Geodesics on an ellipsoid

A geodesic on an oblate 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 Template:Harv.

If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Template:Harvtxt 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 (with three distinct semi-axes), only three geodesics are closed.

Geodesics on an ellipsoid of revolution

There are several ways of defining geodesics Template:Harv. 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 (i.e. shortest among all possible paths). Every globally-shortest path is a geodesic, but not vice versa.

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 Template:Harv Template:Harv.

Fig. 1. A geodesic AB on an ellipsoid of revolution. N is the north pole and EFH lie on the equator.

It is possible to reduce the various geodesic problems into one of two types. Consider two points: Template:Math at latitude Template:Math and longitude Template:Math and Template:Math at latitude Template:Math and longitude Template:Math (see Fig. 1). The connecting geodesic (from Template:Math to Template:Math) is Template:Math, of length Template:Math, which has azimuths Template:Math and Template:Math at the two endpoints.Template:Refn The two geodesic problems usually considered are:

1. the direct geodesic problem or first geodesic problem, given Template:Math, Template:Math, and Template:Math, determine Template:Math and Template:Math;
2. the inverse geodesic problem or second geodesic problem, given Template:Math and Template:Math, determine Template:Math, Template:Math, and Template:Math.

As can be seen from Fig. 1, these problems involve solving the triangle Template:Math given one angle, Template:Math for the direct problem and Template:Math 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 Template:Harvtxt. A systematic solution for the paths of geodesics was given by Template:Harvtxt and Template:Harvtxt (and subsequent papers in [[#Template:Harvid|1808]] and [[#Template:Harvid|1810]]). The full solution for the direct problem (complete with computational tables and a worked out example) is given by Template:Harvtxt.

During the 18th century geodesics were typically referred to as "shortest lines". The term "geodesic line" (actually, a curve) was coined by Template:Harvtxt:

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 Template:Harv,

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.

Equations for a geodesic

Here the equations for a geodesic are developed; the derivation closely follows that of Template:Harvtxt. Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, and Template:Harvtxt also provide derivations of these equations.

Consider an ellipsoid of revolution with equatorial radius Template:Math and polar semi-axis Template:Math. Define the flattening Template:Math, the eccentricity Template:Math, and the second eccentricity Template:Math:

$f=\frac{a-b}{a},e=\frac{\sqrt{a^2-b^2}}{a}=\sqrt{f(2-f)},e^{\prime }=\frac{\sqrt{a^2-b^2}}{b}=\frac{e}{1-f}.$

(In most applications in geodesy, the ellipsoid is taken to be oblate, Template:Math; however, the theory applies without change to prolate ellipsoids, Template:Math, in which case Template:Math, Template:Math, and Template:Math are negative.)

Let an elementary segment of a path on the ellipsoid have length Template:Math. From Figs. 2 and 3, we see that if its azimuth is Template:Mvar, then Template:Math is related to Template:Math and Template:Math by

$\cos \alpha ds=\rho d\phi =-\frac{dR}{\sin \phi },\sin \alpha ds=Rd\lambda ,$ Template:EquationRef

where Template:Mvar is the meridional radius of curvature, Template:Math is the radius of the circle of latitude Template:Mvar, and Template:Mvar is the normal radius of curvature. The elementary segment is therefore given by

$d{s}^2={\rho }^{2}d{\phi }^2+R^{2}d{\lambda }^2$

or

$\begin{array}{cc}ds& =\sqrt{{\rho }^2\phi {\text{'}}^2+R^2}d\lambda \\ & \equiv L(\phi ,{\phi }^{\prime })d\lambda ,\end{array}$

where Template:Math and the [[Lagrangian mechanics#Euler–Lagrange equations and Hamilton's principle|Lagrangian function Template:Math]] depends on Template:Mvar through Template:Math and Template:Math. The length of an arbitrary path between Template:Math and Template:Math is given by

$s_{12}=\underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}L(\phi ,{\phi }^{\prime })d\lambda ,$

where Template:Mvar is a function of Template:Mvar satisfying Template:Math and Template:Math. The shortest path or geodesic entails finding that function Template:Math which minimizes Template:Math. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity,

$L-{\phi }^{\prime }\frac{\partial L}{\partial {\phi }^{\prime }}=\text{const.}$

Substituting for Template:Math and using Eqs. Template:EquationNote gives

$R\sin \alpha =\text{const.}$

Template:Harvtxt found this relation, using a geometrical construction; a similar derivation is presented by Template:Harvtxt.Template:Refn Differentiating this relation gives

$d\alpha =\sin \phi d\lambda .$

This, together with Eqs. Template:EquationNote, leads to a system of ordinary differential equations for a geodesic

$\frac{d\phi }{ds}=\frac{\cos \alpha }{\rho };\frac{d\lambda }{ds}=\frac{\sin \alpha }{\nu \cos \phi };\frac{d\alpha }{ds}=\frac{\tan \phi \sin \alpha }{\nu }.$

We can express Template:Math in terms of the parametric latitude, Template:Mvar, using

$R=a\cos \beta ,$

and Clairaut's relation then becomes

$\sin {\alpha }_1\cos {\beta }_1=\sin {\alpha }_2\cos {\beta }_2.$

Template:Multiple image This is the sine rule of spherical trigonometry relating two sides of the triangle Template:Math (see Fig. 4), Template:Math, and Template:Math and their opposite angles Template:Math and Template:Math.

In order to find the relation for the third side Template:Math, the spherical arc length, and included angle Template:Math, the spherical longitude, it is useful to consider the triangle Template:Math 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 Template:Math; Template:Math, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for Template:Mvar, Template:Mvar and Template:Mvar.

Fig. 6. Differential element of a geodesic on a sphere.

If the side Template:Math is extended by moving Template:Math infinitesimally (see Fig. 6), we obtain

$\cos \alpha d\sigma =d\beta ,\sin \alpha d\sigma =\cos \beta d\omega .$ Template:EquationRef

Combining Eqs. Template:EquationNote and Template:EquationNote gives differential equations for Template:Math and Template:Mvar

$\frac{1}{a}\frac{ds}{d\sigma }=\frac{d\lambda }{d\omega }=\frac{\sin \beta }{\sin \phi }.$

The relation between Template:Mvar and Template:Mvar is

$\tan \beta =\sqrt{1-e^2}\tan \phi =(1-f)\tan \phi ,$

which gives

$\frac{\sin \beta }{\sin \phi }=\sqrt{1-e^2{\cos }^2\beta },$

so that the differential equations for the geodesic become

$\frac{1}{a}\frac{ds}{d\sigma }=\frac{d\lambda }{d\omega }=\sqrt{1-e^2{\cos }^2\beta }.$

The last step is to use Template:Mvar as the independent parameter in both of these differential equations and thereby to express Template:Mvar and Template:Mvar as integrals. Applying the sine rule to the vertices Template:Math and Template:Math in the spherical triangle Template:Math in Fig. 5 gives

$\sin \beta =\sin \beta (\sigma ;{\alpha }_0)=\cos {\alpha }_0\sin \sigma ,$

where Template:Math is the azimuth at Template:Math. Substituting this into the equation for Template:Math and integrating the result gives

$\frac{s}{b}=\underset{0}{\overset{\sigma }{\int }}\sqrt{1+k^2{\sin }^2{\sigma }^{\prime }}d{\sigma }^{\prime },$ Template:EquationRef

where

$k=e^{\prime }\cos {\alpha }_0,$

and the limits on the integral are chosen so that Template:Math. Template:Harvtxt pointed out that the equation for Template:Mvar is the same as the equation for the arc on an ellipse with semi-axes Template:Math and Template:Math. In order to express the equation for Template:Mvar in terms of Template:Mvar, we write

$d\omega =\frac{\sin {\alpha }_0}{{\cos }^2\beta }d\sigma ,$

which follows from Template:EquationNote and Clairaut's relation. This yields

$\lambda -{\lambda }_0=\omega -f\sin {\alpha }_0\underset{0}{\overset{\sigma }{\int }}\frac{2-f}{1+(1-f)\sqrt{1+k^2{\sin }^2{\sigma }^{\prime }}}d{\sigma }^{\prime },$ Template:EquationRef

and the limits on the integrals are chosen so that Template:Math at the equator crossing, Template:Math.

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 (with small flattening) Template:Harv; 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 Template:Harv.

Behavior of geodesics

Fig. 7. Meridians and the equator are the only closed geodesics. (For the very flattened ellipsoids, there are other closed geodesics; see Figs. 11 and 12).

Fig. 10. Geodesic on a prolate ellipsoid (Template:Math) with Template:Math. Compare with Fig. 8.

Fig. 7 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.

All other geodesics are typified by Figs. 8 and 9 which show a geodesic starting on the equator with Template:Math. 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 Template:Math. The geodesic completes one full oscillation in latitude before the longitude has increased by Template:Val. Thus, on each successive northward crossing of the equator (see Fig. 8), Template:Mvar falls short of a full circuit of the equator by approximately Template:Math (for a prolate ellipsoid, this quantity is negative and Template:Mvar completes more that a full circuit; see Fig. 10). For nearly all values of Template:Math, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. 9).

Template:Multiple image If the ellipsoid is sufficiently oblate, i.e., Template:Math, another class of simple closed geodesics is possible Template:Harv. Two such geodesics are illustrated in Figs. 11 and 12. Here Template:Math and the equatorial azimuth, Template:Math, for the green (resp. blue) geodesic is chosen to be Template:Val (resp. Template:Val), so that the geodesic completes 2 (resp. 3) complete oscillations about the equator on one circuit of the ellipsoid.

Fig. 13. Geodesics (blue) from a single point for Template:Math, Template:Math; geodesic circles are shown in green and the cut locus in red.

Fig. 13 shows geodesics (in blue) emanating Template:Math with Template:Math a multiple of Template:Val up to the point at which they cease to be shortest paths. (The flattening has been increased to Template:Frac in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant Template:Math, which are the geodesic circles centered Template:Math. Template:Harvtxt 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 Template:Math. 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 Template:Math, Template:Math. The longitudinal extent of cut locus is approximately Template:Math. If Template:Math lies on the equator, Template:Math, this relation is exact and as a consequence the equator is only a shortest geodesic if Template:Math. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to Template:Math, Template:Math, 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 Template:Harv, determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by Template:Math, and a second geodesic a small distance Template:Math away from it. Template:Harvtxt showed that Template:Math obeys the Gauss-Jacobi equation

$\frac{d^{2}t(s)}{d{s}^2}+K(s)t(s)=0,$

Fig. 14. Definition of reduced length and geodesic scale.

where Template:Math is the Gaussian curvature at Template:Math. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions

$t(s_2)=Cm(s_1,s_2)+DM(s_1,s_2)$

where

$\begin{array}{cc}m(s_1,s_1)& =0,{\frac{dm(s_1,s_2)}{d{s}_2}|}_{s_2=s_1}=1,\\ M(s_1,s_1)& =1,{\frac{dM(s_1,s_2)}{d{s}_2}|}_{s_2=s_1}=0.\end{array}$

The quantity Template:Math is the so-called reduced length, and Template:Math is the geodesic scale.Template:Refn Their basic definitions are illustrated in Fig. 14.

The Gaussian curvature for an ellipsoid of revolution is

$K=\frac{1}{\rho \nu }=\frac{(1-e^2{\sin }^2\phi {)}^2}{b^2}=\frac{b^2}{a^4(1-e^2{\cos }^2\beta {)}^2}.$

Template:Harvtxt solved the Gauss-Jacobi equation for this case enabling Template:Math and Template:Math to be expressed as integrals.

As we see from Fig. 14 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by Template:Math is Template:Math. On a closed surface such as an ellipsoid, Template:Math oscillates about zero. The point at which Template:Math becomes zero is the point conjugate to the starting point. In order for a geodesic between Template:Math and Template:Math, of length Template:Math, to be a shortest path it must satisfy the Jacobi condition Template:Harv Template:Harv Template:Harv Template:Harv, that there is no point conjugate to Template:Math between Template:Math and Template:Math. 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:

• for an oblate ellipsoid, Template:Math;
• for a prolate ellipsoid, Template:Math, if Template:Math; if Template:Math, the supplemental condition Template:Math is required if Template:Math.

Envelope of geodesics

The geodesics from a particular point Template:Math if continued past the cut locus form an envelope illustrated in Fig. 15. Here the geodesics for which Template:Math is a multiple of Template:Val 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 Template:Math; points on the envelope may be computed by finding the point at which Template:Math on a geodesic. Template:Harvtxt 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 Template:Math 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. (This figure uses the same position for Template:Math as Fig. 13 and is drawn in the same projection.) The two shorter geodesics are stable, i.e., Template:Math, 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 Template:Math. 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 Template:Math. Likewise, the geodesic circles are involutes of the astroid.

Area of a geodesic polygon Template:Anchor

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 Template:Math in Fig. 1 Template:Harv. 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 Template:Math of Template:Math is developed following Template:Harvtxt. The area of any closed region of the ellipsoid is

$T=\int dT=\int \frac{1}{K}\cos \phi d\phi d\lambda ,$

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

$\mathrm{\Gamma }=\int KdT=\int \cos \phi d\phi d\lambda ,$

where

$\mathrm{\Gamma }=2\pi -\sum _j{\theta }_j$

is the geodesic excess and Template:Math is the exterior angle at vertex Template:Math. Multiplying the equation for Template:Math by Template:Math, where Template:Math is the authalic radius, and subtracting this from the equation for Template:Math gives

$\begin{array}{cc}T& =R_2^2\mathrm{\Gamma }+\int (\frac{1}{K}-R_2^2)\cos \phi d\phi d\lambda \\ & =R_2^2\mathrm{\Gamma }+\int (\frac{b^2}{(1-e^2{\sin }^2\phi {)}^2}-R_2^2)\cos \phi d\phi d\lambda ,\end{array}$

where the [[Spheroid#Curvature|value of Template:Math for an ellipsoid]] has been substituted. Applying this formula to the quadrilateral Template:Math, noting that Template:Math, and performing the integral over Template:Mvar gives

$S_{12}=R_2^2({\alpha }_2-{\alpha }_1)+b^2\underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}(\frac{1}{2(1-e^2{\sin }^2\phi )}+\frac{{\tanh }^{-1}(e\sin \phi )}{2e\sin \phi }-\frac{R_2^2}{b^2})\sin \phi d\lambda ,$

where the integral is over the geodesic line (so that Template:Mvar is implicitly a function of Template:Mvar). The integral can be expressed as a series valid for small Template:Math Template:Harv Template:Harv.

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

$\tan \frac{E_{12}}{2}=\frac{\sin \frac{1}{2}({\beta }_2+{\beta }_1)}{\cos \frac{1}{2}({\beta }_2-{\beta }_1)}\tan \frac{{\omega }_{12}}{2}.$

Solution of the direct and inverse problems

Template:Further Template:See also 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 Template:Math in Fig. 5, Napier's rules for quadrantal triangles can be employed,

$\begin{array}{cc}\sin {\alpha }_0& =\sin \alpha \cos \beta =\tan \omega \cot \sigma ,\\ \cos \sigma & =\cos \beta \cos \omega =\tan {\alpha }_0\cot \alpha ,\\ \cos \alpha & =\cos \omega \cos {\alpha }_0=\cot \sigma \tan \beta ,\\ \sin \beta & =\cos {\alpha }_0\sin \sigma =\cot \alpha \tan \omega ,\\ \sin \omega & =\sin \sigma \sin \alpha =\tan \beta \tan {\alpha }_0.\end{array}$

The mapping of the geodesic involves evaluating the integrals for the distance, Template:Math, and the longitude, Template:Mvar, Eqs. Template:EquationNote and Template:EquationNote and these depend on the parameter Template:Math.

Handling the direct problem is straightforward, because Template:Math can be determined directly from the given quantities Template:Math and Template:Math; for a sample calculation, see Template:Harvtxt.

In the case of the inverse problem, Template:Math is given; this cannot be easily related to the equivalent spherical angle Template:Math because Template:Math is unknown. Thus, the solution of the problem requires that Template:Math be found iteratively (root finding); see Template:Harvtxt for details.

In geodetic applications, where Template:Math is small, the integrals are typically evaluated as a series Template:Harv Template:Harv Template:Harv Template:Harv Template:Harv Template:Harv. For arbitrary Template:Math, the integrals (3) and (4) can be found by numerical quadrature or by expressing them in terms of elliptic integrals Template:Harv Template:Harv Template:Harv.

Template:Harvtxt 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 Template:Val for the WGS84 ellipsoid; however the inverse method fails to converge for nearly antipodal points.

Template:Harvtxt continues the expansions to sixth order which suffices to provide full double precision accuracy for Template:Math and improves the solution of the inverse problem so that it converges in all cases. Template:Harvtxt extends the method to use elliptic integrals which can be applied to ellipsoids with arbitrary flattening.

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 (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, Template:Harvtxt discovered a constant of the motion allowing this problem to be reduced to quadrature also Template:Harv.Template:Refn

Triaxial ellipsoid coordinate system

Fig. 17. Triaxial ellipsoidal coordinates.

Consider the ellipsoid defined by

$h=\frac{X^2}{a^2}+\frac{Y^2}{b^2}+\frac{Z^2}{c^2}=1,$

where Template:Math are Cartesian coordinates centered on the ellipsoid and, without loss of generality, Template:Math.Template:Refn Template:Anchor Template:Harvtxt employed the (triaxial) ellipsoidal coordinates (with triaxial ellipsoidal latitude and triaxial ellipsoidal longitude, Template:Math) defined by

$\begin{array}{cc}X& =a\cos \omega \frac{\sqrt{a^2-b^2{\sin }^2\beta -c^2{\cos }^2\beta }}{\sqrt{a^2-c^2}},\\ Y& =b\cos \beta \sin \omega ,\\ Z& =c\sin \beta \frac{\sqrt{a^2{\sin }^2\omega +b^2{\cos }^2\omega -c^2}}{\sqrt{a^2-c^2}}.\end{array}$

In the limit Template:Math, Template:Mvar becomes the parametric latitude for an oblate ellipsoid, so the use of the symbol Template:Mvar is consistent with the previous sections. However, Template:Mvar is different from the spherical longitude defined above.Template:Refn

Grid lines of constant Template:Mvar (in blue) and Template:Mvar (in green) 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 Template:Math and Template:Math are shown in red. The third principal section, Template:Math, is covered by the lines Template:Math and Template:Math or Template:Math. 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 Template:Math, and it is viewed in an orthographic projection from a point above Template:Math, Template:Math.

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 Template:Harv.
2. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets Template:Harv.
3. Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points Template:Harv. For example, the lines of constant Template:Mvar 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 Template:Harv,

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 Template:Harv Template:Harv is

$\begin{array}{cc}\delta & =\int \frac{\sqrt{b^2{\sin }^2\beta +c^2{\cos }^2\beta }d\beta }{\sqrt{a^2-b^2{\sin }^2\beta -c^2{\cos }^2\beta }\sqrt{(b^2-c^2){\cos }^2\beta -\gamma }}\\ & -\int \frac{\sqrt{a^2{\sin }^2\omega +b^2{\cos }^2\omega }d\omega }{\sqrt{a^2{\sin }^2\omega +b^2{\cos }^2\omega -c^2}\sqrt{(a^2-b^2){\sin }^2\omega +\gamma }}.\end{array}$

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

$\gamma =(b^2-c^2){\cos }^2\beta {\sin }^2\alpha -(a^2-b^2){\sin }^2\omega {\cos }^2\alpha ,$

where Template:Mvar is the angle the geodesic makes with lines of constant Template:Mvar. In the limit Template:Math, this reduces to Template:Math, the familiar Clairaut relation. A derivation of Jacobi's result is given by Template:Harvtxt; he gives the solution found by Template:Harvtxt for general quadratic surfaces.

Survey of triaxial geodesics

Template:Multiple image On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by Template:Math, Template:Math, and Template:Math.Template:Refn To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section, Template:Math, 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 Template:Math, Template:Math, and Template:Math, then Template:Math 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 Template:Math. Two examples are given in Figs. 18 and 19. Figure 18 shows practically the same behavior as for an oblate ellipsoid of revolution (because Template:Math); compare to Fig. 9. However, if the starting point is at a higher latitude (Fig. 18) the distortions resulting from Template:Math are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at Template:Math Template:Harv Template:Harv.

Template:Multiple image If the starting point is Template:Math, Template:Math, and Template:Math, then Template:Math and the geodesic encircles the ellipsoid in a "transpolar" sense. The geodesic oscillates east and west of the ellipse Template:Math; 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 Template:Math and Template:Math. If Template:Math, all meridians are geodesics; the effect of Template:Math 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 Template:Math; in this case, the ellipsoid becomes a prolate ellipsoid and Fig. 20 would resemble Fig. 10 (rotated on its side). All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at Template:Math.

In Figs. 18–21, the geodesics are (very nearly) closed. As noted above, in the typical case, the geodesics are not closed, but fill the area bounded by the limiting lines of latitude (in the case of Figs. 18–19) or longitude (in the case of Figs. 20–21).

Fig. 22. An umbilical geodesic, Template:Math, Template:Math, Template:Math.

If the starting point is Template:Math, Template:Math (an umbilical point), and Template:Math (the geodesic leaves the ellipse Template:Math at right angles), then Template:Math 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 Template:Math becomes closer to Template:Val or Template:Val so that asymptotically the geodesic lies on the ellipse Template:Math Template:Harv Template:Harv, 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 geodesic 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 Template:Math and Template:Math are stable (a geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse Template:Math, which goes through all 4 umbilical points, is exponentially unstable. If it is perturbed, it will swing out of the plane Template:Math and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.)

If the starting point Template:Math of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on Template:Math and the other two on Template:Math. The cut locus for Template:Math is the portion of the line Template:Math 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 Template:Harv. Nevertheless, terrestrial geodesics still play an important role in several areas:

• for measuring distances and areas in geographic information systems;
• the definition of maritime boundaries Template:Harv;
• in the rules of the Federal Aviation Administration for area navigation Template:Harv;
• the method of measuring distances in the FAI Sporting Code Template:Harv.
• help Muslims find their direction toward Mecca

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 Template:Harv Template:Harv. 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 Template:Harv and elliptic functions Template:Harv;
• the development of differential geometry Template:Harv Template:Harv;
• methods for solving systems of differential equations by a change of independent variables Template:Harv;
• the study of caustics Template:Harv;
• investigations into the number and stability of periodic orbits Template:Harv;
• in the limit Template:Math, geodesics on a triaxial ellipsoid reduce to a case of dynamical billiards;
• extensions to an arbitrary number of dimensions Template:Harv;
• geodesic flow on a surface Template:Harv.

See also

• Earth section paths
• Figure of the Earth
• Geographical distance
• Great-circle navigation
• Great ellipse
• Geodesic
• Geodesy
• Map projection
  • Map projection of the triaxial ellipsoid
• Meridian arc
• Rhumb line
• Vincenty's formulae

References

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External links

• Online geodesic bibliography of books and articles on geodesics on ellipsoids.
• Test set for geodesics, a set of 500000 geodesics for the WGS84 ellipsoid, computed using high-precision arithmetic.
• NGS tool implementing Template:Harvtxt.
• geod(1), man page for the PROJ utility for geodesic calculations.
• GeographicLib implementation of Template:Harvtxt.
• Drawing geodesics on Google Maps.