodesics on a terrestrial ellipsoid ; some of these are described in the article on geographical distance. However, these are typically comparable in complexity to the method for the exact solution.

Behavior of geodesics

Fig. 7 shows the simple closed geodesics which consist of the meridians and the equator. This follows from the equations for the geodesics given in the previous section.
All other geodesics are typified by Figs. 8 and 9 which show a geodesic starting on the equator with. The geodesic oscillates about the equator. The equatorial crossings are called nodes and the points of maximum or minimum latitude are called vertices; the parametric latitudes of the vertices are given by. The geodesic completes one full oscillation in latitude before the longitude has increased by. Thus, on each successive northward crossing of the equator, falls short of a full circuit of the equator by approximately . For nearly all values of, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes.
If the ellipsoid is sufficiently oblate, i.e.,, another class of simple closed geodesics is possible. Two such geodesics are illustrated in Figs. 11 and 12. Here and the equatorial azimuth,, for the green geodesic is chosen to be , so that the geodesic completes 2 complete oscillations about the equator on one circuit of the ellipsoid.
Fig. 13 shows geodesics emanating with a multiple of up to the point at which they cease to be shortest paths. Also shown are curves of constant, which are the geodesic circles centered. showed that, on any surface, geodesics and geodesic circle intersect at right angles.
The red line is the cut locus, the locus of points which have multiple shortest geodesics from. On a sphere, the cut locus is a point. On an oblate ellipsoid, it is a segment of the circle of latitude centered on the point antipodal to,. The longitudinal extent of cut locus is approximately. If
lies on the equator,, this relation is exact and as a consequence the equator is only a shortest geodesic if. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to,, and this means that meridional geodesics stop being shortest paths before the antipodal point is reached.

Differential properties of geodesics

Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments, determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by, and a second geodesic a small distance away from it. showed that obeys the Gauss-Jacobi equation
where is the Gaussian curvature at. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions
where
The quantity is the so-called reduced length, and is the geodesic scale.
Their basic definitions are illustrated in Fig. 14.
The Gaussian curvature for an ellipsoid of revolution is
solved the Gauss-Jacobi equation for this case enabling and to be expressed as integrals.
As we see from Fig. 14, the separation of two geodesics starting at the same point with azimuths differing by is. On a closed surface such as an ellipsoid, oscillates about zero. The point at which becomes zero is the point conjugate to the starting point. In order for a geodesic between and, of length, to be a shortest path it must satisfy the Jacobi condition , that there is no point conjugate to between and. If this condition is not satisfied, then there is a nearby path which is shorter. Thus, the Jacobi condition is a local property of the geodesic and is only a necessary condition for the geodesic being a global shortest path. Necessary and sufficient conditions for a geodesic being the shortest path are:

for an oblate ellipsoid, ;
for a prolate ellipsoid,, if ; if, the supplemental condition is required if.