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:
  1. the direct geodesic problem or first geodesic problem, given,, and, determine and ;
  2. 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 Latitude#Parametric [latitude |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.

    Envelope of geodesics

The geodesics from a particular point if continued past the cut locus form an envelope illustrated in Fig. 15. Here the geodesics for which is a multiple of are shown in light blue. Some geodesic circles are shown in green; these form cusps on the envelope. The cut locus is shown in red. The envelope is the locus of points which are conjugate to ; points on the envelope may be computed by finding the point at which on a geodesic. calls this star-like figure produced by the envelope an astroid.
Outside the astroid two geodesics intersect at each point; thus there are two geodesics between and these points. This corresponds to the situation on the sphere where there are "short" and "long" routes on a great circle between two points. Inside the astroid four geodesics intersect at each point. Four such geodesics are shown in Fig. 16 where the geodesics are numbered in order of increasing length. The two shorter geodesics are stable, i.e.,, so that there is no nearby path connecting the two points which is shorter; the other two are unstable. Only the shortest line has. All the geodesics are tangent to the envelope which is shown in green in the figure.
The astroid is the evolute of the geodesic circles centered at. Likewise, the geodesic circles are involutes of the astroid.

Area of a geodesic polygon

A geodesic polygon is a polygon whose sides are geodesics. It is analogous to a spherical polygon, whose sides are great circles. The area of such a polygon may be found by first computing the area between a geodesic segment and the equator, i.e., the area of the quadrilateral in Fig. 1. Once this area is known, the area of a polygon may be computed by summing the contributions from all the edges of the polygon.
Here an expression for the area of is developed following. The area of any closed region of the ellipsoid is
where is an element of surface area and is the Gaussian curvature. Now the Gauss–Bonnet theorem applied to a geodesic polygon states
where
is the geodesic excess and is the exterior angle at vertex. Multiplying the equation for by, where is the authalic radius, and subtracting this from the equation for gives
where the value of for an ellipsoid has been substituted.
Applying this formula to the quadrilateral, noting that, and performing the integral over gives
where the integral is over the geodesic line. The integral can be expressed as a series valid for small .
The area of a geodesic polygon is given by summing over its edges. This result holds provided that the polygon does not include a pole; if it does, must be added to the sum. If the edges are specified by their vertices, then a convenient expression for the geodesic excess is

Solution of the direct and inverse problems

Solving the geodesic problems entails mapping the geodesic onto the auxiliary sphere and solving the corresponding problem in great-circle navigation.
When solving the "elementary" spherical triangle for in Fig. 5, Napier's rules for quadrantal triangles can be employed,
The mapping of the geodesic involves evaluating the integrals for the distance,, and the longitude,, Eqs. and and these depend on the parameter.
Handling the direct problem is straightforward, because can be determined directly from the given quantities and ; for a sample calculation, see.
In the case of the inverse problem, is given; this cannot be easily related to the equivalent spherical angle because is unknown. Thus, the solution of the problem requires that be found iteratively ; see for details.
In geodetic applications, where is small, the integrals are typically evaluated as a series . For arbitrary, the integrals and can be found by numerical quadrature or by expressing them in terms of elliptic integrals .
provides solutions for the direct and inverse problems; these are based on a series expansion carried out to third order in the flattening and provide an accuracy of about for the WGS84 ellipsoid; however the inverse method fails to converge for nearly antipodal points.
continues the expansions to sixth order which suffices to provide full double precision accuracy for and improves the solution of the inverse problem so that it converges in all cases.

Geodesics on a triaxial ellipsoid

Solving the geodesic problem for an ellipsoid of revolution is mathematically straightforward: because of symmetry, geodesics have a constant of motion, given by Clairaut's relation allowing the problem to be reduced to quadrature. By the early 19th century, there was a complete understanding of the properties of geodesics on an ellipsoid of revolution.
On the other hand, geodesics on a triaxial ellipsoid have no obvious constant of the motion and thus represented a challenging unsolved problem in the first half of the 19th century. In a remarkable paper, discovered a constant of the motion allowing this problem to be reduced to quadrature also.

Triaxial ellipsoid coordinate system

Consider the ellipsoid defined by
where are Cartesian coordinates centered on the ellipsoid and, without loss of generality,.
employed the ellipsoidal coordinates defined by
In the limit, becomes the parametric latitude for an oblate ellipsoid, so the use of the symbol is consistent with the previous sections. However, is different from the spherical longitude defined above.
Grid lines of constant and are given in Fig. 17. These constitute an orthogonal coordinate system: the grid lines intersect at right angles. The principal sections of the ellipsoid, defined by and are shown in red. The third principal section,, is covered by the lines and or. These lines meet at four umbilical points where the principal radii of curvature are equal. Here and in the other figures in this section the parameters of the ellipsoid are, and it is viewed in an orthographic projection from a point above,.
The grid lines of the ellipsoidal coordinates may be interpreted in three
different ways:
  1. They are "lines of curvature" on the ellipsoid: they are parallel to the directions of principal curvature.
  2. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets.
  3. Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points. For example, the lines of constant in Fig. 17 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.

    Jacobi's solution

Jacobi showed that the geodesic equations, expressed in ellipsoidal coordinates, are separable. Here is how he recounted his discovery to his friend and neighbor Bessel,
The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal.
Königsberg, 28th Dec. '38.

The solution given by Jacobi is
As Jacobi notes "a function of the angle equals a function of the angle. These two functions are just Abelian integrals..." Two constants and appear in the solution. Typically is zero if the lower limits of the integrals are taken to be the starting point of the geodesic and the direction of the geodesics is determined by. However, for geodesics that start at an umbilical points, we have and determines the direction at the umbilical point.
The constant may be expressed as
where is the angle the geodesic makes with lines of constant. In the limit, this reduces to, the familiar Clairaut relation. A derivation of Jacobi's result is given by ; he gives the solution found by for general quadratic surfaces.

Survey of triaxial geodesics

On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by,, and.
To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section,, at right angles. Such geodesics are shown in Figs. 18–22, which use the same ellipsoid parameters and the same viewing direction as Fig. 17. In addition, the three principal ellipses are shown in red in each of these figures.
If the starting point is,, and, then and the geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic oscillates north and south of the equator; on each oscillation it completes slightly less than a full circuit around the ellipsoid resulting, in the typical case, in the geodesic filling the area bounded by the two latitude lines. Two examples are given in Figs. 18 and 19. Figure 18 shows practically the same behavior as for an oblate ellipsoid of revolution ; compare to Fig. 9.
However, if the starting point is at a higher latitude the distortions resulting from are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at .
If the starting point is,, and, then and the geodesic encircles the ellipsoid in a "transpolar" sense. The geodesic oscillates east and west of the ellipse ; on each oscillation it completes slightly more than a full circuit around the ellipsoid. In the typical case, this results in the geodesic filling the area bounded by the two longitude lines and.
If, all meridians are geodesics; the effect of causes such geodesics to oscillate east and west.
Two examples are given in Figs. 20 and 21. The constriction of the geodesic near the pole disappears in the limit ; in this case, the ellipsoid becomes a prolate ellipsoid and Fig. 20 would resemble Fig. 10. All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at.
In Figs. 18–21, the geodesics are closed. As noted above, in the typical case, the geodesics are not closed, but fill the area bounded by the limiting lines of latitude or longitude.
If the starting point is, , and , then and the geodesic repeatedly intersects the opposite umbilical point and returns to its starting point. However, on each circuit the angle at which it intersects becomes closer to or so that asymptotically the geodesic lies on the ellipse , as shown in Fig. 22. A single geodesic does not fill an area on the ellipsoid. All tangents to umbilical geodesics touch the confocal hyperbola that intersects the ellipsoid at the umbilic points.
Umbilical geodesics enjoy several interesting properties:
  • Through any point on the ellipsoid, there are two umbilical geodesics.
  • The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic.
  • Whereas the closed geodesics on the ellipses and are stable, the closed geodesic on the ellipse, which goes through all 4 umbilical points, is exponentially unstable. If it is perturbed, it will swing out of the plane and flip around before returning to close to the plane.
If the starting point of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on and the other two on. The cut locus for is the portion of the line between the cusps.

Applications

The direct and inverse geodesic problems no longer play the central role in geodesy that they once did. Instead of solving adjustment of geodetic networks as a two-dimensional problem in spheroidal trigonometry, these problems are now solved by three-dimensional methods.
Nevertheless, terrestrial geodesics still play an important role in several areas:
By the principle of least action, many problems in physics can be formulated as a variational problem similar to that for geodesics. Indeed, the geodesic problem is equivalent to the motion of a particle constrained to move on the surface, but otherwise subject to no forces .
For this reason, geodesics on simple surfaces such as ellipsoids of revolution or triaxial ellipsoids are frequently used as "test cases" for exploring new methods. Examples include:
  • the development of elliptic integrals and elliptic functions ;
  • the development of differential geometry ;
  • methods for solving systems of differential equations by a change of independent variables ;
  • the study of caustics ;
  • investigations into the number and stability of periodic orbits ;
  • in the limit, geodesics on a triaxial ellipsoid reduce to a case of dynamical billiards;
  • extensions to an arbitrary number of dimensions ;
  • geodesic flow on a surface.