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{{Short description|Shortest paths on a bounded deformed sphere-like quadric surface}}
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[[File:Long geodesic on an oblate ellipsoid.svg|thumb|200px|A geodesic on an oblate ellipsoid]]
{{Geodesy}}

The study of '''geodesics on an ellipsoid''' arose in connection with [[geodesy]] specifically with the solution of [[triangulation network]]s.  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 {{harv|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, {{harvtxt|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 [[meridian (geography)|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 {{harv|Hilbert|Cohn-Vossen|1952|loc=[https://archive.org/details/geometryimaginat0000dhil/page/220/ 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 network]]s entailed reducing all the measurements to a [[reference ellipsoid]] and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry {{harv|Bomford|1952|loc=Chap. 3}} {{harv|Leick et al.|2015|loc=§4.5}}.

[[File:Geodesic problem on an ellipsoid.svg|thumb|right|
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: {{math|''A''}} at [[latitude]] {{math|''φ''<sub>1</sub>}} and [[longitude]] {{math|''λ''<sub>1</sub>}} and {{math|''B''}} at latitude {{math|''φ''<sub>2</sub>}} and longitude {{math|''λ''<sub>2</sub>}} (see Fig.&nbsp;1).  The connecting geodesic (from {{math|''A''}} to {{math|''B''}}) is {{math|''AB''}}, of length {{math|''s''<sub>12</sub>}}, which has [[azimuth]]s {{math|&alpha;<sub>1</sub>}} and {{math|&alpha;<sub>2</sub>}} at the two endpoints.{{refn|
Here {{math|&alpha;<sub>2</sub>}} is the ''forward'' azimuth at {{math|''B''}}. Some authors calculate the ''back'' azimuth instead; this is given by {{math|&alpha;<sub>2</sub> &plusmn; ''π''}}.}}  The two geodesic problems usually considered are:
# the ''direct geodesic problem'' or ''first geodesic problem'', given {{math|''A''}}, {{math|&alpha;<sub>1</sub>}}, and {{math|''s''<sub>12</sub>}}, determine {{math|''B''}} and {{math|&alpha;<sub>2</sub>}};
# the ''inverse geodesic problem'' or ''second geodesic problem'', given {{math|''A''}} and {{math|''B''}}, determine {{math|''s''<sub>12</sub>}}, {{math|&alpha;<sub>1</sub>}}, and {{math|&alpha;<sub>2</sub>}}.
As can be seen from Fig. 1, these problems involve solving the triangle {{math|''NAB''}} given one angle, {{math|&alpha;<sub>1</sub>}} for the direct problem and {{math|''λ''<sub>12</sub> {{=}} ''λ''<sub>2</sub> &minus; ''λ''<sub>1</sub>}} 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 [[Solution of triangles#Two sides and the included angle given|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 {{harvtxt|Clairaut|1735}}.  A systematic solution for the paths of geodesics was given by {{harvtxt|Legendre|1806}} and {{harvtxt|Oriani|1806}} (and subsequent papers in [[#{{harvid|Oriani|1808|}}|1808]] and [[#{{harvid|Oriani|1810|}}|1810]]).
The full solution for the direct problem (complete with computational tables and a worked out example) is given by {{harvtxt|Bessel|1825}}.

During the 18th century geodesics were typically referred to as "shortest lines".
The term "geodesic line" (actually, a [[curve]]) was coined by {{harvtxt|Laplace|1799b}}:
<blockquote>
Nous désignerons cette ligne sous le nom de ''ligne géodésique'' [We will call this line the ''geodesic line''].
</blockquote>
This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example {{harv|Hutton|1811|loc=[https://archive.org/details/courseofmathemat03hutt/page/115/ p. 115]}},
<blockquote>
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.
</blockquote>
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 ===
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Differential element of a meridian ellipse.svg
|caption1=Fig. 2. Differential element of a meridian ellipse.
|image2=Differential element of a geodesic on an ellipsoid.svg
|caption2=Fig. 3.  Differential element of a geodesic on an ellipsoid.
}}

Here the equations for a geodesic are developed; the derivation closely follows that of {{harvtxt|Bessel|1825}}. {{harvtxt|Jordan|Eggert|1941}}, {{harvtxt|Bagratuni|1962|loc=§15}}, {{harvtxt|Gan'shin|1967|loc=Chap. 5}}, {{harvtxt|Krakiwsky|Thomson|1974|loc=§4}}, {{harvtxt|Rapp|1993|loc=§1.2}}, {{harvtxt|Jekeli|2012}}, and {{harvtxt|Borre|Strang|2012}} also provide derivations of these equations.

Consider an ellipsoid of revolution with equatorial radius {{math|''a''}} and polar semi-axis {{math|''b''}}.  Define the flattening {{math|''f''}}, the [[Eccentricity (mathematics)|eccentricity]] {{math|''e''}}, and the second eccentricity {{math|''e''&prime;}}:

:<math>
f = \frac{a - b}{a}, \quad e = \frac{\sqrt{a^2 - b^2}}{a} = \sqrt{f(2-f)},\quad e' = \frac{\sqrt{a^2-b^2}}{b} = \frac{e}{1-f}.
</math>

(In most applications in geodesy, the ellipsoid is taken to be oblate, {{math|''a'' &gt; ''b''}}; however, the theory applies without change to prolate ellipsoids, {{math|''a'' &lt; ''b''}}, in which case {{math|''f''}}, {{math|''e''<sup>2</sup>}}, and {{math|''e''&prime;<sup>2</sup>}} are negative.)

Let an elementary segment of a path on the ellipsoid have length {{math|''ds''}}.  From Figs. 2 and 3, we see that if its azimuth is {{mvar|&alpha;}}, then {{math|''ds''}} is related to {{math|''dφ''}} and {{math|''dλ''}} by
:<math>
\cos\alpha\,ds = \rho\,d\varphi = - \frac{dR}{\sin\varphi}, \quad
\sin\alpha\,ds = R\,d\lambda,</math> {{EquationRef|1|(1)}}
where {{mvar|&rho;}} is the [[Earth radius#Meridional|meridional radius of curvature]], {{math|''R'' {{=}} ''&nu;'' cos''φ''}} is the radius of the circle of latitude {{mvar|φ}}, and {{mvar|&nu;}} is the [[Earth radius#Prime vertical|normal radius of curvature]].
The elementary segment is therefore given by
:<math>ds^2 = \rho^2\,d\varphi^2 + R^2\,d\lambda^2</math>
or
:<math>\begin{align}
ds &= \sqrt{\rho^2\varphi'^2 + R^2}\,d\lambda \\
&\equiv L(\varphi,\varphi')\,d\lambda,
\end{align}</math>
where {{math|''φ''&prime; {{=}} ''dφ''/''dλ''}} and the [[Lagrangian mechanics#Euler–Lagrange equations and Hamilton's principle|Lagrangian function {{math|''L''}}]] depends on {{mvar|φ}} through {{math|&rho;(''φ'')}} and {{math|''R''(''φ'')}}.  The length of an arbitrary path between {{math|(''φ''<sub>1</sub>, ''λ''<sub>1</sub>)}} and {{math|(''φ''<sub>2</sub>, ''λ''<sub>2</sub>)}} is given by
:<math> s_{12} = \int_{\lambda_1}^{\lambda_2} L(\varphi, \varphi')\,d\lambda,</math>
where {{mvar|φ}} is a function of {{mvar|λ}} satisfying {{math|''φ''(''λ''<sub>1</sub>) {{=}} ''φ''<sub>1</sub>}} and {{math|''φ''(''λ''<sub>2</sub>) {{=}} ''φ''<sub>2</sub>}}.  The shortest path or geodesic entails finding that function {{math|''φ''(''λ'')}} which minimizes {{math|''s''<sub>12</sub>}}.  This is an exercise in the [[calculus of variations]] and the minimizing condition is given by the [[Beltrami identity]],
:<math>L - \varphi' \frac{\partial L}{\partial \varphi'} = \text{const.}</math>
Substituting for {{math|''L''}} and using Eqs. {{EquationNote|1|(1)}} gives
:<math>R\sin\alpha = \text{const.}</math>
{{harvtxt|Clairaut|1735}} found this [[Clairaut's relation|relation]], using a geometrical construction; a similar derivation is presented by {{harvtxt|Lyusternik|1964|loc=§10}}.{{refn|
{{harvtxt|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.}}  Differentiating this relation gives
:<math>d\alpha=\sin\varphi\,d\lambda.</math>
This, together with Eqs. {{EquationNote|1|(1)}}, leads to a system of [[ordinary differential equations]] for a geodesic
:<math>
\frac{d\varphi}{ds} = \frac{\cos\alpha}{\rho};\quad
\frac{d\lambda}{ds} = \frac{\sin\alpha}{\nu\cos\varphi};\quad
\frac{d\alpha}{ds} = \frac{\tan\varphi\sin\alpha}{\nu}.</math>
We can express {{math|''R''}} in terms of the [[Latitude#Parametric latitude (or reduced latitude)|parametric latitude]], {{mvar|β}}, using
:<math>R = a\cos\beta,</math>
and Clairaut's relation then becomes
:<math>\sin\alpha_1\cos\beta_1 = \sin\alpha_2\cos\beta_2.</math>
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Geodesic problem on a sphere.svg
|caption1=Fig. 4.  Geodesic problem mapped to the auxiliary sphere.
|image2=Geodesic problem mapped to the auxiliary sphere.svg
|caption2=Fig. 5. The elementary geodesic problem on the auxiliary sphere.
}}
This is the [[Spherical trigonometry#Sine rules|sine rule]] of spherical trigonometry relating two sides of the triangle {{math|''NAB''}} (see Fig.&nbsp;4), {{math|''NA'' {{=}} {{frac|1|2}}''π'' &minus; ''β''<sub>1</sub>}}, and {{math|''NB'' {{=}} {{frac|1|2}}''π'' &minus; ''β''<sub>2</sub>}} and their opposite angles {{math|''B'' {{=}} ''π'' &minus; &alpha;<sub>2</sub>}} and {{math|''A'' {{=}} &alpha;<sub>1</sub>}}.

In order to find the relation for the third side {{math|''AB'' {{=}} ''σ''<sub>12</sub>}}, the ''spherical arc length'', and included angle {{math|''N'' {{=}} &omega;<sub>12</sub>}}, the ''spherical longitude'', it is useful to consider the triangle {{math|''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 {{math|''P''}}; {{math|''E''}}, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for {{mvar|σ}}, {{mvar|s}} and {{mvar|ω}}.

[[File:Differential element of a geodesic on a sphere.svg|thumb|
Fig. 6.
Differential element of a geodesic on a sphere.]]
If the side {{math|''EP''}} is extended by moving {{math|''P''}} infinitesimally (see Fig. 6), we obtain
:<math>
\cos\alpha\,d\sigma = d\beta, \quad
\sin\alpha\,d\sigma = \cos\beta\,d\omega.</math> {{EquationRef|2|(2)}}
Combining Eqs. {{EquationNote|1|(1)}} and {{EquationNote|2|(2)}} gives differential equations for {{math|''s''}} and {{mvar|λ}}
:<math>\frac 1 a \frac{ds}{d\sigma}
= \frac{d\lambda}{d\omega}
= \frac{\sin\beta}{\sin\varphi}.</math>

The relation between {{mvar|β}} and {{mvar|φ}} is
:<math>\tan\beta = \sqrt{1-e^2} \tan\varphi = (1-f) \tan\varphi,</math>
which gives
:<math>\frac{\sin\beta}{\sin\varphi} = \sqrt{1-e^2\cos^2\beta},</math>
so that the differential equations for the geodesic become
:<math>\frac1a\frac{ds}{d\sigma} = \frac{d\lambda}{d\omega} = \sqrt{1-e^2\cos^2\beta}.</math>

The last step is to use {{mvar|σ}} as the independent  parameter in both of these differential equations and thereby to express {{mvar|s}} and {{mvar|λ}} as integrals.  Applying the sine rule to the vertices {{math|''E''}} and {{math|''G''}} in the spherical triangle {{math|''EGP''}} in Fig. 5 gives
:<math>\sin\beta = \sin\beta(\sigma;\alpha_0) = \cos\alpha_0 \sin\sigma,</math>
where {{math|&alpha;<sub>0</sub>}} is the azimuth at {{math|''E''}}.
Substituting this into the equation for {{math|''ds''/''d σ''}} and integrating the result gives
:<math>
\frac sb = \int_0^\sigma \sqrt{1 + k^2 \sin^2\sigma'}\,d\sigma',
</math> {{EquationRef|3|(3)}}
where
:<math>k = e'\cos\alpha_0,</math>
and the limits on the integral are chosen so that {{math|1=''s''(''σ'' = 0) = 0}}.  {{harvtxt|Legendre|1811|loc=[https://archive.org/details/exercicescalculi01legerich/page/n185/ p. 180]}} pointed out that the equation for {{mvar|s}} is the same as the equation for the [[Meridian arc#Meridian distance on the ellipsoid|arc on an ellipse]] with semi-axes {{math|''b''{{radical|1 + ''e''&prime;<sup>2</sup> cos<sup>2</sup>&alpha;<sub>0</sub>}}}} and {{math|''b''}}.  In order to express the equation for {{mvar|λ}} in terms of {{mvar|σ}}, we write
:<math>d\omega = \frac{\sin\alpha_0}{\cos^2\beta}\,d\sigma,</math>
which follows from {{EquationNote|2|Eq. 2}} and Clairaut's relation.
This yields
:<math>
\lambda - \lambda_0 = \omega - f\sin\alpha_0
\int_0^\sigma \frac{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}}
\,d\sigma',
</math> {{EquationRef|4|(4)}}
and the limits on the integrals are chosen so that {{math|''λ'' {{=}} ''λ''<sub>0</sub>}} at the equator crossing, {{math|''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) {{harv|Rapp|1991|loc=§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 {{harv|Jekeli|2012|loc=§2.1.4}}.

=== Behavior of geodesics ===
[[File:Closed geodesics on an ellipsoid of revolution.svg|thumb|
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).]]
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Long geodesic on an oblate ellipsoid.svg
|caption1=Fig. 8.  Following the geodesic on the ellipsoid for about 5 circuits.
|image2=Really long geodesic on an oblate ellipsoid.svg
|caption2=Fig. 9.  The same geodesic after about 70 circuits.
|header=Geodesic on an oblate ellipsoid ({{math|''f'' {{=}} {{frac|50}}}}) with {{math|&alpha;<sub>0</sub> {{=}} {{val|45|u=deg}}}}.
}}
[[File:Long geodesic on a prolate ellipsoid.svg|thumb|
Fig. 10.
Geodesic on a prolate ellipsoid ({{math|''f'' {{=}} &minus;{{frac|50}}}}) with {{math|&alpha;<sub>0</sub> {{=}} {{val|45|u=deg}}}}.  Compare with Fig.&nbsp;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 {{math|&alpha;<sub>0</sub> {{=}} 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 {{math|''β'' {{=}} &plusmn;({{frac|1|2}}''π'' &minus; {{!}}&alpha;<sub>0</sub>{{!}})}}. The geodesic completes one full oscillation in latitude before the longitude has increased by {{val|360|u=deg}}. Thus, on each successive northward crossing of the equator (see Fig.&nbsp;8), {{mvar|λ}} falls short of a full circuit of the equator by approximately {{math|2''π'' ''f'' sin&alpha;<sub>0</sub>}} (for a prolate ellipsoid, this quantity is negative and {{mvar|λ}} completes more that a full circuit; see Fig. 10).  For nearly all values of {{math|&alpha;<sub>0</sub>}}, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig.&nbsp;9).

{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Non-standard closed geodesics on an ellipsoid of revolution 1.svg
|caption1=Fig. 11.  Side view.
|image2=Non-standard closed geodesics on an ellipsoid of revolution 2.svg
|caption2=Fig. 12.  Top view.
|header=Two additional closed geodesics for the oblate ellipsoid, {{math|{{frac|''b''|''a''}} {{=}} {{frac|2|7}}}}.
}}
If the ellipsoid is sufficiently oblate, i.e., {{math|{{frac|''b''|''a''}} &lt; {{frac|1|2}}}}, another class of simple closed geodesics is possible {{harv|Klingenberg|1982|loc=§3.5.19}}.  Two such geodesics are illustrated in Figs. 11 and 12.  Here {{math|{{frac|''b''|''a''}} {{=}} {{frac|2|7}}}} and the equatorial azimuth, {{math|&alpha;<sub>0</sub>}}, for the green (resp. blue) geodesic is chosen to be {{val|53.175|u=deg}} (resp. {{val|75.192|u=deg}}), so that the geodesic completes 2 (resp. 3) complete oscillations about the equator on one circuit of the ellipsoid.

[[File:Geodesics and geodesic circles on an oblate ellipsoid.svg|thumb|
Fig. 13.
Geodesics (blue) from a single point for {{math|''f'' {{=}} {{frac|10}}}},
{{math|''φ''<sub>1</sub> {{=}} {{val|-30|u=deg}}}};
geodesic circles are shown in green and the [[cut locus]] in red.]]
Fig.&nbsp;13 shows geodesics (in blue) emanating {{math|''A''}} with {{math|&alpha;<sub>1</sub>}} a multiple of {{val|15|u=deg}} up to the point at which they cease to be shortest paths.  (The flattening has been increased to {{frac|10}} in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant {{math|''s''<sub>12</sub>}}, which are the geodesic circles centered {{math|''A''}}. {{harvtxt|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 {{math|''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 [[antipodes|antipodal]] to {{math|''A''}}, {{math|''φ'' {{=}} &minus;''φ''<sub>1</sub>}}.  The longitudinal extent of cut locus is approximately {{math|''λ''<sub>12</sub> &isin; [''π'' (1 &minus; ''f'' cos''φ''<sub>1</sub>), ''π'' (1 + ''f'' cos''φ''<sub>1</sub>)]}}.  If
{{math|''A''}} lies on the equator, {{math|''φ''<sub>1</sub> {{=}} 0}}, this relation is exact and as a consequence the equator is only a shortest geodesic if {{math|{{!}}''λ''<sub>12</sub>{{!}} &le; ''π'' (1 &minus; ''f'')}}.  For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to {{math|''A''}}, {{math|''λ''<sub>12</sub> {{=}} ''π''}}, 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 {{harv|Ehlert|1993}}, determining the physical properties of signals which follow geodesics, etc.  Consider a reference geodesic, parameterized by {{math|''s''}}, and a second geodesic a small distance {{math|''t''(''s'')}} away from it.  {{harvtxt|Gauss|1828}} showed that {{math|''t''(''s'')}} obeys the [[Differential geometry of surfaces#Gauss–Jacobi equation|Gauss-Jacobi equation]]
:<math>\frac{d^2t(s)}{ds^2} + K(s) t(s) = 0, </math>
[[File:Definition of reduced length and geodesic scale.svg|thumb|
Fig. 14.
Definition of reduced length and geodesic scale.]]
where {{math|''K''(''s'')}} is the [[Gaussian curvature]] at {{math|''s''}}. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions
:<math> t(s_2) = C m(s_1,s_2) + D M(s_1,s_2) </math>
where
:<math>
\begin{align}
m(s_1, s_1) &= 0, \quad
\left.\frac{dm(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 1, \\
M(s_1, s_1) &= 1, \quad
\left.\frac{dM(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 0.
\end{align}
</math>
The quantity {{math|''m''(''s''<sub>1</sub>, ''s''<sub>2</sub>) {{=}} ''m''<sub>12</sub>}} is the so-called ''reduced length'', and {{math|''M''(''s''<sub>1</sub>, ''s''<sub>2</sub>) {{=}} ''M''<sub>12</sub>}} is the ''geodesic scale''.{{refn|
{{harvtxt|Bagratuni|1962|loc=§17}} uses the term "coefficient of convergence of ordinates" for the geodesic scale.}}
Their basic definitions are illustrated in Fig.&nbsp;14.

The [[Ellipsoid of revolution#Curvature|Gaussian curvature for an ellipsoid of revolution]] is
:<math>
K = \frac1{\rho\nu} = \frac{\bigl(1-e^2\sin^2\varphi\bigr)^2}{b^2}
  = \frac{b^2}{a^4\bigl(1-e^2\cos^2\beta\bigr)^2}.
</math>
{{harvtxt|Helmert|1880|loc=Eq. (6.5.1.)}} solved the Gauss-Jacobi equation for this case enabling {{math|''m''<sub>12</sub>}} and {{math|''M''<sub>12</sub>}} 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 {{math|''d''&alpha;<sub>1</sub>}} is {{math|''m''<sub>12</sub> ''d''&alpha;<sub>1</sub>}}.  On a closed surface such as an ellipsoid, {{math|''m''<sub>12</sub>}} oscillates about zero.  The point at which {{math|''m''<sub>12</sub>}} becomes zero is the point [[conjugate point|conjugate]] to the starting point.  In order for a geodesic between {{math|''A''}} and {{math|''B''}}, of length {{math|''s''<sub>12</sub>}}, to be a shortest path it must satisfy the Jacobi condition {{harv|Jacobi|1837}} {{harv|Jacobi|1866|loc=§6}} {{harv|Forsyth|1927|loc=§§26–27}} {{harv|Bliss|1916}}, that there is no point conjugate to {{math|''A''}} between {{math|''A''}} and {{math|''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, {{math|{{!}}''σ''<sub>12</sub>{{!}} &le; ''π''}};
* for a prolate ellipsoid, {{math|{{!}}''λ''<sub>12</sub>{{!}} &le; ''π''}}, if {{math|&alpha;<sub>0</sub> &ne; 0}}; if {{math|&alpha;<sub>0</sub> {{=}} 0}}, the supplemental condition {{math|''m''<sub>12</sub> &ge; 0}} is required if {{math|{{!}}''λ''<sub>12</sub>{{!}} {{=}} ''π''}}.

=== Envelope of geodesics ===
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Envelope of geodesics on an oblate ellipsoid.svg
|caption1=Fig. 15.  The envelope of geodesics from a point {{math|''A''}} at {{math|''φ''<sub>1</sub> {{=}} {{val|-30|u=deg}}}}.
|image2=Four geodesics connecting two points on an oblate ellipsoid.svg
|caption2=Fig. 16.  The four geodesics connecting {{math|''A''}} and a point {{math|''B''}}, {{math|''φ''<sub>2</sub> {{=}} {{val|26|u=deg}}}}, {{math|''λ''<sub>12</sub> {{=}} {{val|175|u=deg}}}}.
|header=Geodesics from a single point ({{math|''f'' {{=}} {{frac|10}}}}, {{math|''φ''<sub>1</sub> {{=}} {{val|-30|u=deg}}}})
}}

The geodesics from a particular point {{math|''A''}} if continued past the cut locus form an envelope illustrated in Fig.&nbsp;15. Here the geodesics for which {{math|&alpha;<sub>1</sub>}} is a multiple of {{val|3|u=deg}} 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 {{math|''A''}}; points on the envelope may be computed by finding the point at which {{math|''m''<sub>12</sub> {{=}} 0}} on a geodesic. {{harvtxt|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 {{math|''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 {{math|''A''}} as Fig. 13 and is drawn in the same projection.) The two shorter geodesics are ''stable'', i.e., {{math|''m''<sub>12</sub> &gt; 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 {{math|''σ''<sub>12</sub> &le; ''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 {{math|''A''}}.  Likewise, the geodesic circles are [[involute]]s of the astroid.

=== Area of a geodesic polygon {{anchor|Area|Geodesic polygon}} ===
{{see also|Equal-area map}}

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 {{math|''AFHB''}} in Fig. 1 {{harv|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 {{math|''S''<sub>12</sub>}} of {{math|''AFHB''}} is developed following {{harvtxt|Sjöberg|2006}}.  The area of any closed region of the ellipsoid is
:<math> T = \int dT = \int \frac 1 K \cos\varphi\,d\varphi\,d\lambda,
</math>
where {{math|''dT''}} is an element of surface area and {{math|''K''}} is the [[Gaussian curvature]].  Now the [[Gauss–Bonnet theorem]] applied to a geodesic polygon states
:<math>
\Gamma = \int K \,dT = \int \cos\varphi\,d\varphi\,d\lambda,
</math>
where
:<math>
\Gamma = 2\pi - \sum_j \theta_j
</math>
is the geodesic excess and {{math|&theta;<sub>''j''</sub>}} is the exterior angle at vertex {{math|''j''}}.  Multiplying the equation for {{math|Γ}} by {{math|''R''<sub>2</sub><sup>2</sup>}}, where {{math|''R''<sub>2</sub>}} is the [[Earth radius#Authalic radius|authalic radius]], and subtracting this from the equation for {{math|''T''}} gives
:<math>
\begin{align}
T &=
R_2^2 \,\Gamma + \int \left(\frac 1 K - R_2^2\right)\cos\varphi\,d\varphi\,d\lambda \\
&=R_2^2 \,\Gamma + \int \left(
\frac{b^2}{\bigl(1 - e^2\sin^2\varphi\bigr)^2} - R_2^2
\right)\cos\varphi\,d\varphi\,d\lambda,
\end{align}
</math>
where the [[Spheroid#Curvature|value of {{math|''K''}} for an ellipsoid]] has been substituted.
Applying this formula to the quadrilateral {{math|''AFHB''}}, noting that {{math|Γ {{=}} &alpha;<sub>2</sub> &minus; &alpha;<sub>1</sub>}}, and performing the integral over {{mvar|φ}} gives
:<math>
S_{12}=R_2^2 (\alpha_2-\alpha_1)
+ b^2 \int_{\lambda_1}^{\lambda_2} \left(
\frac1{2\bigl(1 - e^2\sin^2\varphi\bigr)}+
\frac{\tanh^{-1}(e \sin\varphi)}{2e \sin\varphi}
- \frac{R_2^2}{b^2}\right)\sin\varphi
\,d\lambda,
</math>
where the integral is over the geodesic line (so that {{mvar|φ}} is implicitly a function of {{mvar|λ}}). The integral can be expressed as a series valid for small {{math|''f''}} {{harv|Danielsen|1989}} {{harv|Karney|2013|loc=§6 and addendum}}.

The area of a geodesic polygon is given by summing {{math|''S''<sub>12</sub>}} over its edges.  This result holds provided that the polygon does not include a pole; if it does, {{math|2''π'' ''R''<sub>2</sub><sup>2</sup>}} must be added to the sum.  If the edges are specified by their vertices, then a [[Spherical excess|convenient expression]] for the geodesic excess {{math|''E''<sub>12</sub> {{=}} &alpha;<sub>2</sub> &minus; &alpha;<sub>1</sub>}} is
:<math>
\tan\frac{E_{12}}2 =
\frac{\sin\tfrac12 (\beta_2 + \beta_1)}
{\cos\tfrac12 (\beta_2 - \beta_1)} \tan\frac{\omega_{12}}2.
</math>

=== Solution of the direct and inverse problems ===
{{further|Geographical distance#Ellipsoidal-surface formulae}}
{{see also|Azimuth#In geodesy}}
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 {{math|''NEP''}} in Fig.&nbsp;5, [[Spherical trigonometry#Napier's rules for quadrantal triangles|Napier's rules for quadrantal triangles]] can be employed,
:<math>
\begin{align}
\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{align}
</math>
The mapping of the geodesic involves evaluating the integrals for the distance, {{math|''s''}}, and the longitude, {{mvar|λ}}, Eqs. {{EquationNote|3|(3)}} and {{EquationNote|4|(4)}} and these depend on the parameter {{math|&alpha;<sub>0</sub>}}.

Handling the direct problem is straightforward, because {{math|&alpha;<sub>0</sub>}} can be determined directly from the given quantities {{math|''φ''<sub>1</sub>}} and {{math|&alpha;<sub>1</sub>}}; for a sample calculation, see {{harvtxt|Karney|2013}}.

In the case of the inverse problem, {{math|''λ''<sub>12</sub>}} is given; this cannot be easily related to the equivalent spherical angle {{math|&omega;<sub>12</sub>}} because {{math|&alpha;<sub>0</sub>}} is unknown. Thus, the solution of the problem requires that {{math|&alpha;<sub>0</sub>}} be found iteratively ([[root finding]]); see {{harvtxt|Karney|2013}} for details.

In geodetic applications, where {{math|''f''}} is small, the integrals are typically evaluated as a series {{harv|Legendre|1806}} {{harv|Oriani|1806}} {{harv|Bessel|1825}} {{harv|Helmert|1880}} {{harv|Rainsford|1955}} {{harv|Rapp|1993}}.  For arbitrary {{math|''f''}}, the integrals (3) and (4) can be found by numerical quadrature or by expressing them in terms of [[elliptic integrals]] {{harv|Legendre|1806}} {{harv|Cayley|1870}} {{harv|Karney|2024}}.

{{harvtxt|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 {{val|0.1|u=mm}} for the [[WGS84]] ellipsoid; however the inverse method fails to converge for nearly antipodal points.

{{harvtxt|Karney|2013}} continues the expansions to sixth order which suffices to provide full [[double precision]] accuracy for {{math|{{!}}''f''{{!}} &le; {{frac|1|50}}}} and improves the solution of the inverse problem so that it converges in all cases. {{harvtxt|Karney|2013|loc=addendum}} 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 (mathematics)|quadrature]].  By the early 19th century (with the work of [[Adrien-Marie Legendre|Legendre]], [[Barnaba Oriani|Oriani]], [[Friedrich Bessel|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, {{harvtxt|Jacobi|1839}} discovered a constant of the motion allowing this problem to be reduced to quadrature also {{harv|Klingenberg|1982|loc=§3.5}}.{{refn|
This section is adapted from the documentation for GeographicLib {{harv|Karney|2015|loc=[https://geographiclib.sourceforge.io/1.44/triaxial.html Geodesics on a triaxial ellipsoid]}}}}

=== Triaxial ellipsoid coordinate system ===
{{see also|Geodetic coordinates|Ellipsoidal-harmonic coordinates}}
[[File:Triaxial ellipsoid coordinate system.svg|thumb|
Fig. 17.
Triaxial ellipsoidal coordinates.]]

Consider the ellipsoid defined by
:<math>
  h = \frac{X^2}{a^2} + \frac{Y^2}{b^2} + \frac{Z^2}{c^2} = 1,
</math>
where {{math|(''X'',''Y'',''Z'')}} are Cartesian coordinates centered on the ellipsoid and, without loss of generality, {{math|''a'' &ge; ''b'' &ge; ''c'' &gt; 0}}.{{refn|1=
This notation for the semi-axes is incompatible with that used in the
previous section on ellipsoids of revolution in which {{math|''a''}} and
{{math|''b''}} stood for the equatorial radius and polar semi-axis.
Thus the corresponding inequalities are {{math|''a'' {{=}} ''a'' &ge; ''b'' &gt; 0}} for
an oblate ellipsoid and {{math|''b'' &ge; ''a'' {{=}} ''a'' &gt; 0}} for a prolate
ellipsoid.}}
{{anchor|Triaxial ellipsoidal latitude|Triaxial ellipsoidal longitude}}
{{harvtxt|Jacobi|1866|loc=§§26–27}}
employed the '''(triaxial) ellipsoidal coordinates''' (with '''triaxial ellipsoidal latitude''' and '''triaxial ellipsoidal longitude''', {{math|''β'', ''&omega;''}}) defined by
:<math>
\begin{align}
  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{align}
</math>
In the limit {{math|''b'' &rarr; ''a''}}, {{mvar|β}} becomes the [[parametric latitude]] for an oblate ellipsoid, so the use of the symbol {{mvar|β}} is consistent with the previous sections. However, {{mvar|&omega;}} is ''different'' from the spherical longitude defined above.{{refn|The limit {{math|''b'' &rarr; ''c''}} gives a prolate ellipsoid with {{mvar|&omega;}} playing the role of the parametric latitude.}}

Grid lines of constant {{mvar|β}} (in blue) and {{mvar|&omega;}} (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 {{math|''X'' {{=}} 0}} and {{math|''Z'' {{=}} 0}} are shown in red.  The third principal section, {{math|''Y'' {{=}} 0}}, is covered by the lines {{math|''β'' {{=}} &plusmn;90°}} and {{math|&omega; {{=}} 0°}} or {{math|&plusmn;180°}}.  These lines meet at four [[umbilical point]]s (two of which are visible in this figure) where the [[principal curvature|principal radii of curvature]] are equal.  Here and in the other figures in this section the parameters of the ellipsoid are {{math|''a'':''b'':''c'' {{=}} 1.01:1:0.8}}, and it is viewed in an orthographic projection from a point above {{math|''φ'' {{=}} 40°}}, {{math|''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 {{harv|Monge|1796}}.
# They are also intersections of the ellipsoid with [[confocal ellipsoidal coordinates|confocal systems of hyperboloids of one and two sheets]] {{harv|Dupin|1813|loc = [https://archive.org/details/dveloppementsde00dupigoog/page/n315 <!-- pg=297 --> Part 5]}}.
# Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points {{harv|Hilbert|Cohn-Vossen|1952|loc=[https://archive.org/details/geometryimaginat0000dhil/page/188/ p. 188]}}.  For example, the lines of constant {{mvar|β}} in Fig. 17 can be generated with the familiar [[Gardener's ellipse|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 {{harv|Jacobi|1839|loc=Letter to Bessel}},
<blockquote> 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 integral]]s, which become the well known elliptic integrals if 2 axes are set equal.<br>
[[Königsberg]], 28th Dec. '38.
</blockquote>

The solution given by Jacobi {{harv|Jacobi|1839}} {{harv|Jacobi|1866|loc=§28}} is
:<math>
\begin{align}
\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{\bigl(b^2-c^2\bigr)\cos^2\beta - \gamma}} \\[6pt]
&\quad -
\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{\bigl(a^2-b^2\bigr)\sin^2\omega + \gamma}}.
\end{align}
</math>
As Jacobi notes "a function of the angle {{mvar|β}} equals a function of the angle {{mvar|&omega;}}.  These two functions are just Abelian integrals..."  Two constants {{math|&delta;}} and {{math|&gamma;}} appear in the solution.  Typically {{math|&delta;}} 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 {{math|&gamma;}}.  However, for geodesics that start at an umbilical points, we have {{math|&gamma; {{=}} 0}} and {{math|&delta;}} determines the direction at the umbilical point.
The constant {{math|&gamma;}} may be expressed as
:<math>
\gamma = \bigl(b^2-c^2\bigr)\cos^2\beta\sin^2\alpha-
\bigl(a^2-b^2\bigl)\sin^2\omega\cos^2\alpha,
</math>
where {{mvar|&alpha;}} is the angle the geodesic makes with lines of constant {{mvar|&omega;}}. In the limit {{math|''b'' &rarr; ''a''}}, this reduces to {{math|sin&alpha; cos''β'' {{=}} const.}}, the familiar Clairaut relation.  A derivation of Jacobi's result is given by {{harvtxt|Darboux|1894|loc=[https://archive.org/details/leonssurlathorie03darb/page/9 §§583–584]}}; he gives the solution found by {{harvtxt|Liouville|1846}} for general quadratic surfaces.

=== Survey of triaxial geodesics ===
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Circumpolar geodesic on a triaxial ellipsoid case A.svg
|image2=Circumpolar geodesic on a triaxial ellipsoid case B.svg
|header=Circumpolar geodesics, {{math|&omega;<sub>1</sub> {{=}} {{val|0|u=deg}}}}, {{math|&alpha;<sub>1</sub> {{=}} {{val|90|u=deg}}}}.
|caption1=Fig. 18. {{math|''β''<sub>1</sub> {{=}} {{val|45.1|u=deg}}}}.
|caption2=Fig. 19. {{math|''β''<sub>1</sub> {{=}} {{val|87.48|u=deg}}}}.
}}
On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by {{math|''X'' {{=}} 0}}, {{math|''Y'' {{=}} 0}}, and {{math|''Z'' {{=}} 0}}.{{refn|
If {{math|{{frac|''c''|''a''}} &lt; {{frac|1|2}}}}, there are other simple closed geodesics similar to those shown in Figs. 11 and 12
{{harv|Klingenberg|1982|loc=§3.5.19}}.}}
To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section, {{math|''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 {{math|''β''<sub>1</sub> &isin; (&minus;90°, 90°)}}, {{math|&omega;<sub>1</sub> {{=}} 0}}, and {{math|&alpha;<sub>1</sub> {{=}} 90°}}, then {{math|&gamma; &gt; 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 {{math|''β'' {{=}} &plusmn;''β''<sub>1</sub>}}.  Two examples are given in Figs. 18 and 19.  Figure 18 shows practically the same behavior as for an oblate ellipsoid of revolution (because {{math|''a'' &asymp; ''b''}}); compare to Fig. 9.
However, if the starting point is at a higher latitude (Fig. 18) the distortions resulting from {{math|''a'' &ne; ''b''}} are evident.  All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at {{math|''β'' {{=}} ''β''<sub>1</sub>}} {{harv|Chasles|1846}} {{harv|Hilbert|Cohn-Vossen|1952|loc=[https://archive.org/details/geometryimaginat0000dhil/page/223/ pp. 223–224]}}.

{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Transpolar geodesic on a triaxial ellipsoid case A.svg
|image2=Transpolar geodesic on a triaxial ellipsoid case B.svg
|header=Transpolar geodesics, {{math|''β''<sub>1</sub> {{=}} {{val|90|u=deg}}}}, {{math|&alpha;<sub>1</sub> {{=}} {{val|180|u=deg}}}}.
|caption1=Fig. 20. {{math|&omega;<sub>1</sub> {{=}} {{val|39.9|u=deg}}}}.
|caption2=Fig. 21. {{math|&omega;<sub>1</sub> {{=}} {{val|9.966|u=deg}}}}.
}}
If the starting point is {{math|''β''<sub>1</sub> {{=}} 90°}}, {{math|&omega;<sub>1</sub> &isin; (0°, 180°)}}, and {{math|&alpha;<sub>1</sub> {{=}} 180°}}, then {{math|&gamma; &lt; 0}} and the geodesic encircles the ellipsoid in a "transpolar" sense.  The geodesic oscillates east and west of the ellipse {{math|''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 {{math|&omega; {{=}} &omega;<sub>1</sub>}} and {{math|&omega; {{=}} 180° &minus; &omega;<sub>1</sub>}}.
If {{math|''a'' {{=}} ''b''}}, all meridians are geodesics; the effect of {{math|''a'' &ne; ''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 {{math|''b'' &rarr; ''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 {{math|&omega; {{=}} &omega;<sub>1</sub>}}.

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).

[[File:Unstable umbilical geodesic on a triaxial ellipsoid.svg|thumb|
Fig. 22.  An umbilical geodesic,
{{math|''β''<sub>1</sub> {{=}} {{val|90|u=deg}}}},
{{math|&omega;<sub>1</sub> {{=}} {{val|0|u=deg}}}},
{{math|&alpha;<sub>1</sub> {{=}} {{val|135|u=deg}}}}.]]
If the starting point is {{math|''β''<sub>1</sub> {{=}} 90°}}, {{math|&omega;<sub>1</sub> {{=}} 0°}} (an umbilical point), and {{math|&alpha;<sub>1</sub> {{=}} 135°}} (the geodesic leaves the ellipse {{math|''Y'' {{=}} 0}} at right angles), then {{math|&gamma; {{=}} 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 {{math|''Y'' {{=}} 0}} becomes closer to {{val|0|u=deg}} or {{val|180|u=deg}} so that asymptotically the geodesic lies on the ellipse {{math|''Y'' {{=}} 0}} {{harv|Hart|1849}} {{harv|Arnold|1989|loc=[https://archive.org/details/mathematicalmeth0000arno_q8f7/page/265/ 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 geodesics enjoy several interesting properties:
* Through any point on the ellipsoid, there are two umbilical geodesics.
* The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic.
* Whereas the closed geodesics on the ellipses {{math|''X'' {{=}} 0}} and {{math|''Z'' {{=}} 0}} are stable (a geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse {{math|''Y'' {{=}} 0}}, which goes through all 4 umbilical points, is ''exponentially unstable''.  If it is perturbed, it will swing out of the plane {{math|''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 {{math|''A''}} of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on {{math|''β'' {{=}} &minus;''β''<sub>1</sub>}} and the other two on {{math|&omega; {{=}} &omega;<sub>1</sub> + ''π''}}.  The cut locus for {{math|''A''}} is the portion of the line {{math|''β'' {{=}} &minus;''β''<sub>1</sub>}} 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 observations|adjustment]] of [[geodetic networks]] as a two-dimensional problem in spheroidal trigonometry, these problems are now solved by three-dimensional methods {{harv|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]] {{harv|UNCLOS|2006}};
* in the rules of the [[Federal Aviation Administration]] for area navigation {{harv|RNAV|2007}};
* the method of measuring distances in the [[Fédération Aéronautique Internationale|FAI]] Sporting Code {{harv|FAI|2018}}.
* help Muslims find their [[qibla|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 {{harv|Laplace|1799a}} {{harv|Hilbert|Cohn-Vossen|1952|loc= [https://archive.org/details/geometryimaginat0000dhil/page/222/ 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 {{harv|Legendre|1811}} and [[elliptic functions]] {{harv|Weierstrass|1861}};
* the development of differential geometry {{harv|Gauss|1828}} {{harv|Christoffel|1869}};
* methods for solving systems of differential equations by a change of independent variables {{harv|Jacobi|1839}};
* the study of [[caustic (optics)|caustics]] {{harv|Jacobi|1891}};
* investigations into the number and stability of periodic orbits {{harv|Poincaré|1905}};
* in the limit {{math|''c'' &rarr; 0}}, geodesics on a triaxial ellipsoid reduce to a case of [[dynamical billiards]];
* extensions to an arbitrary number of dimensions {{harv|Knörrer|1980}};
* geodesic flow on a surface {{harv|Berger|2010|loc=Chap. 12}}.

== 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]]

== Notes ==
{{reflist|30em}}

== References ==
{{refbegin|30em}}
*{{cite book
|year = 1989
|last = Arnold
|first = V. I.
|author-link = Vladimir Arnold
|title = Mathematical Methods of Classical Mechanics
|edition = 2nd
|publisher = Springer-Verlag
|translator-last1 = Vogtmann
|translator-first1 = K.
|translator-last2 = Weinstein
|translator-first2 = A.
|oclc = 4037141
|isbn = 978-0-387-96890-2
|url-access = registration
|url = https://archive.org/details/mathematicalmeth0000arno_q8f7/
}}
*{{wikicite
|ref = {{harvid|Bagratuni|1962}}
|reference =
{{cite book |ref = none
|year = 1967 |orig-year = 1962
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|title = Course in Spheroidal Geodesy
|doi = 10.5281/zenodo.32371
|oclc = 6150611
}} Translation from Russian of ''Курс сфероидической геодезии'' (Moscow, 1962) by U.S. Air Force ([https://web.archive.org/web/20140102194510/http://www.dtic.mil/docs/citations/AD0650520 FTD-MT-64-390])
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*{{cite book
|year = 2010
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|ref = {{harvid|Bessel|1825}}
|reference = 
{{cite journal |ref = none
|year = 2010 |orig-year = 1825
|last = Bessel |first = F. W. |author-link = Friedrich Bessel
|title = The calculation of longitude and latitude from geodesic measurements
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|volume = 331 |issue = 8 |pages = 852–861
|translator-last1 = Karney |translator-first1 = C. F. F.
|translator-last2 = Deakin |translator-first2 = R. E.
|doi = 10.1002/asna.201011352
|arxiv = 0908.1824
|bibcode = 2010AN....331..852K
|s2cid = 118760590 }} English translation of [http://adsabs.harvard.edu/abs/1825AN......4..241B ''Astron. Nachr.'' '''4''', 241–254 (1825)]. [https://geographiclib.sourceforge.io/bessel-errata.html Errata].
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*{{cite journal
|year = 1916
|last = Bliss |first = G. A. |author-link = Gilbert Ames Bliss
|title = Jacobi's condition for problems of the calculus of variations in parametric form
|journal = Transactions of the American Mathematical Society
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|doi = 10.1090/S0002-9947-1916-1501037-4
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*{{cite book
|year = 1952
|last = Bomford |first = G. |author-link=Guy Bomford
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|chapter = 11, Geometry of the Ellipsoid
|chapter-url = http://old.gps.aau.dk/downloads/notes.pdf 
|oclc = 795014501
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*{{cite journal
|year = 1870
|last = Cayley |first = A. |author-link = Arthur Cayley
|title = On the geodesic lines on an oblate spheroid
|journal = Philosophical Magazine |series=4th series
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|doi = 10.1080/14786447008640411
|url = https://archive.org/details/londonedinburghd440lond/page/329
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|ref = {{harvid|Chasles|1846}}
|reference =
{{cite journal |ref = none
|year = 1846
|last = Chasles |first = M. |author-link = Michel Chasles
|title = Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré
|language = fr
|trans-title = Geodesic lines and the lines of curvature of the surfaces of the second degree
|journal = Journal de Mathématiques Pures et Appliquées
|volume = 11 |pages = 5–20
|url = https://archive.org/details/s1journaldemat11liou/page/5/
}} [http://sites.mathdoc.fr/JMPA/PDF/JMPA_1846_1_11_A2_0.pdf PDF].
}}
*{{cite journal
|year = 1869
|last = Christoffel |first = E. B. |author-link = Elwin Bruno Christoffel
|title = Allgemeine Theorie der geodätischen Dreiecke
|language = de
|trans-title = General theory of geodesic triangles
|journal = Abhandlungen Königlichen Akademie der Wissenschaft zu Berlin
|pages = 119–176
|url = https://archive.org/details/abhandlungenderk1868deut/page/n274
}}
*{{cite journal
|year = 1735
|last = Clairaut |first = A. C. |author-link = Alexis Claude Clairaut
|title = Détermination géometrique de la perpendiculaire à la méridienne tracée par M. Cassini
|language = fr
|trans-title = Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini
|journal = Mémoires de l'Académie Royale des Sciences de Paris 1733
|pages = 406–416
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|language = fr
|trans-title = Lessons on the general theory of surfaces
|publisher = Gauthier-Villars |location = Paris
|url = https://archive.org/details/leonssurlathorie03darb/
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|year = 1813
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|trans-title = Developments in geometry
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|year = 1993
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|institution = [[:de:Deutsche Geodätische Kommission|Deutsche Geodätische Kommission]]
|series = Reihe B: Angewandte Geodäsie, Heft Nr. 292
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|title = Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits
|language = fr
|trans-title = Elements of spheroidal trigonometry taken from the method of maxima and minima
|journal = Mémoires de l'Académie Royale des Sciences de Berlin 1753
|pages = 258–293
|volume = 9
|url = https://books.google.com/books?id=QIIfAAAAYAAJ&pg=PA258
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|year = 2018
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|year = 1837
|last = Jacobi |first = C. G. J. |author-link = Carl Gustav Jacob Jacobi
|title = Zur Theorie der Variations-Rechnung und der Differential-Gleichungen
|language = de
|trans-title = The theory of the calculus of variations and of differential equations
|journal = Journal für die Reine und Angewandte Mathematik
|volume = 1837 |issue = 17 |pages = 68–82
|doi = 10.1515/crll.1837.17.68
|s2cid = 119469290 |url = https://books.google.com/books?id=FKvxAAAAMAAJ&pg=PA68
}}
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|last = Jacobi |first = C. G. J. |author-link = Carl Gustav Jacob Jacobi
|title = Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution
|language = de
|trans-title = The geodesic on an ellipsoid and various applications of a remarkable analytical substitution
|journal = Journal für die Reine und Angewandte Mathematik
|volume = 1839 |issue = 19 |pages = 309–313
|doi = 10.1515/crll.1839.19.309
|s2cid = 121670851 |url = https://books.google.com/books?id=RbwGAAAAYAAJ&pg=PA309
}} [https://books.google.com/books?id=_09tAAAAMAAJ&pg=PA385 Letter to Bessel], Dec. 28, 1838. [http://sites.mathdoc.fr/JMPA/PDF/JMPA_1841_1_6_A20_0.pdf French translation] (1841).
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|editor = A. Clebsch |editor-link = Alfred Clebsch
|translator-last = Balagangadharan |translator-first = K.
|isbn = 978-81-85931-91-3
|mr = 2569315
|oclc = 440645889
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|ref = {{harvid|Jacobi|1891}}
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|title = Jacobi's Gesammelte Werke
|chapter = Über die Curve, welche alle von einem Punkte ausgehenden geodätischen Linien eines Rotationsellipsoides berührt
|publisher = Reimer |location = Berlin
|language = de
|trans-chapter = The envelope of geodesic lines emanating from a single point on an ellipsoid
|pages = 72–87
|editor = K. T. W. Weierstrass|editor-link = Karl Weierstrass
|volume = 7
|url = https://books.google.com/books?id=_09tAAAAMAAJ&pg=PA72
|oclc = 630416023
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|last2 = Eggert |first2 = O. |author-link2 = Otto Eggert
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|volume = 3
|issue = 2 |publisher = Army Map Service |location = Washington, DC
|translator-last = Carta |translator-first = M. W.
|doi = 10.5281/zenodo.35316
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|ref = {{harvid|Karney|2013}}
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|ref = {{harvid|Laplace|1799a}} |year = 1829 |orig-year = 1799a
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|translator-link = Nathaniel Bowditch
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|last = Laplace |first = P. S. |author-link = Pierre-Simon Laplace
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*{{cite journal
|year = 1806
|last = Legendre |first = A. M. |author-link = Adrien-Marie Legendre
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|language = fr
|trans-title = Analysis of spheroidal triangles
|journal = Mémoires de l'Institut National de France
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|title = Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures
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|trans-title = Exercises in Integral Calculus
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|trans-title = On special cases where the equations of motion of a point particle can be integrated
|journal = Journal de Mathématiques Pures et Appliquées
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|chapter = Sur les lignes de courbure de la surface de l'ellipsoïde
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|trans-chapter = On the lines of curvature on the surface of the ellipsoid
|title = Application de l'Analyse à la Géometrie
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|title = The Mathematical Principles of Natural Philosophy
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*{{cite journal
|year = 1806
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|title = Elementi di trigonometria sferoidica, Pt. 1
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|year = 1808
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|title = Elementi di trigonometria sferoidica, Pt. 2
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|last = Oriani |first = B. |author-link = Barnaba Oriani
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*{{cite journal
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|title = Sur les lignes géodésiques des surfaces convexes
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*{{cite journal
|year = 1955
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|title = Long geodesics on the ellipsoid
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*{{cite tech report
|year = 2007
|last = RNAV |author-link = RNAV
|title = Order 8260.54A, The United States Standard for Area Navigation
|institution = U.S. Federal Aviation Administration
|location = Washington, D.C.
|url = https://www.faa.gov/documentLibrary/media/Order/8260.54A.pdf
|chapter = Appendix 2. 
}}
*{{cite journal
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|title = Determination of areas on the plane, sphere and ellipsoid
|journal = Survey Review
|volume = 38 |issue = 301 |pages = 583–593
|doi = 10.1179/003962606780732100
}}
*{{cite tech report
|year = 2006
|last = UNCLOS
|author-link = UNCLOS
|title = A Manual on Technical Aspects of the United Nations Convention on the Law of the Sea, 1982
|institution = International Hydrographic Bureau
|edition = 4th
|location = Monaco
|url = http://www.iho.int/iho_pubs/CB/C-51_Ed4-EN.pdf
|access-date = 2013-08-15
|archive-date = 2013-05-24
|archive-url = https://web.archive.org/web/20130524163050/http://www.iho.int/iho_pubs/CB/C-51_Ed4-EN.pdf
|url-status = dead
}}
*{{wikicite
|ref = {{harvid|Vincenty|1975}}
|reference = 
{{cite journal | ref = none
|year = 1975
|last = Vincenty |first = T. |author-link = Thaddeus Vincenty
|title = Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations
|journal = Survey Review
|volume = 23 |issue = 176 |pages = 88–93
|doi = 10.1179/sre.1975.23.176.88
|bibcode = 1975SurRv..23...88V
|url = https://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
}}  Addendum: Survey Review '''23''' (180): 294 (1976).
}}
*{{cite tech report
|year = 1978
|last1 = Vincenty |first1 = T. |author-link1 = Thaddeus Vincenty
|last2 = Bowring |first2 = B. R.
|title = Application of three-dimensional geodesy to adjustments of horizontal networks
|institution = NOAA
|number = NOS NGS-13
|url = https://www.ngs.noaa.gov/PUBS_LIB/ApplicationOfThreeDimensionalGeodesyToAdjustmentsOfHorizontalNetworks_TM_NOS_NGS13.pdf
}}
*{{cite journal
|year = 1861
|last = Weierstrass |first = K. T. W. |author-link = Karl Weierstrass
|title = Über die geodätischen Linien auf dem dreiaxigen Ellipsoid
|language = de
|trans-title = Geodesic lines on a triaxial ellipsoid
|journal = Monatsberichte der Königlichen Akademie der Wissenschaft zu Berlin
|pages = 986–997
|url = https://archive.org/details/mathematischewer01weieuoft/page/257
}}
{{refend}}

== External links ==
* [https://geographiclib.sourceforge.io/geodesic-papers/biblio.html Online geodesic bibliography] of books and articles on geodesics on ellipsoids.
* [https://dx.doi.org/10.5281/zenodo.32156 Test set for geodesics], a set of 500000 geodesics for the WGS84 ellipsoid, computed using high-precision arithmetic.
* [https://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html NGS tool] implementing {{harvtxt|Vincenty|1975}}.
* [https://proj.org/apps/geod.html geod(1)], man page for the [[PROJ]] utility for geodesic calculations.
* [https://geographiclib.sourceforge.io/ GeographicLib implementation] of {{harvtxt|Karney|2013}}.
* [https://geographiclib.sourceforge.io/scripts/geod-google.html Drawing geodesics on Google Maps.]

[[Category:Geodesy]]
[[Category:Geodesic (mathematics)]]
[[Category:Differential geometry]]
[[Category:Calculus of variations]]
[[Category:Curves]]
[[Category:Ellipsoids]]