Geodesics on an ellipsoid

Geodesics on an ellipsoid of revolution

There are several ways of defining geodesics (Hilbert & Cohn-Vossen 1952, pp. 220–221). 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 (somewhat more than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points.

By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) 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 (Bomford 1952, Chap. 3) (Leick et al. 2015, §4.5).

It is possible to reduce the various geodesic problems into one of two types. Consider two points: A at latitude φ₁ and longitude λ₁ and B at latitude φ₂ and longitude λ₂ (see Fig. 1). The connecting geodesic (from A to B) is AB, of length s₁₂, which has azimuths α₁ and α₂ at the two endpoints. The two geodesic problems usually considered are:

1. the direct geodesic problem or first geodesic problem, given A, α₁, and s₁₂, determine B and α₂;
2. the inverse geodesic problem or second geodesic problem, given A and B, determine s₁₂, α₁, and α₂.

As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α₁ for the direct problem and λ₁₂ = λ₂ − λ₁ 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. (See the article on great-circle navigation.)

For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by Clairaut (1735). A systematic solution for the paths of geodesics was given by Legendre (1806) and Oriani (1806) (and subsequent papers in 1808 and 1810). The full solution for the direct problem (complete with computational tables and a worked out example) is given by Bessel (1825).

During the 18th century geodesics were typically referred to as "shortest lines". The term "geodesic line" was coined by Laplace (1799b):

Nous désignerons cette ligne sous le nom de ligne géodésique [We will call this line the geodesic line].

This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example (Hutton 1811),

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 (both oblate and prolate). The problem on a triaxial ellipsoid is covered in the next section.

When determining distances on the earth, various approximate methods are frequently used; some of these are described in the article on geographical distance.

Equations for a geodesic

Here the equations for a geodesic are developed; the derivation closely follows that of Bessel (1825). Bagratuni (1962, §15), Gan'shin (1967, Chap. 5), Krakiwsky & Thomson (1974, §4), Rapp (1993, §1.2), and Borre & Strang (2012) also provide derivations of these equations.

Consider an ellipsoid of revolution with equatorial radius a and polar semi-axis b. Define the flattening f = (a − b)/a, the eccentricity e = √a² − b²/a = √f(2 − f), and the second eccentricity e′ = √a² − b²/b = e/(1 − f). (In most applications in geodesy, the ellipsoid is taken to be oblate, a > b; however, the theory applies without change to prolate ellipsoids, a < b, in which case f, e², and e′² are negative.)

Let an elementary segment of a path on the ellipsoid have length ds. From Figs. 2 and 3, we see that if its azimuth is α, then ds is related to dφ and dλ by

where ρ is the meridional radius of curvature, R = ν cosφ is the radius of the circle of latitude φ, and ν is the normal radius of curvature. The elementary segment is therefore given by

or

where φ′ = dφ/dλ and the Lagrangian function L depends on φ through ρ(φ) and R(φ). The length of an arbitrary path between (φ₁, λ₁) and (φ₂, λ₂) is given by

where φ is a function of λ satisfying φ(λ₁) = φ₁ and φ(λ₂) = φ₂. The shortest path or geodesic entails finding that function φ(λ) which minimizes s₁₂. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity,

Substituting for L and using Eqs. (1) gives

Clairaut (1735) found this relation, using a geometrical construction; a similar derivation is presented by Lyusternik (1964, §10). Differentiating this relation and manipulating the result gives (Jekeli 2012, Eq. (2.95))

This, together with Eqs. (1), leads to a system of ordinary differential equations for a geodesic (Jordan & Eggert 1941, §7) (Borre & Strang 2012, Eqs. (11.71) and (11.76))

We can express R in terms of the parametric latitude, β, using

(see Fig. 4 for the geometrical construction), and Clairaut's relation then becomes

This is the sine rule of spherical trigonometry relating two sides of the triangle NAB (see Fig. 5), NA =  ¹⁄₂π − β₁, and NB =  ¹⁄₂π − β₂ and their opposite angles B = π − α₂ and A = α₁.

In order to find the relation for the third side AB = σ₁₂, the spherical arc length, and included angle N = ω₁₂, the spherical longitude, it is useful to consider the triangle NEP representing a geodesic starting at the equator; see Fig. 6. 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 P; E, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for σ, s and ω.

If the side EP is extended by moving P infinitesimally (see Fig. 7), we obtain

Combining Eqs. (1) and (3) gives differential equations for s and λ

Specializing to an ellipsoid, R and Z are related by

where Z is the height above the equator (see Fig. 4). Differentiating this and setting dR/dZ = −sinφ/cosφ gives

eliminating Z from these equations, we obtain

This relation between β and φ can be written as

which is the normal definition of the parametric latitude on an ellipsoid. Furthermore, we have

so that the differential equations for the geodesic become

The last step is to use σ as the independent parameter in both of these differential equations and thereby to express s and λ as integrals. Applying the sine rule to the vertices E and G in the spherical triangle EGP in Fig. 6 gives

where α₀ is the azimuth at E. Substituting this into the equation for ds/dσ and integrating the result gives

where

and the limits on the integral are chosen so that s(σ = 0) = 0. Legendre (1811, p. 180) pointed out that the equation for s is the same as the equation for the arc on an ellipse with semi-axes b√1 + e′² cos²α₀ and b. In order to express the equation for λ in terms of σ, we write

which follows from Eq. (3) and Clairaut's relation. This yields

and the limits on the integrals are chosen so that λ = λ₀ at the equator crossing, σ = 0.

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. However, because the equations for s and λ in terms of the spherical quantities depend on α₀, the mapping is not a consistent mapping of the surface of the sphere to the ellipsoid or vice versa; instead, it is merely a convenient tool for solving for a particular geodesic.

There are also several ways of approximating geodesics on an ellipsoid which usually apply for sufficiently short lines (Rapp 1991, §6); however, these are typically comparable in complexity to the method for the exact solution given above (Jekeli 2012, §2.1.4).

Behavior of geodesics

Fig. 8 shows the simple closed geodesics which consist of the meridians (green) and the equator (red). (Here the qualification "simple" means that the geodesic closes on itself without an intervening self-intersection.) This follows from the equations for the geodesics given in the previous section.

For meridians, we have α₀ = 0 and Eq. (5) becomes λ = ω + λ₀, i.e., the longitude will vary the same way as for a sphere, jumping by π each time the geodesic crosses the pole. The distance, Eq. (4), reduces to the length of an arc of an ellipse with semi-axes a and b (as expected), expressed in terms of parametric latitude, β.

The equator (β = 0 on the auxiliary sphere, φ = 0 on the ellipsoid) corresponds to α₀ =  ¹⁄₂π. The distance reduces to the arc of a circle of radius b, s = bσ, while the longitude simplifies to λ = (1 − f)σ + λ₀. A geodesic that is nearly equatorial will intersect the equator at intervals of πb. As a consequence, the maximum length of a equatorial geodesic which is also a shortest path is πb on an oblate ellipsoid (on a prolate ellipsoid, the maximum length is πa).

All other geodesics are typified by Figs. 9 to 11. Figure 9 shows latitude as a function of longitude for a geodesic starting on the equator with α₀ = 45°. A full cycle of the geodesic, from one northward crossing of the equator to the next, is shown. The equatorial crossings are called nodes and the points of maximum or minimum latitude are called vertices; the vertex latitudes are given by |β| = ±( ¹⁄₂π − |α₀|). The latitude is an odd, resp. even, function of the longitude about the nodes, resp. vertices. The geodesic completes one full oscillation in latitude before the longitude has increased by 7000628318530717960♠360°. Thus, on each successive northward crossing of the equator (see Fig. 10), λ falls short of a full circuit of the equator by approximately 2π f sinα₀ (for a prolate ellipsoid, this quantity is negative and λ completes more that a full circuit; see Fig. 12). For nearly all values of α₀, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. 11).

If the ellipsoid is sufficiently oblate, i.e., b/a <  ¹⁄₂, another class of simple closed geodesics is possible (Klingenberg 1982, §3.5.19). Two such geodesics are illustrated in Figs. 13 and 14. Here b/a =  ²⁄₇ and the equatorial azimuth, α₀, for the green (resp. blue) geodesic is chosen to be 6999928078829747986♠53.175° (resp. 7000131234797115957♠75.192°), so that the geodesic completes 2 (resp. 3) complete oscillations about the equator on one circuit of the ellipsoid.

Evaluation of the integrals

Solving the geodesic problems entails evaluating the integrals for the distance, s, and the longitude, λ, Eqs. (4) and (5). In geodetic applications, where f is small, the integrals are typically evaluated as a series; for this purpose, the second form of the longitude integral is preferred (since it avoids the near singular behavior of the first form when geodesics pass close to a pole). In both integrals, the integrand is an even periodic function of period π. Furthermore, the term dependent on σ is multiplied by a small quantity k² = O(f). As a consequence, the integrals can both be written in the form

where B₀ = 1 + O(f) and B_j = O(f ^j). Series expansions for B_j can readily be found and the result truncated so that only terms which are O(f ^J) and larger are retained. This prescription is followed by many authors (Legendre 1806) (Oriani 1806) (Bessel 1825) (Helmert 1880) (Rainsford 1955) (Rapp 1993). Vincenty (1975a) uses J = 3 which provides an accuracy of about 6996100000000000000♠0.1 mm for the WGS84 ellipsoid. Karney (2013) gives expansions carried out for J = 6 which suffices to provide full double precision accuracy for |f| ≤  ¹⁄₅₀. Trigonometric series of this type can be conveniently summed using Clenshaw summation. In order to solve the direct geodesic problem, it is necessary to find σ given s. Since the integrand in the distance integral is positive, this problem has a unique root, which may be found using series reversion (Oriani 1833) (Helmert 1880) or function iteration (Vincenty 1975a).

For arbitrary f, the integrals (4) and (5) can be found by numerical quadrature (Saito 1970) (Saito 1979) (Sjöberg & Shirazian 2012) or to using numerical techniques for the solution of the ordinary differential equations, Eqs. (2) (Kivioja 1971) (Thomas & Featherstone 2005) (Panou et al. 2013). Alternatively, the integrals can be expressed in terms of elliptic integrals. Legendre (1811) writes the integrals, Eqs. (4) and (5), as

where

and

and F(φ, k), E(φ, k), and Π(φ, α², k), are incomplete elliptic integrals (DLMF 2010, §19.2(ii)). The first formula for the longitude in Eq. (7) follows directly from the first form of Eq. (5). The second formula in Eq. (7), due to Cayley (1870), is more convenient for calculation since the elliptic integral appears in a small term.

Solution of the direct problem

The strategy described by Bessel (1825) and Helmert (1880) for solving the geodesic problems on the ellipsoid is to map the problem onto the auxiliary sphere by converting φ, λ, and s, to β, ω and σ, to solve the corresponding great-circle problem on the sphere, and to transfer the results back to the ellipsoid.

In carrying out this prescription, Napier's rules for quadrantal triangles can be applied to the triangle NEP in Fig. 6 (with P replaced by either A, subscript 1, or B, subscript 2) to give

We can also stipulate that cosβ ≥ 0 and cosα₀ ≥ 0. Implementing this plan for the direct problem is straightforward. We are given φ₁, α₁, and s₁₂. From φ₁ we obtain β₁ (using the formula for the parametric latitude). We now solve the triangle problem with P = A and β₁ and α₁ given to find α₀, σ₁, and ω₁. Use the distance and longitude equations, Eqs. (4) and (5), together with the known value of λ₁, to find s₁ and λ₀. Determine s₂ = s₁ + s₁₂ and invert the distance equation to find σ₂. Solve the triangle problem with P = B and α₀ and σ₂ given to find β₂, ω₂, and α₂. Convert β₂ to φ₂ and substitute σ₂ and ω₂ into the longitude equation to give λ₂.

The overall method follows the procedure for solving the direct problem on a sphere.

Solution of the inverse problem

The direct problem is simple because given φ₁ and α₁, we can immediately find α₀, the parameter appearing in the distance and longitude integrals, Eqs. (4) and (5). In the case of the inverse problem, λ₁₂ is given, but this cannot be easily related to the equivalent spherical angle ω₁₂ because α₀ is unknown. Thus, the solution of the problem requires that α₀ be found iteratively. Before tackling this, it is worth understanding better the behavior of geodesics, this time, keeping the starting point fixed and varying the azimuth.

Suppose point A in the inverse problem has φ₁ = −30° and λ₁ = 0°. Fig. 15 shows geodesics (in blue) emanating A with α₁ a multiple of 6999261799387799150♠15° up to the point at which they cease to be shortest paths. (The flattening has been increased to  ¹⁄₁₀ in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant s₁₂, which are the geodesic circles centered A. Gauss (1828) 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 (two in this case) shortest geodesics from A. On a sphere, the cut locus is a point. On an oblate ellipsoid (shown here), it is a segment of the circle of latitude centered on the point antipodal to A, φ = −φ₁. The longitudinal extent of cut locus is approximately λ₁₂ ∈ [π − f π cosφ₁, π + f π cosφ₁]. If A lies on the equator, φ₁ = 0, this relation is exact and as a consequence the equator is only a shortest geodesic if |λ₁₂| ≤ (1 − f)π. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to A, λ₁₂ = π, and this means that meridional geodesics stop being shortest paths before the antipodal point is reached.

The solution of the inverse problem involves determining, for a given point B with latitude φ₂ and longitude λ₂ which blue and green curves it lies on; this determines α₁ and s₁₂ respectively. In Fig. 16, the ellipsoid has been "rolled out" onto a plate carrée projection. Suppose φ₂ = 20°, the green line in the figure. Then as α₁ is varied between 5000000000000000000♠0° and 7000314159265358980♠180°, the longitude at which the geodesic intersects φ = φ₂ varies between 5000000000000000000♠0° and 7000314159265358980♠180° (see Fig. 17). This behavior holds provided that |φ₂| ≤ |φ₁| (otherwise the geodesic does not reach φ₂ for some values of α₁). Thus, the inverse problem may be solved by determining the value α₁ which results in the given value of λ₁₂ when the geodesic intersects the circle φ = φ₂.

This suggests the following strategy for solving the inverse problem (Karney 2013). Assume that the points A and B satisfy

(There is no loss of generality in this assumption, since the symmetries of the problem can be used to generate any configuration of points from such configurations.)

1. First treat the "easy" cases, geodesics which lie on a meridian or the equator. Otherwise…
2. Guess a value of α₁.
3. Solve the so-called hybrid geodesic problem, given φ₁, φ₂, and α₁ find λ₁₂, s₁₂, and α₂, corresponding to the first intersection of the geodesic with the circle φ = φ₂.
4. Compare the resulting λ₁₂ with the desired value and adjust α₁ until the two values agree. This completes the solution.

Each of these steps requires some discussion.

1. For an oblate ellipsoid, the shortest geodesic lies on a meridian if either point lies on a pole or if λ₁₂ = 0 or ±π. The shortest geodesic follows the equator if φ₁ = φ₂ = 0 and |λ₁₂| ≤ (1 − f)π. For a prolate ellipsoid, the meridian is no longer the shortest geodesic if λ₁₂ = ±π and the points are close to antipodal (this will be discussed in the next section). There is no longitudinal restriction on equatorial geodesics.

2. In most cases a suitable starting value of α₁ is found by solving the spherical inverse problem

with ω₁₂ = λ₁₂. This may be a bad approximation if A and B are nearly antipodal (both the numerator and denominator in the formula above become small); however, this may not matter (depending on how step 4 is handled).

3. The solution of the hybrid geodesic problem is as follows. It starts the same way as the solution of the direct problem, solving the triangle NEP with P = A to find α₀, σ₁, ω₁, and λ₀. Now find α₂ from sinα₂ = sinα₀/cosβ₂, taking cosα₂ ≥ 0 (corresponding to the first, northward, crossing of the circle φ = φ₂). Next, σ₂ is given by tanσ₂ = tanβ₂/cosα₂ and ω₂ by tanω₂ = tanσ₂/sinα₀. Finally, use the distance and longitude equations, Eqs. (4) and (5), to find s₁₂ and λ₁₂.

4. In order to discuss how α₁ is updated, let us define the root-finding problem in more detail. The curve in Fig. 17 shows λ₁₂(α₁; φ₁, φ₂) where we regard φ₁ and φ₂ as parameters and α₁ as the independent variable. We seek the value of α₁ which is the root of

where g(0) ≤ 0 and g(π) ≥ 0. In fact, there is a unique root in the interval α₁ ∈ [0, π]. Any of a number of root-finding algorithms can be used to solve such an equation; Karney (2013) uses Newton's method.

An alternative method for solving the inverse problem is given by Helmert (1880, §5.13). Let us rewrite the Eq. (5) as

Helmert's method entails assuming that ω₁₂ = λ₁₂, solving the resulting problem on auxiliary sphere, and obtaining an updated estimate of ω₁₂ using

This fixed point iteration is repeated until convergence. Rainsford (1955) advocates this method and Vincenty (1975a) adopted it in his solution of the inverse problem. A drawback is that the process fails to converge for nearly antipodal points. In a subsequent report, Vincenty (1975b) attempts to cure this defect; but he is only partially successful—the NGS (2012) implementation which includes Vincenty's fix still fails to converge in some cases.

Differential properties of geodesics

Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments (Ehlert 1993), determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by s the length from the northward equator crossing, and a second geodesic a small distance t(s) away from it. Gauss (1828) showed that t(s) obeys the Gauss-Jacobi equation

where K(s) is the Gaussian curvature at s. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions

where

We shall abbreviate m(s₁, s₂) = m₁₂, the so-called reduced length, and M(s₁, s₂) = M₁₂, the geodesic scale. Their basic definitions are illustrated in Fig. 18. Christoffel (1869) made an extensive study of their properties.

The Gaussian curvature for an ellipsoid of revolution is

Helmert (1880, Eq. (6.5.1.)) solved the Gauss-Jacobi equation for this case obtaining

where

As we see from Fig. 18 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by dα₁ is m₁₂ dα₁. On a closed surface such as an ellipsoid, m₁₂ oscillates about zero. Indeed, if the starting point of a geodesic is a pole, φ₁ =  ¹⁄₂π, then the reduced length is the radius of the circle of latitude, m₁₂ = a cosβ₂ = a sinσ₁₂. Similarly, for a meridional geodesic starting on the equator, φ₁ = α₁ = 0, we have M₁₂ = cosσ₁₂. In the typical case, these quantities oscillate with a period of about 2π in σ₁₂ and grow linearly with distance at a rate proportional to f.

To simplify the discussion of shortest paths in this paragraph we consider only geodesics with s₁₂ > 0. The point at which m₁₂ becomes zero is the point conjugate to the starting point. In order for a geodesic between A and B, of length s₁₂, to be a shortest path it must satisfy the Jacobi condition (Jacobi 1837) (Jacobi 1866, §6) (Forsyth 1927, §§26–27) (Bliss 1916), that there is no point conjugate to A between A and B. If this condition is not satisfied, then there is a nearby path (not necessarily a geodesic) 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:

The latter condition above can be used to determine whether the shortest path is a meridian in the case of a prolate ellipsoid with |λ₁₂| = π. The derivative required to solve the inverse method using Newton's method, ∂λ₁₂(α₁; φ₁, φ₂) / ∂α₁, is given in terms of the reduced length (Karney 2013, Eq. (46)).

Envelope of geodesics

The geodesics from a particular point A if continued past the cut locus form an envelope illustrated in Fig. 19. Here the geodesics for which α₁ is a multiple of 6998523598775598300♠3° are shown in light blue. (The geodesics are only shown for their first passage close to the antipodal point, not for subsequent ones.) 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 A; points on the envelope may be computed by finding the point at which m₁₂ = 0 on a geodesic. Jacobi (1891) 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 (with a length approximately half the circumference of the ellipsoid) between A 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. 20 where the geodesics are numbered in order of increasing length. (This figure uses the same position for A as Fig. 15 and is drawn in the same projection.) The two shorter geodesics are stable, i.e., m₁₂ > 0, so that there is no nearby path connecting the two points which is shorter; the other two are unstable. Only the shortest line (the first one) has σ₁₂ ≤ π. All the geodesics are tangent to the envelope which is shown in green in the figure.

The astroid is the (exterior) evolute of the geodesic circles centered at A. Likewise, the geodesic circles are involutes of the astroid.

Area of a geodesic polygon

A geodesic polygon is a polygon whose sides are geodesics. 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 AFHB in Fig. 1 (Danielsen 1989). 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 we develop the formula for the area S₁₂ of AFHB following Sjöberg (2006). The area of any closed region of the ellipsoid is

where dT is an element of surface area and K is the Gaussian curvature. Now the Gauss–Bonnet theorem applied to a geodesic polygon states

where

is the geodesic excess and θ_j is the exterior angle at vertex j. Multiplying the equation for Γ by R₂², where R₂ is the authalic radius, and subtracting this from the equation for T gives

where the value of K for an ellipsoid has been substituted. Applying this formula to the quadrilateral AFHB, noting that Γ = α₂ − α₁, and performing the integral over φ gives

where the integral is over the geodesic line (so that φ is implicitly a function of λ). Converting this into an integral over σ, we obtain

where

and the notation E₁₂ = α₂ − α₁ is used for the geodesic excess. The integral can be expressed as a series valid for small f (Danielsen 1989) (Karney 2013, §6 and addendum).

The area of a geodesic polygon is given by summing S₁₂ over its edges. This result holds provided that the polygon does not include a pole; if it does 2π R₂² must be added to the sum. If the edges are specified by their vertices, then a convenient expression for E₁₂ is

Geodesics on a triaxial ellipsoid

Solving the geodesic problem for an ellipsoid of revolution is, from the mathematical point of view, relatively simple: because of symmetry, geodesics have a constant of the motion, given by Clairaut's relation allowing the problem to be reduced to quadrature. By the early 19th century (with the work of Legendre, Oriani, Bessel, et al.), there was a complete understanding of the properties of geodesics on an ellipsoid of revolution.

On the other hand, geodesics on a triaxial ellipsoid (with three unequal axes) 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, Jacobi (1839) discovered a constant of the motion allowing this problem to be reduced to quadrature also (Klingenberg 1982, §3.5).

The triaxial coordinate system

Consider the ellipsoid defined by

where (X,Y,Z) are Cartesian coordinates centered on the ellipsoid and, without loss of generality, a ≥ b ≥ c > 0. Jacobi (1866, §§26–27) employed the ellipsoidal latitude and longitude (β, ω) defined by

In the limit b → a, β becomes the parametric latitude for an oblate ellipsoid, so the use of the symbol β is consistent with the previous sections. However, ω is different from the spherical longitude defined above.

Grid lines of constant β (in blue) and ω (in green) are given in Fig. 21. These constitute an orthogonal coordinate system: the grid lines intersect at right angles. The principal sections of the ellipsoid, defined by X = 0 and Z = 0 are shown in red. The third principal section, Y = 0, is covered by the lines β = ±90° and ω = 0° or ±180°. These lines meet at four umbilical points (two of which are visible in this figure) where the principal radii of curvature are equal. Here and in the other figures in this section the parameters of the ellipsoid are a:b:c = 1.01:1:0.8, and it is viewed in an orthographic projection from a point above φ = 40°, λ = 30°.

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 (Monge 1796).
2. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets (Dupin 1813, Part 5).
3. Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points (Hilbert & Cohn-Vossen 1952, p. 188). For example, the lines of constant β in Fig. 21 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.

The element of length on the ellipsoid in ellipsoidal coordinates is given by

and the differential equations for a geodesic are

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 (Jacobi 1839, Letter to 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 (Jacobi 1839) (Jacobi 1866, §28) is

As Jacobi notes "a function of the angle β equals a function of the angle ω. These two functions are just Abelian integrals…" Two constants δ and γ appear in the solution. Typically δ 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 γ. However, for geodesics that start at an umbilical points, we have γ = 0 and δ determines the direction at the umbilical point. The constant γ may be expressed as

where α is the angle the geodesic makes with lines of constant ω. In the limit b → a, this reduces to sinα cosβ = const., the familiar Clairaut relation. A derivation of Jacobi's result is given by Darboux (1894, §§583–584); he gives the solution found by Liouville (1846) for general quadratic surfaces. In this formulation, the distance along the geodesic, s, is found by using

An alternative expression for the distance is

Survey of triaxial geodesics

On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by X = 0, Y = 0, and Z = 0. To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section, Y = 0, at right angles. Such geodesics are shown in Figs. 22–26, which use the same ellipsoid parameters and the same viewing direction as Fig. 21. In addition, the three principal ellipses are shown in red in each of these figures.

If the starting point is β₁ ∈ (−90°, 90°), ω₁ = 0, and α₁ = 90°, then γ > 0 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 that a full circuit around the ellipsoid resulting, in the typical case, in the geodesic filling the area bounded by the two latitude lines β = ±β₁. Two examples are given in Figs. 22 and 23. Figure 22 shows practically the same behavior as for an oblate ellipsoid of revolution (because a ≈ b); compare to Fig. 11. However, if the starting point is at a higher latitude (Fig. 22) the distortions resulting from a ≠ b are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at β = β₁ (Chasles 1846) (Hilbert & Cohn-Vossen 1952, pp. 223–224).

If the starting point is β₁ = 90°, ω₁ ∈ (0°, 180°), and α₁ = 180°, then γ < 0 and the geodesic encircles the ellipsoid in a "transpolar" sense. The geodesic oscillates east and west of the ellipse X = 0; 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 ω = ω₁ and ω = 180° − ω₁. If a = b, all meridians are geodesics; the effect of a ≠ b causes such geodesics to oscillate east and west. Two examples are given in Figs. 24 and 25. The constriction of the geodesic near the pole disappears in the limit b → c; in this case, the ellipsoid becomes a prolate ellipsoid and Fig. 24 would resemble Fig. 12 (rotated on its side). All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at ω = ω₁.

If the starting point is β₁ = 90°, ω₁ = 0° (an umbilical point), and α₁ = 135° (the geodesic leaves the ellipse Y = 0 at right angles), then γ = 0 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 Y = 0 becomes closer to 5000000000000000000♠0° or 7000314159265358980♠180° so that asymptotically the geodesic lies on the ellipse Y = 0 (Hart 1849) (Arnold 1989, p. 265), as shown in Fig. 26. 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 geodesic enjoy several interesting properties.

If the starting point A of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on β = −β₁ and the other two on ω = ω₁ + π {{}}. The cut locus for A is the portion of the line β = −β₁ between the cusps (Itoh & Kiyohara 2004).

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 (Vincenty & Bowring 1978). 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 (Laplace 1799a) (Hilbert & Cohn-Vossen 1952, p. 222). 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: