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{{User:Alexander_Misel/沙盒}}
[[File:Long geodesic on an oblate ellipsoid.svg|thumb|200px|A geodesic on an oblate ellipsoid]]
The study of '''geodesics on an ellipsoid''' arose in connection with geodesy
specifically with the solution of [[三角測量|triangulation network]]s.  The
[[figure of the Earth]] is well approximated by an
''[[類球面|oblate ellipsoid]]'', a slightly flattened sphere.  A ''[[测地线|geodesic]]''
is the shortest path between two points on a curved surface, i.e., the analogue
of a [[直线|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}}.

[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|150px|[[艾萨克·牛顿|Isaac Newton]]]]
[[File:Leonhard Euler.jpg|thumb|150px|[[萊昂哈德·歐拉|Leonhard Euler]]]]
If the Earth is treated as a [[球面|sphere]], the geodesics are
[[大圆|great circles]] (all of which are closed) and the problems reduce to
ones in [[球面三角學|spherical trigonometry]].  However, {{harvtxt|Newton|1687}}
showed that the effect of the rotation of the Earth results in its
resembling a slightly oblate ellipsoid and, in this case, the
[[赤道|equator]] and the [[经线|meridians]] are the only
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 [[椭球|triaxial ellipsoid]] (with
three distinct semi-axes), then only three geodesics are closed and one
of these is unstable.

The problems in geodesy are usually reduced to two main cases: the
''direct problem'', given a starting point and an initial heading, find
the position after traveling a certain distance along the geodesic; and
the ''inverse problem'', given two points on the ellipsoid find the
connecting geodesic and hence the shortest distance between them.
Because the flattening of the Earth is small, the geodesic distance
between two points on the Earth is well approximated by the great-circle
distance using the
[[地球半径|mean Earth radius]]&mdash;the relative error is
less than 1%.  However, the course of the geodesic can differ
dramatically from that of the great circle.  As an extreme example,
consider two points on the equator with a longitude difference of
179&deg;59&prime;; while the connecting great circle follows the
equator, the shortest geodesics pass within
180&nbsp;km of either [[两极|pole]] (the
flattening makes two symmetric paths passing close to the poles shorter
than the route along the equator).

Aside from their use in geodesy and related fields such as navigation,
terrestrial geodesics arise in the study of the propagation of signals
which are confined (approximately) to the surface of the Earth, for
example, sound waves in the ocean {{harv|Munk|Forbes|1989}} and the
radio signals from lightning {{harv|Casper|Bent|1991}}.  Geodesics are
used to define some [[maritime boundaries]], which in turn determine the
allocation of valuable resources as such
[[mineral rights|oil and mineral rights]].  Ellipsoidal geodesics also
arise in other applications; for example, the propagation of radio waves
along the fuselage of an aircraft, which can be roughly modeled as a
[[類球面|prolate (elongated)]] ellipsoid
{{harv|Kim|Burnside|1986}}.

Geodesics are an important intrinsic characteristic of curved surfaces.
The sequence of progressively more complex surfaces, the sphere, an
[[類球面|ellipsoid of revolution]], and a triaxial ellipsoid, provide a useful
family of surfaces for investigating the general theory of surfaces.
Indeed, Gauss's work on the
[[:de:Gaußsche Landesaufnahme|survey of Hanover]], which involved
geodesics on an oblate ellipsoid, was a key motivation for his
[[Differential geometry of surfaces|study of surfaces]]
{{harv|Gauss|1828}}.  Similarly, the existence of three closed geodesics
on a triaxial ellipsoid turns out to be a general property of
[[曲面|closed]], [[單連通|simply connected]] surfaces; this was
conjectured by {{harvtxt|Poincaré|1905}} and proved by
{{harvtxt|Lyusternik|Schnirelmann|1929}}
{{harv|Klingenberg|1982|loc=&sect;3.7}}.

== Geodesics on an ellipsoid of revolution ==
There are several ways of defining geodesics
{{harv|Hilbert|Cohn-Vossen|1952|pp=220&ndash;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
[[测地曲率|geodesic curvature]]&mdash;i.e., the analogue of [[直线|straight lines]] on a
curved surface.  This definition encompasses geodesics traveling so far
across the ellipsoid's surface (somewhat less than half the
circumference) that other distinct routes require less distance.
Locally, these geodesics are still identical to the shortest distance
between two points.

By the end of the 18th century, an ellipsoid of revolution (the term
[[類球面|spheroid]] is also used) was a well-accepted approximation to the
[[figure of the Earth]].  The adjustment of [[三角測量|triangulation network]]s
entailed reducing all the measurements to a [[参考椭球体|reference ellipsoid]] and
solving the resulting two-dimensional problem as an exercise in
spheroidal trigonometry {{harv|Bomford|1952|loc=Chap. 3}}.

[[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: ''A'' at latitude
&phi;<sub>1</sub> and longitude &lambda;<sub>1</sub> and
''B'' at latitude &phi;<sub>2</sub> and longitude
&lambda;<sub>2</sub> (see Fig.  1).  The connecting geodesic
(from ''A'' to ''B'') is ''AB'', of length
''s''<sub>12</sub>, which has [[方位角|azimuth]]s &alpha;<sub>1</sub> and
&alpha;<sub>2</sub> at the two endpoints.{{refn|
Here &alpha;<sub>2</sub> is the ''forward'' azimuth at ''B''.
Some authors calculate the ''back'' azimuth instead; this is given by
&alpha;<sub>2</sub>&nbsp;&plusmn;&nbsp;&pi;.}}  The two geodesic problems usually
considered are:
# the ''direct geodesic problem'' or ''first geodesic problem'', given ''A'', &alpha;<sub>1</sub>, and ''s''<sub>12</sub>, determine ''B'' and &alpha;<sub>2</sub>;
# the ''inverse geodesic problem'' or ''second geodesic problem'', given ''A'' and ''B'', determine ''s''<sub>12</sub>, &alpha;<sub>1</sub>, and &alpha;<sub>2</sub>.
As can be seen from Fig. 1, these problems involve solving the triangle
''NAB'' given one angle, &alpha;<sub>1</sub> for the direct
problem and &lambda;<sub>12</sub>&nbsp;=&nbsp;&lambda;<sub>2</sub>&nbsp;&minus;&nbsp;&lambda;<sub>1</sub> for the
inverse problem, and its two adjacent sides.
In the course of the 18th century these problems were elevated
(especially in literature in the German language) to the
[[:de:geodätische Hauptaufgabe|''principal geodesic problems'']]
{{harv|Hansen|1865|
p=[https://books.google.com/books?id=WlsOAAAAYAAJ&pg=PA69 69]}}.

For a sphere the solutions to these problems are simple exercises in
[[球面三角學|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]].)

[[File:Alexis Clairault.jpg|thumb|150px|[[亚历克西斯·克劳德·克莱罗|Alexis Clairaut]]]]
[[File:Barnaba Oriani.jpg|thumb|150px|[[Barnaba Oriani]]]]
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}}.{{refn|
This prompted a courteous note by {{harvtxt|Oriani|1826}} noting his
previous work, of which, presumably, Bessel was unaware, and also a
thinly veiled accusation of [[抄袭|plagiarism]] from {{harvtxt|Ivory|1826}}
(his phrase was "second-hand from Germany"), which resulted in an angry
rebuttal by {{harvtxt|Bessel|1827}}.}}

Much of the early work on these problems was carried out by
mathematicians&mdash;for example, [[阿德里安-马里·勒让德|Legendre]],
[[弗里德里希·威廉·贝塞尔|Bessel]], and [[卡爾·弗里德里希·高斯|Gauss]]&mdash;who
were also heavily involved in the practical aspects of [[測量|surveying]].
Beginning in about 1830, the disciplines diverged: those with an
interest in geodesy concentrated on the practical aspects such as
approximations suitable for field work, while mathematicians pursued the
solution of geodesics on a triaxial ellipsoid, the analysis of the
stability of closed geodesics, etc.

During the 18th century geodesics were typically referred to as "shortest
lines".{{refn|
{{harvtxt|Clairaut|1735}} uses the [[circumlocution]] "perpendiculars to
the meridian"; this refers to Cassini's proposed map projection for
France {{harv|Cassini|1735}} where one of the coordinates was the
distance from the Paris meridian.}}
The term "geodesic line" 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}},
<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.{{refn|
{{harvtxt|Kummell|1883}} attempted to introduce the word "brachisthode"
for geodesic.  This effort failed.}}

This section treats the problem on an ellipsoid of revolution (both
oblate and prolate).  The problem on a triaxial ellipsoid is covered in
the next section.

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

=== Equations for a geodesic ===
[[File:Friedrich Wilhelm Bessel (1839 painting).jpg|thumb|150px|[[弗里德里希·威廉·贝塞尔|Friedrich Bessel]]]]
{{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; these allow the
geodesics of any length to be computed accurately.  The following
derivation closely follows that of {{harvtxt|Bessel|1825}}.
{{harvtxt|Bagratuni|1962|loc=&sect;15}},
{{harvtxt|Krakiwsky|Thomson|1974|loc=&sect;4}},
{{harvtxt|Rapp|1993|loc=&sect;1.2}}, and
{{harvtxt|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''&nbsp;=&nbsp;(''a''&nbsp;&minus;&nbsp;''b'')/''a'', the eccentricity
''e''<sup>2</sup>&nbsp;=&nbsp;''f''(2&nbsp;&minus;&nbsp;''f''), and the second eccentricity
''e''&prime;&nbsp;=&nbsp;''e''/(1&nbsp;&minus;&nbsp;''f'').  (In most applications in geodesy, the
ellipsoid is taken to be oblate, ''a''&nbsp;&gt;&nbsp;''b''; however, the theory
applies without change to prolate ellipsoids, ''a''&nbsp;&lt;&nbsp;''b'', in
which case ''f'', ''e''<sup>2</sup>, and ''e''&prime;<sup>2</sup> 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 &alpha;, then ''ds''
is related to ''d''&phi; and ''d''&lambda; by
:(1)<math>{\color{white}.}\qquad
\cos\alpha\,ds = \rho\,d\phi = - dR/\sin\phi, \quad
\sin\alpha\,ds = R\,d\lambda,</math>
where &rho; is the
[[地球半径|meridional radius of curvature]],
''R''&nbsp;=&nbsp;&nu;&nbsp;cos&phi; is the radius of the circle of latitude
&phi;, and &nu; is the
[[地球半径|normal radius of curvature]].
The elementary segment is therefore given by
:<math>ds^2 = \rho^2\,d\phi^2 + R^2\,d\lambda^2</math>
or
:<math>\begin{align}ds &= \sqrt{\rho^2\phi'^2 + R^2}\,d\lambda \\
&\equiv L(\phi,\phi')\,d\lambda,
\end{align}
</math>
where &phi;&prime;&nbsp;=&nbsp;''d''&phi;/''d''&lambda; and ''L'' depends on
&phi; through &rho;(&phi;) and
''R''(&phi;).  The length of an arbitrary path between
(&phi;<sub>1</sub>,&nbsp;&lambda;<sub>1</sub>) and (&phi;<sub>2</sub>,&nbsp;&lambda;<sub>2</sub>) is
given by
:<math> s_{12} =
\int_{\lambda_1}^{\lambda_2} L(\phi, \phi')\,d\lambda,</math>
where &phi; is a function of &lambda; satisfying
&phi;(&lambda;<sub>1</sub>)&nbsp;=&nbsp;&phi;<sub>1</sub> and
&phi;(&lambda;<sub>2</sub>)&nbsp;=&nbsp;&phi;<sub>2</sub>.  The shortest path or geodesic
entails finding that function &phi;(&lambda;) which minimizes
''s''<sub>12</sub>.  This is an exercise in the
[[变分法|calculus of variations]] and the minimizing condition is given by the
[[貝爾特拉米等式|Beltrami identity]],
:<math>L - \phi' \frac{\partial L}{\partial \phi'} = \text{const.}
</math>
[[File:Construction for parametric latitude.svg|thumb|
Fig. 4.
The geometric construction for parametric latitude, &beta;.  A point
''P'' at latitude &phi; on the meridian (red) is mapped to a point
''P&prime;'' on a sphere of radius ''a'' (shown as a blue circle) by
keeping the radius ''R'' constant.]]
Substituting for ''L'' and using Eqs. (1) gives
:<math>R\sin\alpha = \text{const.}</math>
{{harvtxt|Clairaut|1735}} first found this [[Clairaut's relation|relation]],
using a geometrical construction; a similar derivation is presented by
{{harvtxt|Lyusternik|1964|loc=&sect;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
[[角动量|conservation of angular momentum]] for a particle on a surface of
revolution.  A similar proof is given by
{{harvtxt|Bomford|1952|loc=&sect;8.06}}.}}  Differentiating this
relation and manipulating the result gives
{{harv|Jekeli|2012|loc=Eq. (2.95)}}
:<math>d\alpha=\sin\phi\,d\lambda.</math>
This, together with Eqs. (1), leads to a system of
[[常微分方程|ordinary differential equations]] for a geodesic
{{harv|Borre|Strang|2012|loc=Eqs. (11.71) and (11.76)}}
:(2)<math>{\color{white}.}\qquad\displaystyle
\frac{d\phi}{ds} = \frac{\cos\alpha}{\rho};\quad
\frac{d\lambda}{ds} = \frac{\sin\alpha}{\nu\cos\phi};\quad
\frac{d\alpha}{ds} = \frac{\tan\phi\sin\alpha}{\nu}.</math>
We can express ''R'' in terms of the
[[纬度|parametric latitude]],
&beta;,{{refn|1=
In terms of &beta;, the element of distance on the ellipsoid
is given by
''ds''<sup>2</sup>&nbsp;=&nbsp;(''a''<sup>2</sup>&nbsp;sin<sup>2</sup>&beta;<sup>2</sup>&nbsp;+&nbsp;''b''<sup>2</sup>&nbsp;cos<sup>2</sup>&beta;)&nbsp;d&beta;<sup>2</sup>&nbsp;+&nbsp;''a''<sup>2</sup>&nbsp;cos<sup>2</sup>&beta;&nbsp;d&lambda;<sup>2</sup>.}}
using
:<math>R = a\cos\beta</math>
(see Fig. 4 for the geometrical construction), 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. 5.  Geodesic problem mapped to the auxiliary sphere.
|image2=Geodesic problem mapped to the auxiliary sphere.svg
|caption2=Fig. 6. The elementary geodesic problem on the auxiliary sphere.
}}
This is the [[球面三角學|sine rule]] of spherical
trigonometry relating two sides of the triangle ''NAB'' (see
Fig. 5), ''NA''&nbsp;=&nbsp;&frac12;&pi;&nbsp;&minus;&nbsp;&beta;<sub>1</sub>, and
''NB''&nbsp;=&nbsp;&frac12;&pi;&nbsp;&minus;&nbsp;&beta;<sub>2</sub> and their opposite angles
''B''&nbsp;=&nbsp;&pi;&nbsp;&minus;&nbsp;&alpha;<sub>2</sub> and ''A''&nbsp;=&nbsp;&alpha;<sub>1</sub>.

In order to find the relation for the third side
''AB''&nbsp;=&nbsp;&sigma;<sub>12</sub>, the ''spherical arc length'', and included
angle ''N''&nbsp;=&nbsp;&omega;<sub>12</sub>, the ''spherical longitude'', it is
useful to consider the triangle ''NEP'' representing a geodesic
starting at the equator; see Fig.  6.  In this figure, the
variables referred to the auxiliary sphere are shown with the
corresponding quantities for the ellipsoid shown in parentheses.
Quantities without subscripts refer to the arbitrary point
''P''; ''E'', the point at which the geodesic crosses
the equator in the northward direction, is used as the origin for
&sigma;, ''s'' and &omega;.

[[File:Differential element of a geodesic on a sphere.svg|thumb|
Fig. 7.
Differential element of a geodesic on a sphere.]]
If the side ''EP'' is extended by
moving ''P'' infinitesimally (see Fig. 7), we
obtain
:(3)<math>{\color{white}.}\qquad
\cos\alpha\,d\sigma = d\beta, \quad
\sin\alpha\,d\sigma = \cos\beta\,d\omega.</math>
Combining Eqs. (1) and (3) gives differential
equations for ''s'' and &lambda;
:<math>\frac1a\frac{ds}{d\sigma}
= \frac{d\lambda}{d\omega}
= \frac{\sin\beta}{\sin\phi}.</math>

Up to this point, we have not made use of the specific equations for an
ellipsoid, and indeed the derivation applies to an arbitrary surface of
revolution.{{refn|It may be useful to impose the restriction that the
surface have a positive curvature everywhere so that the latitude be
single valued function of ''Z''.}}
Bessel now specializes to an ellipsoid in which
''R'' and ''Z'' are related by
:<math>\frac{R^2}{a^2} + \frac{Z^2}{b^2} = 1,</math>
where ''Z'' is the height above the equator (see Fig. 4).
Differentiating this and setting
''dR''/''dZ''&nbsp;=&nbsp;&minus;sin&phi;/cos&phi; gives
:<math>\frac{R\sin\phi}{a^2} - \frac{Z\cos\phi}{b^2} = 0;</math>
eliminating ''Z'' from these equations, we obtain
:<math>\frac Ra = \cos\beta = \frac{\cos\phi}{\sqrt{1-e^2\sin^2\phi}}.</math>
This relation between &beta; and &phi; can be
written as
:<math>\tan\beta = \sqrt{1-e^2} \tan\phi = (1-f) \tan\phi,</math>
which is the normal definition of the
[[纬度|parametric latitude]]
on an ellipsoid.  Furthermore, we have
:<math>\frac{\sin\beta}{\sin\phi} = \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 &sigma; as the independent
parameter{{refn| Other choices of independent parameter are possible.
In particular many authors use the vertex of a geodesic (the point of
maximum latitude) as the origin for &sigma;.}} in both of
these differential equations and thereby to express ''s'' and
&lambda; as integrals.  Applying the sine rule to the vertices
''E'' and ''G'' in the spherical triangle
''EGP'' in Fig. 6 gives
:<math>\sin\beta = \sin\beta(\sigma;\alpha_0) =
\cos\alpha_0 \sin\sigma,</math>
where &alpha;<sub>0</sub> is the azimuth at ''E''.
Substituting this into the equation for ''ds''/''d''&sigma; and
integrating the result gives
:(4)<math>{\color{white}.}\qquad
\begin{align}
\frac sb &= \int_0^\sigma
\frac{\sqrt{1 - e^2 \cos^2\beta(\sigma';\alpha_0)}}{1-f}\,d\sigma'\\
&= \int_0^\sigma \sqrt{1 + k^2 \sin^2\sigma'}\,d\sigma',
\end{align}
</math>
where
:<math>k = e'\cos\alpha_0,</math>
and the limits on the integral are chosen so that
''s''(&sigma;&nbsp;=&nbsp;0)&nbsp;=&nbsp;0.  {{harvtxt|Legendre|1811|p=180}} pointed out
that the equation for ''s'' is the same as the equation for the
[[Meridian arc#Meridian distance on the ellipsoid|arc on an ellipse]]
with semi-axes ''b''(1&nbsp;+&nbsp;''e''&prime;<sup>2</sup>&nbsp;cos<sup>2</sup>&alpha;<sub>0</sub>)<sup>1/2</sup> and
''b''.  In order to express the equation for
&lambda; in terms of &sigma;, we write
:<math>d\omega = \frac{\sin\alpha_0}{\cos^2\beta}\,d\sigma,</math>
which follows from Eq. (3) and Clairaut's relation.
This yields
:(5)<math>{\color{white}.}\qquad
\begin{align}
\lambda - \lambda_0 &= (1-f) \sin\alpha_0
\int_0^\sigma\frac
{\sqrt{1 + k^2\sin^2\sigma'}}
{1 - \cos^2\alpha_0\sin^2\sigma'}\,d\sigma'\\
&= \omega - \sin\alpha_0
\int_0^\sigma\frac
{e^2}{1 + \sqrt{1 - e^2\cos^2\beta(\sigma';\alpha_0)}}\,d\sigma'\\
&= \omega - f\sin\alpha_0
\int_0^\sigma\frac
{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}}
\,d\sigma',
\end{align}</math>
and the limits on the integrals are chosen
so that &lambda;&nbsp;=&nbsp;&lambda;<sub>0</sub> at the equator crossing,
&sigma;&nbsp;=&nbsp;0.

In using these integral relations, we allow &sigma; to
increase continuously (not restricting it to a range
[&minus;&pi;,&nbsp;&pi;], for example) as the great circle,
resp. geodesic, encircles the auxiliary sphere, resp. ellipsoid.  The
quantities &omega;, &lambda;, and ''s''
are likewise allowed to increase without limit.  Once the problem is
solved, &lambda; can be reduced to the conventional range.

This completes the solution of the path of a geodesic using the
auxiliary sphere.  By this device a great circle can be mapped exactly
to a geodesic on an ellipsoid of revolution.  However, because the
equations for ''s'' and &lambda; in terms of the
spherical quantities depend on &alpha;<sub>0</sub>, the mapping is not
a consistent mapping of the surface of the sphere to the ellipsoid or
vice versa; instead, it should be viewed merely as a convenient tool for
solving for a particular geodesic.

There are also several ways of approximating geodesics on an ellipsoid
which usually apply for sufficiently short lines
{{harv|Rapp|1991|loc=&sect;6}}; however, these are typically comparable
in complexity to the method for the exact solution given above
{{harv|Jekeli|2012|loc=&sect;2.1.4}}.

=== Behavior of geodesics ===
[[File:Closed geodesics on an ellipsoid of revolution.svg|thumb|
Fig. 8.  Meridians and the equator are the only closed
geodesics.  (For the very flattened ellipsoids, there are other closed
geodesics; see Figs. 13 and 14).]]
{{multiple image
|align=right
|direction=vertical
|width=220
|image1=Latitude vs longitude for geodesic on an oblate ellipsoid.svg
|caption1=Fig. 9.  Latitude as a function of longitude for a single cycle of the geodesic from one northward equatorial crossing to the next.
|image2=Long geodesic on an oblate ellipsoid.svg
|caption2=Fig. 10.  Following the geodesic on the ellipsoid for about 5 circuits.
|image3=Really long geodesic on an oblate ellipsoid.svg
|caption3=Fig. 11.  The same geodesic after about 70 circuits.
|header=Geodesic on an oblate ellipsoid (''f'' = 1/50) with &alpha;<sub>0</sub> = 45&deg;.
}}
[[File:Long geodesic on a prolate ellipsoid.svg|thumb|
Fig. 12.
Geodesic on a prolate ellipsoid (''f'' = &minus;1/50) with &alpha;<sub>0</sub> = 45&deg;.  Compare with
Fig. 10.]]
Before solving for the geodesics, it is worth reviewing their behavior.
Fig. 8 shows the simple closed geodesics which consist of the
meridians (green) and the equator (red).  (Here the qualification
"simple" means that the geodesic closes on itself without an intervening
self-intersection.)  This follows from the equations for the geodesics
given in the previous section.

For meridians, we have &alpha;<sub>0</sub>&nbsp;=&nbsp;0 and Eq. (5)
becomes &lambda;&nbsp;=&nbsp;&omega;&nbsp;+&nbsp;&lambda;<sub>0</sub>, i.e., the longitude will
vary the same way as for a sphere, jumping by &pi; each time
the geodesic crosses the pole.  The distance, Eq. (4), reduces to
the length of an arc of an ellipse with semi-axes ''a'' and
''b'' (as expected), expressed in terms of parametric latitude,
&beta;.

The equator (&beta;&nbsp;=&nbsp;0</sub> on the auxiliary sphere,
&phi;&nbsp;=&nbsp;0 on the ellipsoid) corresponds to
&alpha;<sub>0</sub>&nbsp;=&nbsp;&frac12;&pi;.  The distance reduces to the arc of a
circle of radius ''b'' (and ''not'' ''a''),
''s''&nbsp;=&nbsp;''b''&sigma;, while the longitude simplifies to
&lambda;&nbsp;=&nbsp;(1&nbsp;&minus;&nbsp;''f'')&sigma;&nbsp;+&nbsp;&lambda;<sub>0</sub>.  A geodesic that is nearly
equatorial will intersect the equator at intervals of
&pi;''b''.  As a consequence, the maximum length of a
equatorial geodesic which is also a shortest path is &pi;''b''
on an oblate ellipsoid (on a prolate ellipsoid, the maximum length is
&pi;''a'').

All other geodesics are typified by Figs. 9 to 11.
Figure 9 shows latitude as a function of longitude for a geodesic
starting on the equator with &alpha;<sub>0</sub> = 45&deg;.  A full
cycle of the geodesic, from one northward crossing of the equator to the
next, is shown.  The equatorial crossings are called ''nodes'' and the
points of maximum or minimum latitude are called ''vertices''; the
vertex latitudes are given by
|&beta;|&nbsp;=&nbsp;&plusmn;(&frac12;&pi;&nbsp;&minus;&nbsp;|&alpha;<sub>0</sub>|).
The latitude is an odd, resp. even, function of the longitude about the
nodes, resp. vertices.  The geodesic completes one full oscillation in
latitude before the longitude has increased by 360&deg;.
Thus, on each successive northward crossing of the equator (see
Fig. 10), &lambda; falls short of a full circuit of
the equator by approximately 2&pi;&nbsp;''f''&nbsp;sin&alpha;<sub>0</sub> (for a
prolate ellipsoid, this quantity is negative and &lambda;
completes more that a full circuit; see Fig. 12).  For nearly all
values of &alpha;<sub>0</sub>, the geodesic will fill that portion of
the ellipsoid between the two vertex latitudes (see
Fig. 11).

{{multiple image
|align=right
|direction=vertical
|width=220
|image1=Non-standard closed geodesics on an ellipsoid of revolution 1.svg
|caption1=Fig. 13.  Side view.
|image2=Non-standard closed geodesics on an ellipsoid of revolution 2.svg
|caption2=Fig. 14.  Top view.
|header=Two additional closed geodesics for the oblate ellipsoid, ''b''/''a'' = 2/7.
}}
If the ellipsoid is sufficiently oblate, i.e.,
''b''/''a''&nbsp;&lt;&nbsp;&frac12;, another class of simple closed geodesics is
possible {{harv|Klingenberg|1982|loc=&sect;3.5.19}}.  Two such geodesics
are illustrated in Figs. 13 and 14.  Here
''b''/''a''&nbsp;=&nbsp;2/7 and the equatorial azimuth,
&alpha;<sub>0</sub>, for the green (resp. blue) geodesic is chosen to
be 53.175&deg; (resp. 75.192&deg;), so that the geodesic completes 2
(resp. 3) complete oscillations about the equator on one circuit of the
ellipsoid.

=== Evaluation of the integrals ===
Solving the geodesic problems entails evaluating the integrals for the
distance, ''s'', and the longitude, &lambda;,
Eqs. (4) and (5).  In geodetic applications, where
''f'' is small, the integrals are typically evaluated as a
series; for this purpose, the second form of the longitude integral is
preferred (since it avoids the near singular behavior of the first form
when geodesics pass close to a pole).  In both integrals, the integrand
is an even periodic function of period &pi;.  Furthermore, the
term dependent on &sigma; is multiplied by a small quantity
''k''<sup>2</sup>&nbsp;=&nbsp;''O''(''f'').  As a consequence, the integrals can both be
written in the form
:<math>
I = B_0 \sigma + \sum_{j=1}^\infty B_j \sin 2j\sigma
</math>
where ''B''<sub>0</sub>&nbsp;=&nbsp;1&nbsp;+&nbsp;''O''(''f'') and ''B''<sub>''j''</sub>&nbsp;=&nbsp;''O''(''f''&nbsp;<sup>''j''</sup>).  Series
expansions for ''B''<sub>''j''</sub> can readily be found and the result
truncated so that only terms which are ''O''(''f''&nbsp;<sup>''J''</sup>) and larger are
retained.{{refn| Nowadays, the necessary algebraic manipulations,
expanding in a Taylor series, integration, and performing trigonometric
simplifications, can be carrying using a [[計算機代數系統|computer algebra system]].
Earlier, {{harvtxt|Levallois|Dupuy|1952}} gave recurrence relations for
the series in terms of [[Wallis' integrals]] and
{{harvtxt|Pittman|1986}} describes a similar method.}}
(Because the longitude integral is multiplied by
''f'', it is typically only necessary to retain terms up to
''O''(''f''&nbsp;<sup>''J''&minus;1</sup>) in that integral.)  This prescription is
followed by many authors {{harv|Legendre|1806}} {{harv|Oriani|1806}}
{{harv|Bessel|1825}} {{harv|Helmert|1880}} {{harv|Rainsford|1955}}
{{harv|Rapp|1993}}.  {{harvtxt|Vincenty|1975a}} uses ''J''&nbsp;=&nbsp;3
which provides an accuracy of about 0.1&nbsp;mm for the [[WGS84]]
ellipsoid.  {{harvtxt|Karney|2013}} gives expansions carried out for
''J''&nbsp;=&nbsp;6 which suffices to provide full [[雙精度浮點數|double precision]]
accuracy for |''f''|&nbsp;&le;&nbsp;1/50.  Trigonometric
series of this type can be conveniently summed using
[[Clenshaw递推公式|Clenshaw summation]].

In order to solve the direct geodesic problem, it is necessary to find
&sigma; given ''s''.  Since the integrand in the
distance integral is positive, this problem has a unique root, which
may be found using [[牛顿法|Newton's method]], noting that the required
derivative is just the integrand of the distance integral.
{{harvtxt|Oriani|1833}} instead uses [[series reversion]] so that
&sigma; can be found without iteration;
{{harvtxt|Helmert|1880}} gives a similar series.{{refn|
{{harvtxt|Legendre|1806|loc=Art. 13}} also gives a series
for &sigma; in terms of ''s''; but this is
not suitable for large distances.}}  The reverted series
converges somewhat slower that the direct series and, if
|''f''|&nbsp;&gt;&nbsp;1/100,
{{harvtxt|Karney|2013|loc=addenda}} supplements the reverted series with
one step of Newton's method to maintain accuracy.
{{harvtxt|Vincenty|1975a}} instead relies on a simpler (but slower)
function iteration to solve for &sigma;.

It is also possible to evaluate the integrals (4) and (5)
by numerical quadrature {{harv|Saito|1970}} {{harv|Saito|1979}}
{{harv|Sjöberg|Shirazian|2012}} or to apply numerical techniques for the
solution of the ordinary differential equations, Eqs. (2)
{{harv|Kivioja|1971}} {{harv|Thomas|Featherstone|2005}} {{harv|Panou et
al.|2013}}.  Such techniques can be used for arbitrary flattening
''f''.  However, if ''f'' is small, e.g.,
|''f''|&nbsp;&le;&nbsp;1/50, they do not offer the speed
and accuracy of the series expansions described above.  Furthermore, for
arbitrary ''f'', the evaluation of the integrals in terms of
elliptic integrals (see below) also provides a fast and accurate
solution.  On the other hand, {{harvtxt|Mathar|2007}} has tackled the
more complex problem of geodesics on the surface at a constant altitude,
''h'', above the ellipsoid by solving the corresponding
ordinary differential equations, Eqs. (2) with
[&rho;,&nbsp;&nu;] replaced by [&rho;&nbsp;+&nbsp;''h'',&nbsp;&nu;&nbsp;+&nbsp;''h''].

[[File:Legendre.jpg|thumb|150px|[[阿德里安-马里·勒让德|A. M. Legendre]]]]
[[File:Arthur Cayley.jpg|thumb|150px|[[阿瑟·凱萊|Arthur Cayley]]]]
Geodesics on an ellipsoid was an early application of
[[椭圆积分|elliptic integrals]].  In particular,
{{harvtxt|Legendre|1811|log=&sect;&sect;126&ndash;129}} writes the
integrals, Eqs. (4) and (5), as
:(6)<math>{\color{white}.}\qquad
\displaystyle
  \frac sb = E(\sigma, ik),
</math>
:(7)<math>{\color{white}.}\qquad
\begin{align}
  \lambda &= (1 - f) \sin\alpha_0  G(\sigma, \cos^2\alpha_0, ik) \\
          &= \chi
           - \frac{e'^2}{\sqrt{1+e'^2}}\sin\alpha_0 H(\sigma, -e'^2, ik), \\
\end{align}
</math>
where
:<math>
\tan\chi = \sqrt{\frac{1+e'^2}{1+k^2\sin^2\sigma}}\tan\omega,
</math>
and
:<math>
\begin{align}
 G(\phi,\alpha^2,k) &= \int_0^\phi
 \frac{\sqrt{1 - k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta\\
 &=\frac{k^2}{\alpha^2}F(\phi, k)
 +\biggl(1-\frac{k^2}{\alpha^2}\biggr)\Pi(\phi, \alpha^2, k),\\
 H(\phi, \alpha^2, k)
 &= \int_0^\phi
   \frac{\cos^2\theta}{(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}
   \,d\theta \\
 &=
 \frac1{\alpha^2} F(\phi, k) +
      \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k),
 \end{align}
</math>
and ''F''(&phi;,&nbsp;''k''), ''E''(&phi;,&nbsp;''k''), and
&Pi;(&phi;,&nbsp;&alpha;<sup>2</sup>,&nbsp;''k''), are
[[椭圆积分|incomplete elliptic integrals]] in the
notation of
{{harvtxt|DLMF|2010|loc=[http://dlmf.nist.gov/19.2.ii &sect;19.2(ii)]}}.{{refn|
1=Despite the presence of ''i''&nbsp;=&nbsp;&radic;&minus;1, the elliptic
integrals in Eqs. (6) and (7) are real.}}{{refn|
{{harvtxt|Rollins|2010}} obtains different, but equivalent, expressions
in terms of elliptic integrals.}}
The first formula for the longitude in Eq. (7) follows directly
from the first form of Eq. (5).  The second formula in
Eq. (7), due to {{harvtxt|Cayley|1870}}, is more
convenient for calculation since the elliptic integral appears in a
small term.  The equivalence of the two forms follows from
{{harvtxt|DLMF|2010|loc=[http://dlmf.nist.gov/19.7.E8 Eq. (19.7.8)]}}.
Fast algorithms for computing elliptic integrals are given by
{{harvtxt|Carlson|1995}} in terms of
[[Carlson symmetric form|symmetric elliptic integrals]].
Equation (6) is conveniently inverted using
[[牛顿法|Newton's method]].  The use of elliptic integrals provides a good
method of solving the geodesic problem for
|''f''|&nbsp;&gt;&nbsp;1/50.{{refn|1=
It is also possible to express the integrals in terms of
[[雅可比橢圓函數|Jacobi elliptic functions]] {{harv|Jacobi|1855}} {{harv|Luther|1855}}
{{harv|Forsyth|1896}} {{harv|Thomas|1970|loc=Appendix 1}}.
{{harvtxt|Halphen|1888}} gives the solution for the complex quantities
''R''&nbsp;exp(&plusmn;''i''&lambda;)&nbsp;=&nbsp;''X''&nbsp;&plusmn;&nbsp;''iY'' in terms of
[[Weierstrass functions|Weierstrass sigma and zeta]] functions.  This
form is of interest because the separate periods of latitude and
longitude of the geodesic are captured in a single
[[双周期函数|doubly periodic function]]; see also
{{harvtxt|Forsyth|1927|loc=&sect;75.}}}}

=== Solution of the direct problem ===
The basic strategy for solving the geodesic problems on the ellipsoid is
to map the problem onto the auxiliary sphere by converting
&phi;, &lambda;, and ''s'', to
&beta;, &omega; and &sigma;, to solve
the corresponding great-circle problem on the sphere, and to transfer the
results back to the ellipsoid.

In implementing this program, we will frequently need to solve the
"elementary" spherical triangle for ''NEP'' in Fig.
6 with ''P'' replaced by either ''A'' (subscript
1) or ''B'' (subscript 2).  For this purpose, we can apply
[[球面三角學|
Napier's rules for quadrantal triangles]] to the triangle ''NEP''
on the auxiliary sphere which give
:<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>
We can also stipulate that cos&beta;&nbsp;&ge;&nbsp;0 and
cos&alpha;<sub>0</sub>&nbsp;&ge;&nbsp;0.{{refn|name=atan|
When solving for &sigma;, &alpha;, or
&omega; using a formula for its tangent, the quadrant should
be determined from the signs of the numerator of the expression for the
tangent, e.g., using the [[Atan2|atan2]] function.}}
Implementing this plan for the direct problem is straightforward.  We
are given &phi;<sub>1</sub>, &alpha;<sub>1</sub>, and
''s''<sub>12</sub>.  From &phi;<sub>1</sub> we obtain
&beta;<sub>1</sub> (using the formula for the parametric latitude).
We now solve the triangle problem with ''P''&nbsp;=&nbsp;''A'' and
&beta;<sub>1</sub> and &alpha;<sub>1</sub> given to find
&alpha;<sub>0</sub>, &sigma;<sub>1</sub>, and
&omega;<sub>1</sub>.{{refn|1=
If &beta;<sub>1</sub>&nbsp;=&nbsp;0 and &alpha;<sub>1</sub>&nbsp;=&nbsp;&plusmn;&frac12;&pi;, the
equation for &sigma;<sub>1</sub> is indeterminate and
&sigma;<sub>1</sub>&nbsp;=&nbsp;0 may be used.}}  Use
the distance and longitude equations, Eqs. (4) and
(5), together with the known value of &lambda;<sub>1</sub>, to
find ''s''<sub>1</sub> and &lambda;<sub>0</sub>.  Determine
''s''<sub>2</sub>&nbsp;=&nbsp;''s''<sub>1</sub>&nbsp;+&nbsp;''s''<sub>12</sub> and invert the distance equation to find
&sigma;<sub>2</sub>.  Solve the triangle problem with ''P''&nbsp;=&nbsp;''B''
and &alpha;<sub>0</sub> and &sigma;<sub>2</sub> given to find
&beta;<sub>2</sub>, &omega;<sub>2</sub>, and &alpha;<sub>2</sub>.
Convert &beta;<sub>2</sub> to &phi;<sub>2</sub> and substitute
&sigma;<sub>2</sub> and &omega;<sub>2</sub> into the longitude
equation to give &lambda;<sub>2</sub>.{{refn|1=Because
tan&omega;&nbsp;=&nbsp;sin&alpha;<sub>0</sub>&nbsp;tan&sigma;, &omega;
changes quadrants in step with &sigma;.  It is therefore
straightforward to express &lambda;<sub>2</sub> so that
&lambda;<sub>12</sub> indicates how often and in what sense the
geodesic has encircled the ellipsoid.}}

The overall method follows the procedure for
[[Great-circle navigation#Finding way-points|solving the direct
problem on a sphere]].  It is essentially the program laid out by
{{harvtxt|Bessel|1825}},{{refn|
{{harvtxt|Bessel|1825}} treated the longitude integral approximately in
order to reduce the number of parameters in the equation from two to one
so that it could be tabulated conveniently.}}
{{harvtxt|Helmert|1880|loc=&sect;5.9}}, and most subsequent
authors.

Intermediate points, [[way-point]]s, on a geodesic can be found by
holding &phi;<sub>1</sub> and &alpha;<sub>1</sub> fixed and solving
the direct problem for several values of ''s''<sub>12</sub>.  Once the
first waypoint is found, only the last portion of the solution (starting
with the determination of ''s''<sub>2</sub>) needs to be repeated for
each new value of ''s''<sub>12</sub>.

=== Solution of the inverse problem ===
The ease with which the direct problem can be solved results from the
fact that given &phi;<sub>1</sub> and &alpha;<sub>1</sub>, we can
immediately find &alpha;<sub>0</sub>, the parameter in the distance
and longitude integrals, Eqs. (4) and (5).  In
the case of the inverse problem, we are given &lambda;<sub>12</sub>,
but we cannot easily relate this to the equivalent spherical angle
&omega;<sub>12</sub> because &alpha;<sub>0</sub> is unknown.
Thus, the solution of the problem requires that &alpha;<sub>0</sub> be
found iteratively.  Before tackling this, it is worth understanding
better the behavior of geodesics, this time, keeping the starting point
fixed and varying the azimuth.

{{multiple image
|align=right
|direction=vertical
|width=220
|image1=Geodesics and geodesic circles on an oblate ellipsoid.svg
|caption1=Fig. 15.  Geodesics, geodesic circles, and the cut locus.
|image2=Unrolled geodesics on an oblate ellipsoid.svg
|caption2=Fig. 16.  The geodesics shown on a [[plate carrée projection]].
|image3=Geodesic longitude variation for an ellipsoid.svg
|caption3=Fig. 17.  &lambda;<sub>12</sub> as a function of &alpha;<sub>1</sub> for &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg; and &phi;<sub>2</sub>&nbsp;=&nbsp;20&deg;.
|header=Geodesics from a single point (''f''&nbsp;=&nbsp;1/10, &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg;)
}}

Suppose point ''A'' in the inverse problem has
&phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg; and &lambda;<sub>1</sub>&nbsp;=&nbsp;0&deg;.  Fig.
15 shows geodesics (in blue) emanating
''A'' with &alpha;<sub>1</sub> a multiple of
15&deg; up to the point at which they cease to be shortest
paths.  (The flattening has been increased to
1/10 in order to accentuate the ellipsoidal effects.)
Also shown (in green) are curves of constant ''s''<sub>12</sub>,
which are the geodesic circles centered ''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 ''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 [[對蹠點|antipodal]]
to ''A'', &phi;&nbsp;=&nbsp;&minus;&phi;<sub>1</sub>.  The longitudinal
extent of cut locus is approximately
&lambda;<sub>12</sub>&nbsp;&isin;&nbsp;[&pi;&nbsp;&minus;&nbsp;''f''&nbsp;&pi;&nbsp;cos&phi;<sub>1</sub>,&nbsp;&pi;&nbsp;+&nbsp;''f''&nbsp;&pi;&nbsp;cos&phi;<sub>1</sub>].  If
''A'' lies on the equator, &phi;<sub>1</sub>&nbsp;=&nbsp;0, this relation
is exact and as a consequence the equator is only a shortest geodesic if
|&lambda;<sub>12</sub>|&nbsp;&le;&nbsp;(1&nbsp;&minus;&nbsp;''f'')&pi;.  For a prolate
ellipsoid, the cut locus is a segment of the anti-meridian centered on
the point antipodal to ''A'', &lambda;<sub>12</sub>&nbsp;=&nbsp;&pi;,
and this means that
meridional geodesics stop being shortest paths before the antipodal
point is reached.

The solution of the inverse problem involves determining, for a given
point ''B'' with latitude &phi;<sub>2</sub> and longitude
&lambda;<sub>2</sub> which blue and green curves it lies on; this
determines &alpha;<sub>1</sub> and ''s''<sub>12</sub> respectively.
In Fig. 16, the ellipsoid has been "rolled out" onto a
[[plate carrée projection]].  Suppose &phi;<sub>2</sub>&nbsp;=&nbsp;20&deg;, the
green line in the figure.  Then as &alpha;<sub>1</sub> is varied
between 0&deg; and 180&deg;, the longitude
at which the geodesic intersects &phi;&nbsp;=&nbsp;&phi;<sub>2</sub> varies between
0&deg; and 180&deg; (see Fig. 17).
This behavior holds provided that
|&phi;<sub>2</sub>|&nbsp;&le;&nbsp;|&phi;<sub>1</sub>| (otherwise the
geodesic does not reach &phi;<sub>2</sub> for some values of
&alpha;<sub>1</sub>).  Thus, the inverse problem may be solved by
determining the value &alpha;<sub>1</sub> which results in the given
value of &lambda;<sub>12</sub> when the geodesic intersects the circle
&phi;&nbsp;=&nbsp;&phi;<sub>2</sub>.

This suggests the following strategy for solving the inverse problem
{{harv|Karney|2013}}.
Assume that the points ''A'' and ''B'' satisfy
:(8)<math>{\color{white}.}\qquad
\phi_1 \le 0, \quad \left|\phi_2\right| \le \left|\phi_1\right|,
\quad 0 \le \lambda_{12} \le \pi. </math>
(There is no loss of generality in this assumption, since the symmetries
of the problem can be used to generate any configuration of points from
such configurations.)
# First treat the "easy" cases, geodesics which lie on a meridian or the equator.  Otherwise...
# Guess a value of &alpha;<sub>1</sub>.
# Solve the so-called ''hybrid geodesic problem'', given &phi;<sub>1</sub>, &phi;<sub>2</sub>, and &alpha;<sub>1</sub> find &lambda;<sub>12</sub>, ''s''<sub>12</sub>, and &alpha;<sub>2</sub>, corresponding to the ''first'' intersection of the geodesic with the circle &phi;&nbsp;=&nbsp;&phi;<sub>2</sub>.
# Compare the resulting &lambda;<sub>12</sub> with the desired value and adjust &alpha;<sub>1</sub> until the two values agree.  This completes the solution.

Each of these steps requires some discussion.

1. For an oblate ellipsoid, the shortest geodesic lies on a meridian if
either point lies on a pole or if &lambda;<sub>12</sub>&nbsp;=&nbsp;0 or
&plusmn;&pi;.  The shortest geodesic follows the equator if
&phi;<sub>1</sub>&nbsp;=&nbsp;&phi;<sub>2</sub>&nbsp;=&nbsp;0 and
|&lambda;<sub>12</sub>|&nbsp;&le;&nbsp;(1&nbsp;&minus;&nbsp;''f'')&pi;.  For a prolate
ellipsoid, the meridian is no longer the shortest geodesic if
&lambda;<sub>12</sub>&nbsp;=&nbsp;&plusmn;&pi; and the points are close to antipodal
(this will be discussed in the next section).  There is no longitudinal
restriction on equatorial geodesics.

2. In most cases a suitable starting value of &alpha;<sub>1</sub> is
found by solving
[[Solution of triangles#Two sides and the included angle given|
the spherical inverse problem]]{{refn|name=atan}}
:<math>\tan\alpha_1 = \frac
{\cos\beta_2\sin\omega_{12}}
{\cos\beta_1\sin\beta_2 - \sin\beta_1\cos\beta_2 \cos\omega_{12}},</math>
with &omega;<sub>12</sub>&nbsp;=&nbsp;&lambda;<sub>12</sub>.  This may be a bad
approximation if ''A'' and ''B'' are nearly antipodal
(both the numerator and denominator in the formula above become small);
however, this may not matter (depending on how step 4 is handled).

3. The solution of the hybrid geodesic problem is as follows.  It starts
the same way as the solution of the direct problem, solving the triangle
''NEP'' with ''P''&nbsp;=&nbsp;''A'' to find &alpha;<sub>0</sub>,
&sigma;<sub>1</sub>, &omega;<sub>1</sub>, and
&lambda;<sub>0</sub>.{{refn|1=If &phi;<sub>1</sub>&nbsp;=&nbsp;&phi;<sub>2</sub>&nbsp;=&nbsp;0, take
sin&sigma;<sub>1</sub>&nbsp;=&nbsp;sin&omega;<sub>1</sub>&nbsp;=&nbsp;&minus;0, consistent with the relations
(8); this gives &sigma;<sub>1</sub>&nbsp;=&nbsp;&omega;<sub>1</sub>&nbsp;=&nbsp;&minus;&pi;.}}  Now find
&alpha;<sub>2</sub> from
sin&alpha;<sub>2</sub>&nbsp;=&nbsp;sin&alpha;<sub>0</sub>/cos&beta;<sub>2</sub>, taking
cos&alpha;<sub>2</sub>&nbsp;&ge;&nbsp;0 (corresponding to the first, northward,
crossing of the circle &phi;&nbsp;=&nbsp;&phi;<sub>2</sub>).  Next,
&sigma;<sub>2</sub> is given by
tan&sigma;<sub>2</sub>&nbsp;=&nbsp;tan&beta;<sub>2</sub>/cos&alpha;<sub>2</sub> and
&omega;<sub>2</sub> by
tan&omega;<sub>2</sub>&nbsp;=&nbsp;tan&sigma;<sub>2</sub>/sin&alpha;<sub>0</sub>.{{refn|name=atan}}
Finally, use the distance and longitude equations, Eqs. (4)
and (5), to find ''s''<sub>12</sub> and
&lambda;<sub>12</sub>.{{refn|
The ordering in relations (8) automatically results in
&sigma;<sub>12</sub>&nbsp;&gt;&nbsp;0.}}

4. In order to discuss how &alpha;<sub>1</sub> is updated, let us define
the root-finding problem in more detail.  The curve in Fig. 17
shows &lambda;<sub>12</sub>(&alpha;<sub>1</sub>;&nbsp;&phi;<sub>1</sub>,&nbsp;&phi;<sub>2</sub>) where we regard
&phi;<sub>1</sub> and &phi;<sub>2</sub> as parameters and
&alpha;<sub>1</sub> as the independent variable.  We seek the value of
&alpha;<sub>1</sub> which is the root of
:<math> g(\alpha_1) \equiv
\lambda_{12}(\alpha_1;\phi_1,\phi_2) - \lambda_{12} = 0,
</math>
where ''g''(0)&nbsp;&le;&nbsp;0 and ''g''(&pi;)&nbsp;&ge;&nbsp;0.  In fact,
there is a unique root in the interval &alpha;<sub>1</sub>&nbsp;&isin;&nbsp;[0,&nbsp;&pi;].
Any of a number of [[介值定理#零点定理|root-finding algorithms]] can be used to solve such
an equation.  {{harvtxt|Karney|2013}} uses [[牛顿法|Newton's method]], which
requires a good starting guess; however it may be supplemented by a
[[fail-safe]] method, such as the [[二分法 (數學)|bisection method]], to guarantee
convergence.

[[File:F-R Helmert 1.jpg|thumb|150px|[[Friedrich Robert Helmert|F. R. Helmert]]]]
An alternative method for solving the inverse problem is given by
{{harvtxt|Helmert|1880|loc=&sect;5.13}}.  Let us rewrite the
Eq. (5) as
:<math>\begin{align}
\lambda_{12} &= \omega_{12}
- f\sin\alpha_0
\int_{\sigma_1}^{\sigma_2}\frac
{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}}
\,d\sigma'\\
&= \omega_{12}
- f\sin\alpha_0 I(\sigma_1, \sigma_2; \alpha_0).
\end{align}
</math>

Helmert's method entails assuming that
&omega;<sub>12</sub>&nbsp;=&nbsp;&lambda;<sub>12</sub>, solving the resulting problem on
auxiliary sphere, and obtaining an updated estimate of
&omega;<sub>12</sub> using
:<math>
\omega_{12} = \lambda_{12} + f\sin\alpha_0 I(\sigma_1, \sigma_2; \alpha_0).
</math>
This [[fixed point iteration]] is repeated until convergence.
{{harvtxt|Rainsford|1955}} advocates this method and
{{harvtxt|Vincenty|1975a}} adopted it in his solution of the inverse
problem.  The drawbacks of this method are that convergence is slower
than obtained using Newton's method (as described above) and, more
seriously, that the process fails to converge at all for nearly
antipodal points.  In a subsequent report, {{harvtxt|Vincenty|1975b}}
attempts to cure this defect; but he is only partially
successful&mdash;the {{harvtxt|NGS|2012}} implementation still includes
Vincenty's fix still fails to converge in some cases.
{{harvtxt|Lee|2011}} has compared 17 methods for solving the inverse
problem against the method given by {{harvtxt|Karney|2013}}.

The shortest distance returned by the solution of the inverse problem is
(obviously) uniquely defined.  However, if ''B'' lies on the
cut locus of ''A'' there are multiple azimuths which yield
the same shortest distance.  Here is a catalog of those cases:
* &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;&phi;<sub>2</sub> (with neither point at a pole).  If &alpha;<sub>1</sub>&nbsp;=&nbsp;&alpha;<sub>2</sub>, the geodesic is unique.  Otherwise there are two geodesics and the second one is obtained by interchanging &alpha;<sub>1</sub> and &alpha;<sub>2</sub>.  (This occurs when &lambda;<sub>12</sub>&nbsp;&asymp;&nbsp;&plusmn;&pi; for oblate ellipsoids.)
* &lambda;<sub>12</sub>&nbsp;=&nbsp;&plusmn;&pi; (with neither point at a pole).  If &alpha;<sub>1</sub>&nbsp;=&nbsp;0 or &plusmn;&pi;, the geodesic is unique.  Otherwise there are two geodesics and the second one is obtained by negating &alpha;<sub>1</sub> and &alpha;<sub>2</sub>. (This occurs when &phi;<sub>1</sub>&nbsp;+&nbsp;&phi;<sub>2</sub>&nbsp;&asymp;&nbsp;0 for prolate ellipsoids.)
* ''A'' and ''B'' are at opposite poles.  There are infinitely many geodesics which can be generated by varying the azimuths so as to keep &alpha;<sub>1</sub>&nbsp;+&nbsp;&alpha;<sub>2</sub> constant.  (For spheres, this prescription applies when ''A'' and ''B'' are antipodal.)

=== Differential behavior of geodesics ===
[[File:Carl Friedrich Gauss.jpg|thumb|150px|[[卡爾·弗里德里希·高斯|C. F. Gauss]]]]
[[File:Elwin Bruno Christoffel.JPG|thumb|150px|[[埃尔温·布鲁诺·克里斯托费尔|E. B. Christoffel]]]]
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 ''s'' the
length from the northward equator crossing, and a second geodesic a small
distance ''t''(''s'') away from it.  {{harvtxt|Gauss|1828}} showed that
''t''(''s'') obeys the
[[Differential geometry of surfaces#Gauss–Jacobi equation|
Gauss-Jacobi equation]]
:(9)<math>{\color{white}.}\qquad
\displaystyle\frac{d^2t(s)}{ds^2} = K(s) t(s), </math>
[[File:Definition of reduced length and geodesic scale.svg|thumb|
Fig. 18.
Definition of reduced length and geodesic scale.]]
where ''K''(''s'') is the [[高斯曲率|Gaussian curvature]] at ''s''.
The 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>
We shall abbreviate ''m''(''s''<sub>1</sub>,&nbsp;''s''<sub>2</sub>)&nbsp;=&nbsp;''m''<sub>12</sub>, the so-called
''reduced length'', and ''M''(''s''<sub>1</sub>,&nbsp;''s''<sub>2</sub>)&nbsp;=&nbsp;''M''<sub>12</sub>, the
''geodesic scale''.{{refn|
{{harvtxt|Bagratuni|1962|loc=&sect;17}} uses the term "coefficient of
convergence of ordinates" for the geodesic scale.}}
Their basic definitions are illustrated in
Fig. 18.  {{harvtxt|Christoffel|1869}} made an extensive study of
their properties.  The reduced length obeys a reciprocity relation,
:<math>m_{12} + m_{21} = 0.</math>
Their derivatives are
:<math>
\begin{align}
\frac{d m_{12}}{d s_2} &= M_{21},\\
\frac{d M_{12}}{d s_2} &= -\frac{1 - M_{12}M_{21}}{m_{12}}.
\end{align}
</math>
Assuming that points 1, 2, and 3, lie on the same geodesic, then the
following addition rules apply {{harv|Karney|2013}},
:<math>
\begin{align}
m_{13} &= m_{12} M_{23} + m_{23} M_{21},\\
M_{13} &= M_{12} M_{23} - (1 - M_{12} M_{21}) \frac{m_{23}}{m_{12}},\\
M_{31} &= M_{32} M_{21} - (1 - M_{23} M_{32}) \frac{m_{12}}{m_{23}}.
\end{align}
</math>
The reduced length and the geodesic scale are components of the
[[Jacobi field]].

The [[類球面|Gaussian curvature for an ellipsoid of revolution]]
is
:<math>
K = \frac1{\rho\nu} = \frac{(1-e^2\sin^2\phi)^2}{b^2}
  = \frac{b^2}{a^4(1-e^2\cos^2\beta)^2}.
</math>
{{harvtxt|Helmert|1880|loc=Eq. (6.5.1.)}} solved the Gauss-Jacobi
equation for this case obtaining
:<math>
\begin{align}
m_{12}/b &= \sqrt{1 + k^2\sin^2\sigma_2}\, \cos\sigma_1 \sin\sigma_2
 - \sqrt{1 + k^2\sin^2\sigma_1}\, \sin\sigma_1 \cos\sigma_2 \\
 &\quad - \cos\sigma_1 \cos\sigma_2 \bigl(J(\sigma_2) - J(\sigma_1)\bigr),\\
M_{12} &= \cos\sigma_1 \cos\sigma_2
+ \frac{\sqrt{1 + k^2\sin^2\sigma_2}}{\sqrt{1 + k^2\sin^2\sigma_1}}
\sin\sigma_1 \sin\sigma_2 \\
&\quad - \frac{\sin\sigma_1 \cos\sigma_2
\bigl(J(\sigma_2) - J(\sigma_1)\bigr)}
{\sqrt{1 + k^2\sin^2\sigma_1}},
\end{align}
</math>
where
:<math>
\begin{align}
J(\sigma) &=
\int_0^\sigma \frac{k^2\sin^2\sigma'}{\sqrt{1 + k^2\sin^2\sigma'}}\,d\sigma'\\
&= E(\sigma, ik) - F(\sigma, ik).
\end{align}
</math>

As we see from Fig. 18 (top sub-figure), the separation of two
geodesics starting at the same point with azimuths differing by
''d''&alpha;<sub>1</sub> is ''m''<sub>12</sub>&nbsp;''d''&alpha;<sub>1</sub>.  On a closed
surface such as an ellipsoid, we expect ''m''<sub>12</sub> to oscillate
about zero.  Indeed, if the starting point of a geodesic is a pole,
&phi;<sub>1</sub>&nbsp;=&nbsp;&frac12;&pi;, then the reduced length is the radius
of the circle of latitude,
''m''<sub>12</sub>&nbsp;=&nbsp;''a''&nbsp;cos&beta;<sub>2</sub>&nbsp;=&nbsp;''a''&nbsp;sin&sigma;<sub>12</sub>.  Similarly, for a
meridional geodesic starting on the equator,
&phi;<sub>1</sub>&nbsp;=&nbsp;&alpha;<sub>1</sub>&nbsp;=&nbsp;0, we have
''M''<sub>12</sub>&nbsp;=&nbsp;cos&sigma;<sub>12</sub>.  In the typical case, these
quantities oscillate with a period of about 2&pi; in
&sigma;<sub>12</sub> and grow linearly with distance at a rate
proportional to ''f''.  In trigonometric adjustments over small
areas, it may be possible to approximate ''K''(''s'') in
Eq. (9) by a constant ''K''.  In this limit, the
solutions for ''m''<sub>12</sub> and ''M''<sub>12</sub> are the same
as for a sphere of radius 1/&radic;''K'', namely,
:<math>m_{12} = \sin(\sqrt K s_{12})/\sqrt K, \quad
M_{12} = \cos(\sqrt K s_{12}).</math>

To simplify the discussion of shortest paths in this paragraph we
consider only geodesics with ''s''<sub>12</sub>&nbsp;&gt;&nbsp;0.  The point at
which ''m''<sub>12</sub> becomes zero is the point
[[conjugate point|conjugate]] to the starting point.  In order
for a geodesic between ''A'' and ''B'', of length
''s''<sub>12</sub>, to be a shortest path it must satisfy the
Jacobi condition {{harv|Jacobi|1837}} {{harv|Jacobi|1866|loc=&sect;6}}
{{harv|Forsyth|1927|loc=&sect;&sect;26&ndash;27}}
{{harv|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, |&sigma;<sub>12</sub>|&nbsp;&le;&nbsp;&pi;;
* for a prolate ellipsoid, |&lambda;<sub>12</sub>|&nbsp;&le;&nbsp;&pi;, if &alpha;<sub>0</sub>&nbsp;&ne;&nbsp;0; if &alpha;<sub>0</sub>&nbsp;=&nbsp;0, the supplemental condition ''m''<sub>12</sub>&nbsp;&ge;&nbsp;0 is required if |&lambda;<sub>12</sub>|&nbsp;=&nbsp;&pi;.
The latter condition above can be used to determine whether the shortest
path is a meridian in the case of a prolate ellipsoid with
|&lambda;<sub>12</sub>|&nbsp;=&nbsp;&pi;.  The derivative required to
solve the inverse method using Newton's method,
&part;&lambda;<sub>12</sub>(&alpha;<sub>1</sub>;&nbsp;&phi;<sub>1</sub>,&nbsp;&phi;<sub>2</sub>)&nbsp;/&nbsp;&part;&alpha;<sub>1</sub>,
is given in terms of the reduced length
{{harv|Karney|2013|loc=Eq. (46)}}.

=== Geodesic map projections ===
Two map projections are defined in terms of geodesics.  They are based
on polar and rectangular geodesic coordinates on the surface
{{harv|Gauss|1828}}.  The polar coordinate system
(''r'',&nbsp;&theta;) is centered on some point ''A''.  The
coordinates of another point ''B'' are given by
''r''&nbsp;=&nbsp;''s''<sub>12</sub> and &theta;&nbsp;=&nbsp;&frac12;&pi;&nbsp;&minus;&nbsp;&alpha;<sub>1</sub> and
these coordinates are used to find the projected coordinates on a plane
map, ''x''&nbsp;=&nbsp;''r''&nbsp;cos&theta; and ''y''&nbsp;=&nbsp;''r''&nbsp;sin&theta;.  The
result is the familiar [[azimuthal equidistant projection]]; in the
field of the [[differential geometry of surfaces]], it is called the
[[exponential map (Riemannian geometry)|exponential map]].  Due to the
basic properties of geodesics {{harv|Gauss|1828}}, lines of constant
''r'' and lines of constant &theta; intersect at
right angles on the surface.  The scale of the projection in the radial
direction is unity, while the scale in the azimuthal direction is
''s''<sub>12</sub>/''m''<sub>12</sub>.

The rectangular coordinate system (''x'',&nbsp;''y'') uses a reference
geodesic defined by ''A'' and &alpha;<sub>1</sub> as the
''x'' axis.  The point (''x'',&nbsp;''y'') is found by traveling
a distance ''s''<sub>13</sub>&nbsp;=&nbsp;''x'' from ''A'' along the reference
geodesic to an intermediate point ''C'' and then turning
&frac12;&pi; counter-clockwise and traveling along a
geodesic a distance ''s''<sub>32</sub>&nbsp;=&nbsp;''y''.  If ''A'' is on the
equator and &alpha;<sub>1</sub>&nbsp;=&nbsp;&frac12;&pi;, this gives the
[[equidistant cylindrical projection]].  If &alpha;<sub>1</sub>&nbsp;=&nbsp;0,
this gives the [[Cassini projection|Cassini-Soldner projection]].
[[French cartography#Cassini_maps|Cassini's map of France]] placed
''A'' at the [[巴黎天文台|Paris Observatory]].  Due to the basic
properties of geodesics {{harv|Gauss|1828}}, lines of constant
''x'' and lines of constant ''y'' intersect at right
angles on the surface.  The scale of the projection in the
''y'' direction is unity, while the scale in the ''x''
direction is 1/''M''<sub>32</sub>.

The [[gnomonic projection]] is a projection of the sphere where all
geodesics (i.e., great circles) map to straight lines (making it a
convenient [[Great-circle navigation#Gnomonic chart|aid to navigation]]).
Such a projection is only possible for surfaces of constant
[[高斯曲率|Gaussian curvature]] {{harv|Beltrami|1865}}.  Thus a projection in
which geodesics map to straight lines is not possible for an ellipsoid.
However, it is possible to construct an ellipsoidal gnomonic projection
in which this property ''approximately'' holds
{{harv|Karney|2013|loc=&sect;8}}.  On the sphere, the gnomonic
projection is the limit of a doubly azimuthal projection, a projection
preserving the azimuths from two points ''A'' and
''B'', as ''B'' approaches ''A''.  Carrying
out this limit in the case of a general surface yields an azimuthal
projection in which the distance from the center of projection is given
by &rho;&nbsp;=&nbsp;''m''<sub>12</sub>/''M''<sub>12</sub>.  Even though geodesics are only
approximately straight in this projection, all geodesics through the
center of projection ''are'' straight.  The projection can then be used to
give an iterative but rapidly converging method of solving some problems
involving geodesics, in particular, finding the intersection of two
geodesics and finding the shortest path from a point to a geodesic.

The [[Hammer retroazimuthal projection]] is a variation of the azimuthal
equidistant projection {{harv|Hammer|1910}}.  A geodesic is constructed
from a central point ''A'' to some other point ''B''.
The polar coordinates of the projection of ''B'' are
''r''&nbsp;=&nbsp;''s''<sub>12</sub> and &theta;&nbsp;=&nbsp;&frac12;&pi;&nbsp;&minus;&nbsp;&alpha;<sub>2</sub>
(which depends on the azimuth at ''B'', instead of at
''A'').  This can be used to determine the direction from an
arbitrary point to some fixed center.  {{harvtxt|Hinks|1929}} suggested
another application: if the central point ''A'' is a beacon,
such as the [[Rugby Clock]], then at an unknown location ''B''
the range and the bearing to ''A'' can be measured and the
projection can be used to estimate the location of ''B''.

=== Envelope of geodesics ===
{{multiple image
|align=right
|direction=horizontal
|width=220
|image1=Envelope of geodesics on an oblate ellipsoid.svg
|caption1=Fig. 19.  The envelope of geodesics from a point ''A'' at &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg;.
|image2=Four geodesics connecting two points on an oblate ellipsoid.svg
|caption2=Fig. 20.  The four geodesics connecting ''A'' and a point ''B'', &phi;<sub>2</sub>&nbsp;=&nbsp;26&deg;, &lambda;<sub>12</sub>&nbsp;=&nbsp;175&deg;.
|header=Geodesics from a single point (''f''&nbsp;=&nbsp;1/10, &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg;)
}}

The geodesics from a particular point ''A'' if continued
past the cut locus form an envelope illustrated in Fig. 19.
Here the geodesics for which &alpha;<sub>1</sub> is a multiple of
3&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 ''A''; points on the
envelope may be computed by finding the point at which
''m''<sub>12</sub>&nbsp;=&nbsp;0 on a geodesic (and Newton's method can be used to
find this point).  {{harvtxt|Jacobi|1891}} calls this star-like figure
produced by the envelope an [[星形线|astroid]].

Outside the astroid two geodesics intersect at each point; thus there
are two geodesics (with a length approximately half the
circumference of the ellipsoid) between ''A'' and these points.
This corresponds to the situation on the sphere where there are "short"
and "long" routes on a great circle between two points.  Inside the
astroid four geodesics intersect at each point.  Four such geodesics are
shown in Fig. 20 where the geodesics are numbered in order of
increasing length.  (This figure uses the same position for
''A'' as Fig. 15 and is drawn in the same projection.)
The two shorter geodesics are ''stable'', i.e., ''m''<sub>12</sub>&nbsp;&gt;&nbsp;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 &sigma;<sub>12</sub>&nbsp;&le;&nbsp;&pi;.  All the geodesics are tangent
to the envelope which is shown in green in the figure.  A similar set of
geodesics for the WGS84 ellipsoid is given in this table
{{harv|Karney|2011|loc=Table 1}}:
{| class="wikitable" style="text-align: right;"
|+ Geodesics for &phi;<sub>1</sub>&nbsp;=&nbsp;&minus;30&deg;, &phi;<sub>2</sub>&nbsp;=&nbsp;29.9&deg;, &lambda;<sub>12</sub>&nbsp;=&nbsp;179.8&deg; (WGS84)
|-
!No.!!&alpha;<sub>1</sub> (&deg;)!!&alpha;<sub>2</sub> (&deg;)!!''s''<sub>12</sub> (m)!!&sigma;<sub>12</sub> (&deg;)!!''m''<sub>12</sub> (m)
|-
|1||161.890524736|| 18.090737246||19989832.8276||179.894971388|| 57277.3769
|-
|2|| 30.945226882||149.089121757||20010185.1895||180.116378785|| 24240.7062
|-
|3|| 68.152072881||111.990398904||20011886.5543||180.267429871||&minus;22649.2935
|-
|4||&minus;81.075605986||&minus;99.282176388||20049364.2525||180.630976969||&minus;68796.1679
|}

The approximate shape of the astroid is given by
:<math> x^{2/3} + y^{2/3} = 1</math>
or, in parametric form,
:<math> x = \cos^3\theta, \quad y = \sin^3\theta.</math>
The astroid is also the envelope of the family of lines
:<math> \frac x{\cos\gamma} + \frac y{\sin\gamma} = 1, </math>
where &gamma; is a parameter.  (These are
generated by the rod of the [[trammel of Archimedes]].)  This aids
in finding a good starting guess for &alpha;<sub>1</sub> for Newton's
method for in inverse problem in the case of nearly antipodal points
{{harv|Karney|2013|loc=&sect;5}}.

The astroid is the (exterior) [[渐屈线|evolute]] of the geodesic circles
centered at ''A''.  Likewise, the geodesic circles are
[[漸伸線|involute]]s of the astroid.

=== Area of a geodesic polygon ===
A ''geodesic polygon'' is a polygon whose sides are geodesics.  The area of
such a polygon may be found by first computing the area between a
geodesic segment and the equator, i.e., the area of the quadrilateral
''AFHB'' in Fig. 1 {{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 we develop the formula for the area ''S''<sub>12</sub> of
''AFHB'' following {{harvtxt|Sjöberg|2006}}.  The area of any
closed region of the ellipsoid is
:<math> T = \int dT = \int \frac1K \cos\phi\,d\phi\,d\lambda,
</math>
where ''dT'' is an element of surface area and ''K''
is the [[高斯曲率|Gaussian curvature]].  Now the
[[高斯-博内定理|Gauss&ndash;Bonnet theorem]] applied to a geodesic polygon states
:<math>
\Gamma = \int K \,dT = \int \cos\phi\,d\phi\,d\lambda,
</math>
where
:<math>
\Gamma = 2\pi - \sum_j \theta_j
</math>
is the geodesic excess and &theta;<sub>''j''</sub> is the exterior angle at
vertex ''j''.  Multiplying the equation for &Gamma;
by ''R''<sub>2</sub><sup>2</sup>, where ''R''<sub>2</sub> is the
[[地球半径|authalic radius]], and subtracting this
from the equation for ''T'' gives{{refn|
{{harvtxt|Sjöberg|2006}} multiplies &Gamma; by
''b''<sup>2</sup> instead of ''R''<sub>2</sub><sup>2</sup>.  However, this leads to a
singular integrand {{harv|Karney|2011|loc=&sect;15}}.}}
:<math>
\begin{align}
T &=
R_2^2 \,\Gamma + \int \biggl(\frac1K - R_2^2\biggr)\cos\phi\,d\phi\,d\lambda\\
&=R_2^2 \,\Gamma + \int \biggl(
\frac{b^2}{(1 - e^2\sin^2\phi)^2} - R_2^2
\biggr)\cos\phi\,d\phi\,d\lambda,
\end{align}
</math>
where the [[類球面|value of ''K'' for an ellipsoid]]
has been substituted.
Applying this formula to the quadrilateral ''AFHB'', noting
that &Gamma;&nbsp;=&nbsp;&alpha;<sub>2</sub>&nbsp;&minus;&nbsp;&alpha;<sub>1</sub>, and performing
the integral over &phi; gives
:<math>
S_{12}=R_2^2 (\alpha_2-\alpha_1)
+ b^2 \int_{\lambda_1}^{\lambda_2} \biggl(
\frac1{2(1 - e^2\sin^2\phi)}+
\frac{\tanh^{-1}(e \sin\phi)}{2e \sin\phi}
- \frac{R_2^2}{b^2}\biggr)\sin\phi
\,d\lambda,
</math>
where the integral is over the geodesic line (so that &phi;
is implicitly a function of &lambda;).  Converting this into
an integral over &sigma;, we obtain
:<math>
S_{12} = R_2^2 E_{12} - e^2a^2\cos\alpha_0 \sin\alpha_0
\int_{\sigma_1}^{\sigma_2}
\frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma}
\frac{\sin\sigma}2 \,d\sigma,
</math>
where
:<math>
t(x) = 1 + x + \sqrt{1 + x}\frac{\sinh^{-1}\!\sqrt x}{\sqrt x},
</math>
and the notation ''E''<sub>12</sub>&nbsp;=&nbsp;&alpha;<sub>2</sub>&nbsp;&minus;&nbsp;&alpha;<sub>1</sub> is used for
the geodesic excess.
The integral can be expressed as a series valid for small ''f''
{{harv|Danielsen|1989}} {{harv|Karney|2013|loc=&sect;6 and addendum}}.

The area of a geodesic polygon is given by summing ''S''<sub>12</sub>
over its edges.  This result holds provided that the polygon does not
include a pole; if it does 2&pi;&nbsp;''R''<sub>2</sub><sup>2</sup> must be added to the
sum.   If the edges are specified by their vertices, then a
[[球面三角學|convenient expression]]
for ''E''<sub>12</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>
This result follows from one of [[球面三角學|Napier's analogies]].

=== Software implementations ===
An implementation of Vincenty's algorithm in [[Fortran]] is provided by
{{harvtxt|NGS|2012}}.  Version 3.0 includes Vincenty's treatment of
nearly antipodal points {{harv|Vincenty|1975b}}.
[[Vincenty's formulae|Vincenty's original formulas]] are used in many
geographic information systems.  Except for nearly antipodal points
(where the inverse method fails to converge), this method is accurate to
about 0.1&nbsp;mm for the WGS84 ellipsoid {{harv|Karney|2011|loc=&sect;9}}.

The algorithms given in {{harvtxt|Karney|2013}} are included in
GeographicLib {{harv|Karney|2014}}.  These are accurate to about
15&nbsp;nanometers for the WGS84 ellipsoid.  Implementations in several
languages ([[C++]], [[C语言|C]], Fortran,
[[Java]], [[JavaScript]],
[[Python]], [[MATLAB|Matlab]], and
[[Maxima]]) are provided.  In addition to solving the
basic geodesic problem, this library can return ''m''<sub>12</sub>,
''M''<sub>12</sub>, ''M''<sub>21</sub>, and ''S''<sub>12</sub>.  The
library includes a [[command-line utility]], <code>GeodSolve</code>, for
geodesic calculations.  As of version 4.9.1, the [[PROJ.4]] library for
cartographic projections uses the C implementation for geodesic
calculations.  This is exposed in the command-line utility,
<code>geod</code>, and in the library itself.

The solution of the geodesic problems in terms of elliptic integrals is
included in GeographicLib (in C++ only), e.g., via the <code>-E</code>
option to <code>GeodSolve</code>.  This method of solution is about
2&ndash;3 times slower than using series expansions; however it
provides accurate solutions for ellipsoids of revolution with
''b''/''a''&nbsp;&isin;&nbsp;[0.01,&nbsp;100] {{harv|Karney|2013|loc=addenda}}.

== Geodesics on a triaxial ellipsoid ==
Solving the geodesic problem for an ellipsoid of revolution is, from the
mathematical point of view, relatively simple: because of symmetry,
geodesics have a constant of the motion, given by Clairaut's relation
allowing the problem to be reduced to
[[Quadrature (mathematics)|quadrature]].  By the early 19th century
(with the work of Legendre, [[Barnaba Oriani|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 [[椭球|triaxial ellipsoid]] (with 3 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=&sect;3.5}}.{{refn|
This section is adapted from the documentation for GeographicLib
{{harv|Karney|2014|
loc=[http://geographiclib.sourceforge.net/1.32/triaxial.html geodesics on a triaxial ellipsoid]}}}}{{refn|
Even though Jacobi and {{harvtxt|Weierstrass|1861}}
use terrestrial geodesics as the motivation for their
work, a triaxial ellipsoid approximates the Earth only slightly better
than an ellipsoid of revolution.  A better approximation to the shape of
the Earth is given by the [[大地水准面|geoid]].  However, geodesics on a surface of
the complexity of the geoid are partly [[混沌理论|chaotic]]
{{harv|Waters|2011}}.}}

=== Triaxial coordinate systems ===
[[File:Gaspard monge litho delpech.jpg|thumb|150px|[[加斯帕尔·蒙日|Gaspard Monge]]]]
[[File:Charles Dupin.jpeg|thumb|150px|[[夏尔·迪潘|Charles Dupin]]]]
The key to the solution is expressing the problem in the "right"
coordinate system.  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 (''X'',''Y'',''Z'') are Cartesian coordinates centered on the
ellipsoid and, without loss of generality,
''a''&nbsp;&ge;&nbsp;''b''&nbsp;&ge;&nbsp;''c''&nbsp;&gt;&nbsp;0.{{refn|1=
This notation for the semi-axes is incompatible with that used in the
previous section on ellipsoids of revolution, where ''a'' and
''b'' stood for the equatorial radius and polar semi-axis.
Thus the corresponding inequalities are ''a''&nbsp;=&nbsp;''a''&nbsp;&ge;&nbsp;''b''&nbsp;&gt;&nbsp;0 for
an oblate ellipsoid and ''b''&nbsp;&ge;&nbsp;''a''&nbsp;=&nbsp;''a''&nbsp;&gt;&nbsp;0 for a prolate
ellipsoid.}}
A point on the surface is specified by a latitude and longitude.  The
''geographical'' latitude and longitude (&phi;,&nbsp;&lambda;) are
defined by
:<math>
 \frac{\nabla h}{\left| \nabla h\right|} = \left(
\begin{array}{c} \cos\phi \cos\lambda \\ \cos\phi \sin\lambda \\ \sin\phi
\end{array}\right).
</math>
The ''parametric'' latitude and longitude (&phi;&prime;,&nbsp;&lambda;&prime;)
are defined by
:<math>
\begin{align}
 X &= a \cos\phi' \cos\lambda', \\
 Y &= b \cos\phi' \sin\lambda', \\
 Z &= c \sin\phi'.
\end{align}
</math>
{{harvtxt|Jacobi|1866|loc=&sect;&sect;26&ndash;27}}
employed the ''ellipsoidal'' latitude and longitude
(&beta;,&nbsp;&omega;) defined by
[[File:Triaxial ellipsoid coordinate system.svg|thumb|
Fig. 21.
[[橢球坐標系|Ellipsoidal coordinates]].]]
:<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 ''b''&nbsp;&rarr;&nbsp;''a'', &beta;
becomes the parametric latitude for an oblate ellipsoid, so the use of
the symbol &beta; is consistent with the previous sections.
However, &omega; is ''different'' from the spherical
longitude defined above.{{refn|
The limit ''b''&nbsp;&rarr;&nbsp;''c'' gives a prolate ellipsoid with
&omega; playing the role of the parametric latitude.}}

Grid lines of constant &beta; (in blue) and
&omega; (in green) are given in Fig. 21.  In contrast
to (&phi;,&nbsp;&lambda;) and (&phi;&prime;,&nbsp;&lambda;&prime;),
(&beta;,&nbsp;&omega;) is an [[正交|orthogonal]] coordinate system: the
grid lines intersect at right angles.  The principal sections of the
ellipsoid, defined by ''X''&nbsp;=&nbsp;0 and ''Z''&nbsp;=&nbsp;0 are shown in
red.  The third principal section, ''Y''&nbsp;=&nbsp;0, is covered by the
lines &beta;&nbsp;=&nbsp;&plusmn;90&deg; and &omega;&nbsp;=&nbsp;0&deg; or
&plusmn;180&deg;.  These lines meet at four
[[umbilical point]]s (two of which are visible in this figure) where the
[[主曲率|principal radii of curvature]] are equal.  Here
and in the other figures in this section the parameters of the ellipsoid
are ''a'':''b'':''c''&nbsp;=&nbsp;1.01:1:0.8, and it is viewed in an orthographic
projection from a point above &phi;&nbsp;=&nbsp;40&deg;,
&lambda;&nbsp;=&nbsp;30&deg;.

The grid lines of the ellipsoidal coordinates may be interpreted in three
different ways
# They are "lines of curvature" on the ellipsoid, i.e., they are parallel to the directions of principal curvature {{harv|Monge|1796}}.
# They are also intersections of the ellipsoid with [[橢球坐標系|confocal systems of hyperboloids of one and two sheets]] {{harv|Dupin|1813|loc = [https://books.google.com/books?id=j40AAAAAMAAJ&pg=PA297 Part 5]}}.
# Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points {{harv|Hilbert|Cohn-Vossen|1952|p=188}}.  For example, the lines of constant &beta; in Fig. 21 can be generated with the familiar [[椭圆|string construction for ellipses]] with the ends of the string pinned to the two umbilical points.

Conversions between these three types of latitudes and longitudes
and the Cartesian coordinates are simple algebraic exercises.

The element of length on the ellipsoid in ellipsoidal coordinates is
given by
:<math>
\begin{align}
\frac{ds^2}{(a^2-b^2)\sin^2\omega+(b^2-c^2)\cos^2\beta} &=
\frac{b^2\sin^2\beta+c^2\cos^2\beta}
     {a^2-b^2\sin^2\beta-c^2\cos^2\beta}
 d\beta^2 \\
&\qquad+
\frac{a^2\sin^2\omega+b^2\cos^2\omega}
     {a^2\sin^2\omega+b^2\cos^2\omega-c^2}
 d\omega^2
\end{align}
</math>
and the differential equations for a geodesic are
:<math>
\begin{align}
\frac{d\beta}{ds} &=
\frac1{\sqrt{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta}}
\frac
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}
{\sqrt{b^2 \sin^2\beta + c^2 \cos^2\beta}} \cos\alpha,\\
\frac{d\omega}{ds} &=
\frac1{\sqrt{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta}}
\frac
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}} \sin\alpha,\\
\frac{d\alpha}{ds} &=
\frac1{((a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta)^{3/2}}\times\\
&\quad\biggl(\frac
{(a^2-b^2) \cos\omega\sin\omega
\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}} \cos\alpha \\
&\qquad+\frac
{(b^2-c^2) \cos\beta \sin\beta
\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}} \sin\alpha\biggr).
\end{align}
</math>

=== Jacobi's solution ===
[[File:Carl Jacobi.jpg|thumb|150px|[[卡爾·雅可比|C. G. J. Jacobi]]]]
[[File:Joseph liouville.jpeg|thumb|150px|[[约瑟夫·刘维尔|Joseph Liouville]]]]
[[File:Darboux.jpg|thumb|150px|[[讓·加斯東·達布|J. G. Darboux]]]]
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>
[[柯尼斯堡|Königsberg]], 28th Dec. '38.
</blockquote>

The solution given by Jacobi {{harv|Jacobi|1839}}
{{harv|Jacobi|1866|loc=&sect;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{(b^2-c^2)\cos^2\beta - \gamma}}\\
&\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{(a^2-b^2)\sin^2\omega + \gamma}}.
\end{align}
</math>
As Jacobi notes &quot;a function of the angle &beta; equals
a function of the angle &omega;.  These two functions are
just Abelian integrals...&quot;  Two constants &delta; and
&gamma; appear in the solution.  Typically
&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 &gamma;.  However, for geodesics
that start at an umbilical points, we have &gamma;&nbsp;=&nbsp;0 and
&delta; determines the direction at the umbilical point.
The constant &gamma; may be expressed as
:<math>
\gamma = (b^2-c^2)\cos^2\beta\sin^2\alpha-(a^2-b^2)\sin^2\omega\cos^2\alpha,
</math>
where &alpha; is the angle the geodesic makes with lines of
constant &omega;. In the limit ''b''&nbsp;&rarr;&nbsp;''a'',
this reduces to sin&alpha;&nbsp;cos&beta;&nbsp;=&nbsp;const., the
familiar Clairaut relation.  A nice derivation of Jacobi's result is
given by {{harvtxt|Darboux|1894|loc=&sect;&sect;583&ndash;584}} where he
gives the solution found by {{harvtxt|Liouville|1846}} for general quadratic
surfaces.  In this formulation, the distance along the geodesic,
''s'', is found using
:<math>
\begin{align}
\frac{ds}{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta}
&= \frac
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}\,d\beta}
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}
 \sqrt{(b^2-c^2)\cos^2\beta - \gamma}}\\
&= \frac
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}\,d\omega}
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}
 \sqrt{(a^2-b^2)\sin^2\omega + \gamma}}.
\end{align}
</math>
An alternative expression for the distance is
:<math>
\begin{align}
ds
&= \frac
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}
 \sqrt{(b^2-c^2)\cos^2\beta - \gamma}\,d\beta}
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}\\
&\quad {}+ \frac
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}
 \sqrt{(a^2-b^2)\sin^2\omega + \gamma}\,d\omega}
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}.
\end{align}
</math>

=== 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, &omega;<sub>1</sub> = 0&deg;, &alpha;<sub>1</sub> = 90&deg;.
|caption1=Fig. 22. &beta;<sub>1</sub> = 45.1&deg;.
|caption2=Fig. 23. &beta;<sub>1</sub> = 87.48&deg;.
}}
On a triaxial ellipsoid, there are only 3 simple closed geodesics, the
three principal sections of the ellipsoid given by ''X''&nbsp;=&nbsp;0,
''Y''&nbsp;=&nbsp;0, and ''Z''&nbsp;=&nbsp;0.{{refn|
If ''c''/''a''&nbsp;&lt;&nbsp;&frac12;, there are other simple closed geodesics
similar to those shown in Figs. 13 and 14
{{harv|Klingenberg|1982|loc=&sect;3.5.19}}.}}
To survey the other geodesics, it is convenient to consider geodesics
which intersect the middle principal section, ''Y''&nbsp;=&nbsp;0, at right
angles.  Such geodesics are shown in Figs. 22&ndash;26,
which use the same ellipsoid parameters and the same viewing direction
as Fig. 21.  In addition, the three principal ellipses are shown
in red in each of these figures.

If the starting point is &beta;<sub>1</sub>&nbsp;&isin;&nbsp;(&minus;90&deg;,&nbsp;90&deg;),
&omega;<sub>1</sub>&nbsp;=&nbsp;0, and &alpha;<sub>1</sub>&nbsp;=&nbsp;90&deg;, then
&gamma;&nbsp;&gt;&nbsp;0 and the
geodesic encircles the ellipsoid in a "circumpolar" sense.  The geodesic
oscillates north and south of the equator; on each oscillation it
completes slightly less that a full circuit around the ellipsoid
resulting, in the typical case, in the geodesic filling the area bounded
by the two latitude lines &beta;&nbsp;=&nbsp;&plusmn;&beta;<sub>1</sub>.  Two examples
are given in Figs. 22 and 23.  Figure 22 shows
practically the same behavior as for an oblate ellipsoid of revolution
(because ''a''&nbsp;&asymp;&nbsp;''b''); compare to Fig. 11.
However, if the starting point is at a higher latitude (Fig. 22)
the distortions resulting from ''a''&nbsp;&ne;&nbsp;''b'' are evident.  All
tangents to a circumpolar geodesic touch the confocal single-sheeted
hyperboloid which intersects the ellipsoid at &beta;&nbsp;=&nbsp;&beta;<sub>1</sub>
{{harv|Chasles|1846}}
{{harv|Hilbert|Cohn-Vossen|1952|pp=223&ndash;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, &beta;<sub>1</sub> = 90&deg;, &alpha;<sub>1</sub> = 180&deg;.
|caption1=Fig. 24. &omega;<sub>1</sub> = 39.9&deg;.
|caption2=Fig. 25. &omega;<sub>1</sub> = 9.966&deg;.
}}
If the starting point is &beta;<sub>1</sub>&nbsp;=&nbsp;90&deg;,
&omega;<sub>1</sub>&nbsp;&isin;&nbsp;(0&deg;,&nbsp;180&deg;), and
&alpha;<sub>1</sub>&nbsp;=&nbsp;180&deg;, then
&gamma;&nbsp;&lt;&nbsp;0 and the geodesic encircles the ellipsoid
in a "transpolar" sense.  The geodesic oscillates east and west of the
ellipse ''X''&nbsp;=&nbsp;0; on each oscillation it completes slightly more
that a full circuit around the ellipsoid resulting, in the typical case,
in the geodesic filling the area bounded by the two longitude lines
&omega;&nbsp;=&nbsp;&omega;<sub>1</sub> and &omega;&nbsp;=&nbsp;180&deg;&nbsp;&minus;&nbsp;&omega;<sub>1</sub>.
If ''a''&nbsp;=&nbsp;''b'', all meridians are geodesics; the effect of
''a''&nbsp;&ne;&nbsp;''b'' causes such geodesics to oscillate east and west.
Two examples are given in Figs. 24 and 25.  The constriction
of the geodesic near the pole disappears in the limit
''b''&nbsp;&rarr;&nbsp;''c''; in this case, the ellipsoid becomes a
prolate ellipsoid and Fig. 24 would resemble Fig. 12
(rotated on its side).  All tangents to a transpolar geodesic touch the
confocal double-sheeted hyperboloid which intersects the ellipsoid at
&omega;&nbsp;=&nbsp;&omega;<sub>1</sub>.

[[File:Unstable umbilical geodesic on a triaxial ellipsoid.svg|thumb|
Fig. 26.  An umbilical geodesic, &beta;<sub>1</sub> = 90&deg;,
&omega;<sub>1</sub> = 0&deg;, &alpha;<sub>1</sub> = 135&deg;.]]
If the starting point is &beta;<sub>1</sub>&nbsp;=&nbsp;90&deg;,
&omega;<sub>1</sub>&nbsp;=&nbsp;0&deg; (an umbilical point), and
&alpha;<sub>1</sub>&nbsp;=&nbsp;135&deg; (the geodesic leaves the ellipse
''Y''&nbsp;=&nbsp;0 at right angles), then
&gamma;&nbsp;=&nbsp;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''&nbsp;=&nbsp;0 becomes closer to 0&deg; or
180&deg; so that asymptotically the geodesic lies on the
ellipse ''Y''&nbsp;=&nbsp;0 {{harv|Hart|1849}} {{harv|Arnold|1989|p=265}}.
This is shown in Fig. 26.  Note that a single geodesic does not
fill an area on the ellipsoid.  All tangents to umbilical geodesics
touch the confocal hyperbola which 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''&nbsp;=&nbsp;0 and ''Z''&nbsp;=&nbsp;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''&nbsp;=&nbsp;0, which goes through all 4 umbilical points, is ''exponentially unstable''.  If it is perturbed, it will swing out of the plane ''Y''&nbsp;=&nbsp;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, then its envelope is an astroid with two cusps lying on
&beta;&nbsp;=&nbsp;&minus;&beta;<sub>1</sub> and the other two on
&omega;&nbsp;=&nbsp;&omega;<sub>1</sub>&nbsp;+&nbsp;&pi; {{harv|Sinclair|2003}}.  The cut locus
for ''A'' is the portion
of the line &beta;&nbsp;=&nbsp;&minus;&beta;<sub>1</sub> between the cusps
{{harv|Itoh|Kiyohara|2004}}.

{{harv|Panou|2013}} gives a method for solving the inverse problem for a
triaxial ellipsoid by directly integrating the system of
ordinary differential equations for a geodesic.  (Thus, it does not
utilize Jacobi's solution.)

== Applications ==
[[File:Karl Weierstrass.jpg|thumb|150px|[[卡尔·魏尔斯特拉斯|Karl Weierstrass]]]]
[[File:Henri Poincaré-2.jpg|thumb|150px|[[儒勒·昂利·庞加莱|Henri Poincaré]]]]
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 problem 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 [[地理信息系统|geographic information systems]];
* the definition of [[maritime boundaries]] {{harv|UNCLOS|2006}};
* in the rules of the [[美国联邦航空管理局|Federal Aviation Administration]] for area navigation {{harv|RNAV|2007}};
* the method of measuring distances in the [[国际航空联合会|FAI]] Sporting Code {{harv|FAI|2013}}.

By the [[最小作用量原理|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|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 [[橢圓函數|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 ''c''&nbsp;&rarr;&nbsp;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 ==
* [[Geographical distance]]
* [[Great-circle navigation]]
* [[测地线|Geodesics]]
* [[大地测量学|Geodesy]]
* [[Meridian arc]]
* [[Rhumb line]]
* [[Vincenty's formulae]]

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

== References ==
{{refbegin|30em}}
*{{cite book
|ref = harv |year = 1989
|last = Arnold |first = V. I. |authorlink = Vladimir Arnold
|title = Mathematical Methods of Classical Mechanics
|edition = 2nd
|publisher = Springer-Verlag
|others = Translated by K. Vogtmann & A. Weinstein
|oclc = 4037141
|isbn = 978-0-387-96890-2
}}
*{{cite book
|ref = {{harvid|Bagratuni|1962}} |year = 1967 |origyear = 1962
|last = Bagratuni |first = G. V.
|title = Course in Spheroidal Geodesy
|url = http://geographiclib.sf.net/geodesic-papers/bagratuni67.pdf
|postscript = . Translation from Russian of ''Курс сфероидической геодезии'' (Moscow, 1962) by U.S. Air Force ([http://www.dtic.mil/docs/citations/AD0650520 FTD-MT-64-390])
|oclc = 6150611
}}
*{{cite doi
|10.1007/BF03198517
|comment = Beltrami 1865
|noedit
}}
*{{cite book
|ref = harv |year = 2010
|last = Berger |first = M. |authorlink = Marcel Berger
|title = Geometry Revealed
|publisher = Springer
|others = Translated by L. J. Senechal
|isbn = 978-3-540-70996-1
|doi = 10.1007/978-3-540-70997-8
}}
*{{cite journal
|ref = {{harvid|Bessel|1825}} |year = 2010 |origyear = 1825
|last = Bessel |first = F. W. |authorlink = Friedrich Bessel
|doi = 10.1002/asna.201011352
|title = The calculation of longitude and latitude from geodesic measurements
|journal = Astronomische Nachrichten
|volume = 331 |issue = 8 |pages = 852&ndash;861
|arxiv = 0908.1824
|others = .  Translated by C. F. F. Karney & R. E. Deakin
|postscript = . English translation of [http://adsabs.harvard.edu/abs/1825AN......4..241B ''Astron. Nachr.'' '''4''', 241–254 (1825)]. [http://geographiclib.sourceforge.net/bessel-errata.html Errata].
}}
*{{cite doi
|10.1002/asna.18270051202
|comment = Bessel 1827
|noedit
}}
*{{cite doi
|10.1090/S0002-9947-1916-1501037-4
|comment = Bliss 1916
|noedit
}}
*{{cite book
|ref = harv |year = 1952
|last = Bomford |first = G.
|title = Geodesy
|publisher = Clarendon
|location = Oxford
|oclc = 1396190
}}
*{{cite book
|ref = harv |year = 2012
|last1 = Borre |first1 = K.
|last2 = Strang |first2 = W. G. |author2-link = Gilbert Strang
|title = Algorithms for Global Positioning
|oclc = 795014501
|isbn = 978-0-9802327-3-8
|publisher = Wellesley-Cambridge Press
|postscript = .  Chapter 11, [http://old.gps.aau.dk/downloads/notes.pdf Geometry of the Ellipsoid].
}}
*{{cite doi
|10.1007/BF02198293
|comment = Carlson 1995
|noedit
}}
*{{cite journal
|ref = harv |year = 1991
|last1 = Casper |first1 = P. W.
|last2 = Bent |first2 = R. B.
|title = The effect of the Earth's oblate spheroid shape on the accuracy of a time-of-arrival lightning ground strike locating system
|url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910023392_1991023392.pdf
|postscript = . in Proceedings 1991 International Aerospace and Ground Conference on Lightning and Static Electricity, (Vol. 2).
}}
*{{cite journal
|ref = harv |year = 1735
|last = Cassini |first = J. |authorlink = Jacques Cassini
|title = De la carte de la France et de la perpendiculaire a la méridienne de Paris
|trans_title = The map of France and the perpendicular to the meridian of Paris
|language = French
|journal = Mémoires de l'Académie Royale des Sciences de Paris 1733
|pages = 389&ndash;405
|url = https://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA389
}}
*{{cite journal
|ref = harv |year = 1870
|last = Cayley |first = A. |authorlink = Arthur Cayley
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*{{cite journal
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|language = German
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|language = French
|journal = Mémoires de l'Académie Royale des Sciences de Paris 1733
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{{refend}}

== External links ==
* [http://geographiclib.sourceforge.net/geodesic-papers/biblio.html Online geodesic bibliography], approximately 180 books and articles on geodesics on ellipsoids together with links to online copies.
* Implementations of {{harvtxt|Vincenty|1975a}} for oblate ellipsoids:
** [http://www.ngs.noaa.gov/PC_PROD/Inv_Fwd/ NGS implementation], includes modifications described in {{harvtxt|Vincenty|1975b}}.
** [http://www.ngs.noaa.gov/TOOLS/Inv_Fwd/Inv_Fwd.html NGS online tools].
** [http://www.ga.gov.au/earth-monitoring/geodesy/geodetic-techniques/distance-calculation-algorithms.html Online calculator from Geoscience Australia].
** Javascript implementations of solutions to [http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html direct problem] and [http://www.movable-type.co.uk/scripts/latlong-vincenty.html inverse problem].
* Implementation of {{harvtxt|Karney|2013}} for ellipsoids of revolution in Geographiclib {{harv|Karney|2014}}:
** [http://geographiclib.sourceforge.net/ GeographicLib web site] for downloading library and documentation.
** [http://geographiclib.sourceforge.net/html/GeodSolve.1.html GeodSolve(1)], [[手册页|man page]] for a utility for geodesic calculations.
** [http://geographiclib.sourceforge.net/cgi-bin/GeodSolve An online version of GeodSolve].
** [http://geographiclib.sourceforge.net/html/Planimeter.1.html Planimeter(1)], man page for a utility for calculating the area of geodesic polygons.
** [http://geographiclib.sourceforge.net/cgi-bin/Planimeter An online version of Planimeter].
** [http://trac.osgeo.org/proj/wiki/man_geod geod(1)], man page for the PROJ.4 utility for geodesic calculations.
** [http://geographiclib.sourceforge.net/scripts/geod-calc.html Javascript utility for direct and inverse problems and area calculations].
** [http://geographiclib.sourceforge.net/scripts/geod-google.html Drawing geodesics on Google Maps].
** [http://www.mathworks.com/matlabcentral/fileexchange/50605 Matlab implementation of the geodesic routines] (used for the figures for geodesics on ellipsoids of revolution in this article).
** The first [http://geographiclib.sourceforge.net/2009-03/geodesic.html description of the geodesic algorithms] from {{harv|Karney|2009}}.
* Geodesics on a triaxial ellipsoid:
** [http://geographiclib.sourceforge.net/html/triaxial.html Additional notes about Jacobi's solution].
** [http://www.math.harvard.edu/~knill/caustic/exhibits/ellipsoid/index.html Caustics on an ellipsoid].

[[Category:大地测量学]]
[[Category:Geodesic (mathematics)]]
[[Category:微分几何]]
[[Category:变分法]]