Geodesics On An Ellipsoid

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## Geodesics on an ellipsoid of revolution

### Equations for a geodesic

### Behavior of geodesics

### Differential properties of geodesics

### Envelope of geodesics

### Area of a geodesic polygon

### Solution of the direct and inverse problems

## Geodesics on a triaxial ellipsoid

### The triaxial coordinate system

### Jacobi's solution

### Survey of triaxial geodesics

## Applications

## See also

## Notes

## References

## External links

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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 (Euler 1755).

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, Newton (1687) 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.

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 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 (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
φ_{1} and longitude λ_{1} and
*B* at latitude φ_{2} and longitude
λ_{2} (see Fig. 1). The connecting geodesic
(from *A* to *B*) is *AB*, of length
*s*_{12}, which has azimuths α_{1} and
α_{2} at the two endpoints.^{[1]} The two geodesic problems usually
considered are:

- the
*direct geodesic problem*or*first geodesic problem*, given*A*, α_{1}, and*s*_{12}, determine*B*and α_{2}; - the
*inverse geodesic problem*or*second geodesic problem*, given*A*and*B*, determine*s*_{12}, α_{1}, and α_{2}.

As can be seen from Fig. 1, these problems involve solving the triangle
*NAB* given one angle, α_{1} for the direct
problem and λ_{12} = λ_{2} − λ_{1} 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 thegeodesic 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

geodeticorgeodesic 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.

Here the equations for a geodesic are developed; the derivation closely follows that of Bessel (1825). Jordan & Eggert (1941), Bagratuni (1962, §15), Gan'shin (1967, Chap. 5), Krakiwsky & Thomson (1974, §4), Rapp (1993, §1.2), Jekeli (2012), 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* = /*a* = , and the second
eccentricity *e*′ = /*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*, *e*^{2}, 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 *d*φ and *d*λ by

- (1)

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′ = *d*φ/*d*λ and the
Lagrangian function *L* depends on
φ through ρ(φ) and
*R*(φ). The length of an arbitrary path between
(φ_{1}, λ_{1}) and (φ_{2}, λ_{2}) is
given by

where φ is a function of λ satisfying
φ(λ_{1}) = φ_{1} and
φ(λ_{2}) = φ_{2}. The shortest path or geodesic
entails finding that function φ(λ) which minimizes
*s*_{12}. 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).^{[2]} Differentiating this
relation gives

This, together with Eqs. (1), leads to a system of ordinary differential equations for a geodesic

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

and Clairaut's relation then becomes

This is the sine rule of spherical
trigonometry relating two sides of the triangle *NAB* (see
Fig. 4), *NA* = π − β_{1}, and
*NB* = π − β_{2} and their opposite angles
*B* = π − α_{2} and *A* = α_{1}.

In order to find the relation for the third side
*AB* = σ_{12}, the *spherical arc length*, and included
angle *N* = ω_{12}, the *spherical longitude*, it is
useful to consider the triangle *NEP* 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
*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. 6), we
obtain

- (2)

Combining Eqs. (1) and (2) gives differential
equations for *s* and λ

The relation between β and φ is

which gives

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. 5 gives

where α_{0} is the azimuth at *E*.
Substituting this into the equation for *ds*/*d*σ and
integrating the result gives

- (3)

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* and
*b*. In order to express the equation for
λ in terms of σ, we write

which follows from Eq. (2) and Clairaut's relation. This yields

- (4)

and the limits on the integrals are chosen
so that λ = λ_{0} 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.

There are also several ways of approximating geodesics on a terrestrial ellipsoid (with small flattening) (Rapp 1991, §6); 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 (Jekeli 2012, §2.1.4).

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
α_{0} = 45°. 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
β = ±(π − |α_{0}|).
The geodesic completes one full oscillation in
latitude before the longitude has increased by .
Thus, on each successive northward crossing of the equator (see
Fig. 8), λ falls short of a full circuit of
the equator by approximately 2π *f* sinα_{0} (for a
prolate ellipsoid, this quantity is negative and λ
completes more that a full circuit; see Fig. 10). For nearly all
values of α_{0}, the geodesic will fill that portion of
the ellipsoid between the two vertex latitudes (see
Fig. 9).

If the ellipsoid is sufficiently oblate, i.e.,
< , another class of simple closed geodesics is
possible (Klingenberg 1982, §3.5.19). Two such geodesics
are illustrated in Figs. 11 and 12. Here
= and the equatorial azimuth,
α_{0}, for the green (resp. blue) geodesic is chosen to
be (resp. ), so that the
geodesic completes 2 (resp. 3) complete oscillations about the equator
on one circuit of the ellipsoid.

Fig.
13 shows geodesics (in blue) emanating
*A* with α_{1} a multiple of
up to the point at which they cease to be shortest
paths. (The flattening has been increased to
in order to accentuate the ellipsoidal effects.)
Also shown (in green) are curves of constant *s*_{12},
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 = −φ_{1}. The longitudinal
extent of cut locus is approximately
λ_{12} ∈ [π − *f* π cosφ_{1}, π + *f* π cosφ_{1}]. If
*A* lies on the equator, φ_{1} = 0, this relation
is exact and as a consequence the equator is only a shortest geodesic if
|λ_{12}| ≤ (1 − *f*)π. For a prolate
ellipsoid, the cut locus is a segment of the anti-meridian centered on
the point antipodal to *A*, λ_{12} = π,
and this means that
meridional geodesics stop being shortest paths before the antipodal
point is reached.

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*,
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

The quantity *m*(*s*_{1}, *s*_{2}) = *m*_{12} is the so-called
*reduced length*, and *M*(*s*_{1}, *s*_{2}) = *M*_{12} is the
*geodesic scale*.^{[3]}
Their basic definitions are illustrated in
Fig. 14.

The Gaussian curvature for an ellipsoid of revolution is

Helmert (1880, Eq. (6.5.1.)) solved the Gauss-Jacobi
equation for this case enabling *m*_{12} and
*M*_{12} 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
*d*α_{1} is *m*_{12}*d*α_{1}. On a closed
surface such as an ellipsoid, *m*_{12} oscillates
about zero. The point at
which *m*_{12} becomes zero is the point
conjugate to the starting point. In order
for a geodesic between *A* and *B*, of length
*s*_{12}, 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:

- for an oblate ellipsoid, |σ
_{12}| ≤ π; - for a prolate ellipsoid, |λ
_{12}| ≤ π, if α_{0}≠ 0; if α_{0}= 0, the supplemental condition*m*_{12}≥ 0 is required if |λ_{12}| = π.

The geodesics from a particular point *A* if continued
past the cut locus form an envelope illustrated in Fig. 15.
Here the geodesics for which α_{1} is a multiple of
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
*m*_{12} = 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. 16 where the geodesics are numbered in order of
increasing length. (This figure uses the same position for
*A* as Fig. 13 and is drawn in the same projection.)
The two shorter geodesics are *stable*, i.e., *m*_{12} > 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 σ_{12} ≤ 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.

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
*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 an expression for the area *S*_{12} of *AFHB*
is developed 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 *R*_{2}^{2}, where *R*_{2} 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 = α_{2} − α_{1}, and performing
the integral over φ gives

where the integral is over the geodesic line (so that φ
is implicitly a function of λ).
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 *S*_{12}
over its edges. This result holds provided that the polygon does not
include a pole; if it does, 2π *R*_{2}^{2} must be added to the
sum. If the edges are specified by their vertices, then a
convenient expression
for the geodesic excess *E*_{12} = α_{2} − α_{1} is

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 *NEP* in Fig.
5,
Napier's rules for quadrantal triangles can be employed,

The mapping of the geodesic involves evaluating the
integrals for the distance, *s*, and the longitude,
λ, Eqs. (3) and (4) and these depend on
the parameter α_{0}.

Handling the direct problem is straightforward, because
α_{0} can be determined directly from the given
quantities φ_{1} and α_{1}.

In the case of the inverse problem, λ_{12} is given;
this cannot be easily related to the equivalent spherical angle
ω_{12} because α_{0} is unknown.
Thus, the solution of the problem requires that α_{0} be
found iteratively.

In geodetic applications, where *f* is small, the integrals
are typically evaluated as a series (Legendre 1806)
(Oriani 1806) (Bessel 1825) (Helmert 1880)
(Rainsford 1955) (Rapp 1993). For arbitrary
*f*, the integrals (3) and (4) can be found by
numerical quadrature or by expressing them in terms of
elliptic integrals (Legendre 1806) (Cayley 1870).

Vincenty (1975) 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. Karney (2013)
continues the expansions to sixth order which suffices to provide full
double precision accuracy for
|*f*| ≤ and improves the solution of
the inverse problem so that it converges in all cases.
Karney (2013, addendum) extends the method to use elliptic
integrals which can be applied to ellipsoids with arbitrary flattening.

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 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).^{[4]}

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.^{[5]}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.^{[6]}

Grid lines of constant β (in blue) and
ω (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 *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:

- They are "lines of curvature" on the ellipsoid: they are parallel to the directions of principal curvature (Monge 1796).
- They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets (Dupin 1813, Part 5).
- 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. 17 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.

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.

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.^{[7]}
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. 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 β_{1} ∈ (−90°, 90°),
ω_{1} = 0, and α_{1} = 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 than a full circuit around the ellipsoid
resulting, in the typical case, in the geodesic filling the area bounded
by the two latitude lines β = ±β_{1}. Two examples
are given in Figs. 18 and 19. Figure 18 shows
practically the same behavior as for an oblate ellipsoid of revolution
(because *a* ≈ *b*); compare to Fig. 9.
However, if the starting point is at a higher latitude (Fig. 18)
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 = β_{1}
(Chasles 1846)
(Hilbert & Cohn-Vossen 1952, pp. 223-224).

If the starting point is β_{1} = 90°,
ω_{1} ∈ (0°, 180°), and
α_{1} = 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 = ω_{1} and ω = 180° − ω_{1}.
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. 20 and 21. 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. 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
ω = ω_{1}.

If the starting point is β_{1} = 90°,
ω_{1} = 0° (an umbilical point), and
α_{1} = 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 or
so that asymptotically the geodesic lies on the
ellipse *Y* = 0 (Hart 1849) (Arnold 1989, p. 265),
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
*X*= 0 and*Z*= 0 are stable (an geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse*Y*= 0, which goes through all 4 umbilical points, is*exponentially unstable*. If it is perturbed, it will swing out of the plane*Y*= 0 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 *A* of a geodesic is not an umbilical
point, its envelope is an astroid with two cusps lying on
β = −β_{1} and the other two on
ω = ω_{1} + π. The cut locus
for *A* is the portion
of the line β = −β_{1} between the cusps.

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:

- for measuring distances and areas in geographic information systems;
- the definition of maritime boundaries (UNCLOS 2006);
- in the rules of the Federal Aviation Administration for area navigation (RNAV 2007);
- the method of measuring distances in the FAI Sporting Code (FAI 2018).
- 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 (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:

- the development of elliptic integrals (Legendre 1811) and elliptic functions (Weierstrass 1861);
- the development of differential geometry (Gauss 1828) (Christoffel 1869);
- methods for solving systems of differential equations by a change of independent variables (Jacobi 1839);
- the study of caustics (Jacobi 1891);
- investigations into the number and stability of periodic orbits (Poincaré 1905);
- in the limit
*c*→ 0, geodesics on a triaxial ellipsoid reduce to a case of dynamical billiards; - extensions to an arbitrary number of dimensions (Knörrer 1980);
- geodesic flow on a surface (Berger 2010, Chap. 12).

- Figure of the Earth
- Geographical distance
- Great-circle navigation
- Great ellipse
- Geodesics
- Geodesy
- Meridian arc
- Rhumb line
- Vincenty's formulae

**^**Here α_{2}is the*forward*azimuth at*B*. Some authors calculate the*back*azimuth instead; this is given by α_{2}± π.**^**Laplace (1799a) showed that a particle constrained to move on a surface but otherwise subject to no forces moves along a geodesic for that surface. Thus, Clairaut's relation is just a consequence of conservation of angular momentum for a particle on a surface of revolution.**^**Bagratuni (1962, §17) uses the term "coefficient of convergence of ordinates" for the geodesic scale.**^**This section is adapted from the documentation for GeographicLib (Karney 2015, Geodesics on a triaxial ellipsoid)**^**This notation for the semi-axes is incompatible with that used in the previous section on ellipsoids of revolution in which*a*and*b*stood for the equatorial radius and polar semi-axis. Thus the corresponding inequalities are*a*=*a*≥*b*> 0 for an oblate ellipsoid and*b*≥*a*=*a*> 0 for a prolate ellipsoid.**^**The limit*b*→*c*gives a prolate ellipsoid with ω playing the role of the parametric latitude.**^**If < , there are other simple closed geodesics similar to those shown in Figs. 11 and 12 (Klingenberg 1982, §3.5.19).

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- 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 Vincenty (1975).
- geod(1), man page for the PROJ utility for geodesic calculations.
- GeographicLib implementation of Karney (2013).
- Drawing geodesics on Google Maps.

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