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| The study of '''geodesics on an ellipsoid''' arose in connection with geodesy
| |
| specifically with the solution of [[triangulation network]]s. The
| |
| [[figure of the earth]] is well approximated by an
| |
| ''[[oblate ellipsoid]]'', a slightly flattened sphere. A ''[[geodesic]]''
| |
| is the shortest path between two points on a curved surface, i.e., the analogue
| |
| of a [[straight line]] on a plane surface. The solution of a triangulation
| |
| network on an ellipsoid is therefore a set of exercises in spheroidal
| |
| trigonometry {{harv|Euler|1755}}.
| |
|
| |
|
| [[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|150px|[[Isaac Newton]]]]
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| [[File:Leonhard Euler.jpg|thumb|150px|[[Leonhard Euler]]]]
| |
| If the earth is treated as a [[sphere]], the geodesics are
| |
| [[great circles]] (all of which are closed) and the problems reduce to
| |
| ones in [[spherical trigonometry]]. However, {{harvtxt|Newton|1687}}
| |
| showed that the effect of the rotation of the earth results in its
| |
| resembling a slightly oblate ellipsoid and, in this case, the
| |
| [[equator]] and the [[meridian (geography)|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 [[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
| |
| [[Earth radius#Mean radius|mean earth radius]]—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°59′; while the connecting great circle follows the
| |
| equator, the shortest geodesics pass within
| |
| 180 km of either [[geographical pole|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
| |
| [[prolate spheroid|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
| |
| [[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
| |
| [[closed surface|closed]], [[simply connected]] surfaces; this was
| |
| conjectured by {{harvtxt|Poincaré|1905}} and proved by
| |
| {{harvtxt|Lyusternik|Schnirelmann|1929}}
| |
| {{harv|Klingenberg|1982|loc=§3.7}}.
| |
| | |
| == Geodesics on an ellipsoid of revolution ==
| |
| There are several ways of defining geodesics
| |
| {{harv|Hilbert|Cohn-Vossen|1952|pp=220–221}}. A simple definition
| |
| is as the shortest path between two points on a surface. However it is
| |
| frequently more useful to define them as paths with zero
| |
| [[geodesic curvature]]—i.e., the analogue of [[straight lines]] on a
| |
| curved surface. This definition encompasses geodesics traveling so far
| |
| across the ellipsoid's surface (somewhat 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
| |
| [[spheroid]] is also used) was a well-accepted approximation to the
| |
| [[figure of the Earth]]. The adjustment of [[triangulation network]]s
| |
| entailed reducing all the measurements to a [[reference ellipsoid]] and
| |
| solving the resulting two-dimensional problem as an exercise in
| |
| spheroidal trigonometry {{harv|Bomford|1952|loc=Chap. 3}}.
| |
| | |
| [[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
| |
| φ<sub>1</sub> and longitude λ<sub>1</sub> and
| |
| ''B'' at latitude φ<sub>2</sub> and longitude
| |
| λ<sub>2</sub> (see Fig. 1). The connecting geodesic
| |
| (from ''A'' to ''B'') is ''AB'', of length
| |
| ''s''<sub>12</sub>, which has [[azimuth]]s α<sub>1</sub> and
| |
| α<sub>2</sub> at the two endpoints.{{refn|
| |
| Here α<sub>2</sub> is the ''forward'' azimuth at ''B''.
| |
| Some authors calculate the ''back'' azimuth instead; this is given by
| |
| α<sub>2</sub> ± π.}} The two geodesic problems usually
| |
| considered are:
| |
| # the ''direct geodesic problem'' or ''first geodesic problem'', given ''A'', α<sub>1</sub>, and ''s''<sub>12</sub>, determine ''B'' and α<sub>2</sub>;
| |
| # the ''inverse geodesic problem'' or ''second geodesic problem'', given ''A'' and ''B'', determine ''s''<sub>12</sub>, α<sub>1</sub>, and α<sub>2</sub>.
| |
| As can be seen from Fig. 1, these problems involve solving the triangle
| |
| ''NAB'' given one angle, α<sub>1</sub> for the direct | |
| problem and λ<sub>12</sub> = λ<sub>2</sub> − λ<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=[http://books.google.com/books?id=WlsOAAAAYAAJ&pg=PA69 69]}}.
| |
| | |
| For a sphere the solutions to these problems are simple exercises in
| |
| [[spherical trigonometry]], whose solution is given by
| |
| [[Solution of triangles#Two sides and the included angle given|formulas
| |
| for solving a spherical triangle]].
| |
| (See the article on [[great-circle navigation]].)
| |
| | |
| [[File:Alexis Clairault.jpg|thumb|150px|[[Alexis Clairault]]]]
| |
| [[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 [[plagiarism]] from {{harvtxt|Ivory|1826}}
| |
| (his phrase was "second-hand from Germany"), which prompted an angry
| |
| rebuttal by {{harvtxt|Bessel|1827}}.}}
| |
| | |
| Much of the early work on these problems was carried out by
| |
| mathematicians—for example, [[Adrien-Marie Legendre|Legendre]],
| |
| [[Friedrich Bessel|Bessel]], and [[Carl Friedrich Gauss|Gauss]]—who
| |
| were also heavily involved in the practical aspects of [[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.
| |
| | |
| === Equations for a geodesic ===
| |
| [[File:Friedrich Wilhelm Bessel (1839 painting).jpg|thumb|150px|[[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=§15}},
| |
| {{harvtxt|Krakiwsky|Thomson|1974|loc=§4}}, and
| |
| {{harvtxt|Rapp|1993|loc=§1.2}} also provide derivations of these
| |
| equations.
| |
| | |
| Consider an ellipsoid of revolution with equatorial radius
| |
| ''a'' and polar semi-axis ''b''. Define the
| |
| flattening ''f'' = (''a'' − ''b'')/''a'', eccentricity
| |
| ''e''<sup>2</sup> = ''f''(2 − ''f'') second eccentricity
| |
| ''e''′ = ''e''/(1 − ''f''). (In most applications in geodesy, the
| |
| ellipsoid is taken to be oblate, ''a'' > ''b''; however, the theory
| |
| applies without change to prolate ellipsoids, ''a'' < ''b'', in
| |
| which case ''f'', ''e''<sup>2</sup>, and ''e''′<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 α, then ''ds'' can
| |
| is related to ''d''φ and ''d''λ by
| |
| :(1)<math>{\color{white}.}\qquad
| |
| \cos\alpha\,ds = \rho\,d\phi = - dR/\sin\phi, \quad
| |
| \sin\alpha\,ds = R\,d\lambda,</math>
| |
| where ρ is the
| |
| [[Earth radius#Meridional|meridional radius of curvature]],
| |
| ''R'' = ν cosφ is the radius of the circle of latitude
| |
| φ, and ν is the
| |
| [[Earth radius#Normal|normal radius of curvature]].
| |
| The elementary segment can therefore be expressed as
| |
| :<math>\begin{align}ds &= \sqrt{\rho^2\phi'^2 + R^2}\,d\lambda \\
| |
| &\equiv L(\phi,\phi')\,d\lambda,
| |
| \end{align}
| |
| </math>
| |
| where φ′ = ''d''φ/''d''λ and ''L'' depends on
| |
| φ through ρ(φ) and
| |
| ''R''(φ). The length of an arbitrary path between
| |
| (φ<sub>1</sub>, λ<sub>1</sub>) and (φ<sub>2</sub>, λ<sub>2</sub>) is
| |
| given by
| |
| :<math> s_{12} =
| |
| \int_{\lambda_1}^{\lambda_2} L(\phi, \phi')\,d\lambda,</math>
| |
| where φ is a function of λ satisfying
| |
| φ(λ<sub>1</sub>) = φ<sub>1</sub> and
| |
| φ(λ<sub>2</sub>) = φ<sub>2</sub>. The shortest path or geodesic
| |
| entails finding that function φ(λ) which minimizes
| |
| ''s''<sub>12</sub>. This is an exercise in the
| |
| [[calculus of variations]] and the minimizing condition is given by the
| |
| [[Beltrami identity]],
| |
| :<math>L - \phi' \frac{\partial L}{\partial \phi'} = \text{const.}
| |
| </math>
| |
| [[File:Construction for parametric latitude.svg|thumb|
| |
| Fig. 4.
| |
| The geometric construction for parametric latitude, β. A point
| |
| ''P'' at latitude φ on the meridian (red) is mapped to a point
| |
| ''P′'' on a sphere of radius ''a'' (shown as a blue circle) by
| |
| keeping the radius ''R'' constant.]]
| |
| Substituting for ''L'' and using Eq. (1) gives
| |
| :<math>R\sin\alpha = \text{const.}</math>
| |
| {{harvtxt|Clairaut|1735}} first found this [[Clairaut's relation|relation]],
| |
| using a geometrical construction.{{refn|
| |
| {{harvtxt|Laplace|1799a}} showed that a particle constrained to move on
| |
| a surface but otherwise subject to no forces moves along a geodesic for
| |
| that surface. Thus, Clairaut's relation is just a consequence of
| |
| [[conservation of angular momentum]] for a particle on a surface of
| |
| revolution. A similar proof is given by
| |
| {{harvtxt|Bomford|1952|loc=§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
| |
| [[ordinary differential equations]] for a geodesic
| |
| :(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
| |
| [[Latitude#Reduced (or parametric) latitude|parametric latitude]],
| |
| β, 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 [[Spherical trigonometry#Sine rules|sine rule]] of spherical
| |
| trigonometry relating two sides of the triangle ''NAB'' (see
| |
| Fig. 5), ''NA'' = ½π − β<sub>1</sub>, and
| |
| ''NB'' = ½π − β<sub>2</sub> and their opposite angles
| |
| ''B'' = π − α<sub>2</sub> and ''A'' = α<sub>1</sub>.
| |
| | |
| In order to find the relation for the third side
| |
| ''AB'' = σ<sub>12</sub>, the ''spherical arc length'', and included
| |
| angle ''N'' = ω<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
| |
| σ, ''s'' and ω.
| |
| | |
| [[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 λ
| |
| :<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'' = −sinφ/cosφ 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 β and φ 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
| |
| [[Latitude#Reduced (or parametric) latitude|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 σ 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 σ.}} in both of
| |
| these differential equations and thereby to express ''s'' and
| |
| λ as integrals. Applying the sine rule to the vertices
| |
| ''E'' and ''G'' in the spherical triangle
| |
| ''EGP'' in Fig. 6 gives
| |
| :<math>\sin\beta = \sin\beta(\sigma;\alpha_0) =
| |
| \cos\alpha_0 \sin\sigma,</math>
| |
| where α<sub>0</sub> is the azimuth at ''E''.
| |
| Substituting this into the equation for ''ds''/''d''σ 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''(σ = 0) = 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 ''a''(1 − ''e''<sup>2</sup> sin<sup>2</sup>α<sub>0</sub>)<sup>1/2</sup> and
| |
| ''b''. In order to express the equation for
| |
| λ in terms of σ, we write
| |
| :<math>d\omega = \frac{\sin\alpha_0}{\cos^2\beta}\,d\sigma,</math>
| |
| which follows from Eq. (2) 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 λ = λ<sub>0</sub> at the equator crossing,
| |
| σ = 0.
| |
| | |
| In using these integral relations, we allow σ to
| |
| increase continuously (not restricting it to a range
| |
| [−π, π], for example) as the great circle,
| |
| resp. geodesic, encircles the auxiliary sphere, resp. ellipsoid. The
| |
| quantities ω, λ, and ''s''
| |
| are likewise allowed to increase without limit. Once the problem is
| |
| solved, λ 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 λ in terms of the
| |
| spherical quantities depend on α<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=§6}}; however, these are typically comparable
| |
| in complexity to the method for the exact solution given above
| |
| {{harv|Jekeli|2012|loc=§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 α<sub>0</sub> = 45°.
| |
| }}
| |
| [[File:Long geodesic on a prolate ellipsoid.svg|thumb|
| |
| Fig. 12.
| |
| Geodesic on a prolate ellipsoid (''f'' = −1/50) with α<sub>0</sub> = 45°. 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 α<sub>0</sub> = 0 and Eq. (5)
| |
| becomes λ = ω + λ<sub>0</sub>, i.e., the longitude will
| |
| vary the same way as for a sphere, jumping by π each time
| |
| the geodesic crosses the pole. The distance, Eq. (4), reduces to
| |
| the length of an arc of an ellipse with semi-axes ''a'' and
| |
| ''b'' (as expected), expressed in terms of parametric latitude,
| |
| β.
| |
| | |
| The equator (β = 0</sub> on the auxiliary sphere,
| |
| φ = 0 on the ellipsoid) corresponds to
| |
| α<sub>0</sub> = ½π. The distance reduces to the arc of a
| |
| circle of radius ''b'' (and ''not'' ''a''),
| |
| ''s'' = ''b''σ, while the longitude simplifies to
| |
| λ = (1 − ''f'')σ + λ<sub>0</sub>. A geodesic that is nearly
| |
| equatorial will intersect the equator at intervals of
| |
| π''b''. As a consequence, the maximum length of a
| |
| equatorial geodesic which is also a shortest path is π''b''
| |
| on an oblate ellipsoid (on a prolate ellipsoid, the maximum length is
| |
| π''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 α<sub>0</sub> = 45°. A full
| |
| cycle of the geodesic, from one northward crossing of the equator to the
| |
| next is shown. The equatorial crossings are called ''nodes'' and the
| |
| points of maximum or minimum latitude are called ''vertices''; the
| |
| vertex latitudes are given by
| |
| |β| = ±(½π − |α<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°.
| |
| Thus, on each successive northward crossing of the equator (see
| |
| Fig. 10), λ falls short of a full circuit of
| |
| the equator by approximately 2π ''f'' sinα<sub>0</sub> (for a
| |
| prolate ellipsoid, this quantity is negative and λ
| |
| completes more that a full circuit; see Fig. 12). For nearly all
| |
| values of α<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'' < ½, another class of simple closed geodesics is
| |
| possible {{harv|Klingenberg|1982|loc=§3.5.19}}. Two such geodesics
| |
| are illustrated in Figs. 13 and 14. Here
| |
| ''b''/''a'' = 2/7 and the equatorial azimuth,
| |
| α<sub>0</sub>, for the green (resp. blue) geodesic is chosen to
| |
| be 53.175° (resp. 75.192°), so that the geodesic completes 2
| |
| (resp. 3) complete oscillations about the equator on one circuit of the
| |
| ellipsoid.
| |
| | |
| === Evaluation of the integrals ===
| |
| Solving the geodesic problems entails evaluating the integrals for the
| |
| distance, ''s'', and the longitude, λ,
| |
| Eqs. (4) and (5). In geodetic applications where
| |
| ''f'' is small, the integrals are typically evaluated as a
| |
| series; for this purpose the second form of the longitude integral is
| |
| preferred (since it avoids the near singular behavior of the first form
| |
| when geodesics pass close to a pole). In both integrals, the integrand
| |
| is an even periodic function of period π. Furthermore the
| |
| term dependent on σ is multiplied by a small quantity
| |
| ''k''<sup>2</sup> = ''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> = 1 + ''O''(''f'') and ''B''<sub>''j''</sub> = ''O''(''f'' <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'' <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 [[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'' <sup>''J''−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'' = 3
| |
| which provides an accuracy of about 0.1 mm for the [[WGS84]]
| |
| ellipsoid. {{harvtxt|Karney|2013}} gives expansions carried out for
| |
| ''J'' = 6 which suffices to provide full [[double precision]]
| |
| accuracy for |''f''| ≤ 1/50. Trigonometric
| |
| series of this type can be conveniently summed using
| |
| [[Clenshaw summation]].
| |
| | |
| In order to solve the direct geodesic problem, it is necessary to find
| |
| σ given ''s''. Since the integrand in the
| |
| distance integral is positive, this problem has a unique root, which
| |
| may be found using [[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
| |
| σ can be found without iteration;
| |
| {{harvtxt|Helmert|1880}} gives a similar series.{{refn|
| |
| {{harvtxt|Legendre|1806|loc=Art. 13}} also gives a series
| |
| for σ 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''| > 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 σ.
| |
| | |
| 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''| ≤ 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
| |
| [ρ, ν] replaced by [ρ + ''h'', ν + ''h''].
| |
| | |
| [[File:Legendre.jpg|thumb|150px|[[Adrien-Marie Legendre|A. M. Legendre]]]]
| |
| [[File:Arthur Cayley.jpg|thumb|150px|[[Arthur Cayley]]]]
| |
| Geodesics on an ellipsoid was an early application of
| |
| [[elliptic integrals]]. In particular,
| |
| {{harvtxt|Legendre|1811|log=§§126–129}} 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''(φ, ''k''), ''E''(φ, ''k''), and
| |
| Π(φ, α<sup>2</sup>, ''k''), are
| |
| [[Elliptic integral|incomplete elliptic integrals]] in the
| |
| notation of
| |
| {{harvtxt|DLMF|2010|loc=[http://dlmf.nist.gov/19.2.ii §19.2(ii)]}}.{{refn|
| |
| 1=Despite the presence of ''i'' = √−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
| |
| [[Newton's method]]. The use of elliptic integrals provides a good
| |
| method of solving the geodesic problem for
| |
| |''f''| > 1/50.{{refn|1=
| |
| It is also possible to express the integrals in terms of
| |
| [[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'' exp(±''i''λ) = ''X'' ± ''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
| |
| [[doubly periodic function]]; see also
| |
| {{harvtxt|Forsyth|1927|loc=§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
| |
| φ, λ, and ''s'' to
| |
| β, ω and σ, solve
| |
| the corresponding great-circle problem on the sphere and transfer the
| |
| results back to the ellipsoid.
| |
| | |
| In implementing this program, we will frequently need to solve the
| |
| "elementary" spherical triangle problem for ''NEP'' in Fig.
| |
| 6 with ''P'' replaced by either ''A'' (subscript
| |
| 1) or ''B'' (subscript 2). For this purpose, we can apply
| |
| [[Spherical trigonometry#Napier's rules for quadrantal triangles|
| |
| 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 cosβ ≥ 0 and
| |
| cosα<sub>0</sub> ≥ 0.{{refn|name=atan|
| |
| When solving for σ, α, or
| |
| ω 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]] function.}}
| |
| Implementing this plan for the direct problem is straightforward. We
| |
| are given φ<sub>1</sub>, α<sub>1</sub>, and
| |
| ''s''<sub>12</sub>. From φ<sub>1</sub> we obtain
| |
| β<sub>1</sub> (using the formula for the parametric latitude).
| |
| We now solve the triangle problem with ''P'' = ''A'' and
| |
| β<sub>1</sub> and α<sub>1</sub> given to find
| |
| α<sub>0</sub>, σ<sub>1</sub>, and
| |
| ω<sub>1</sub>.{{refn|1=
| |
| If β<sub>1</sub> = 0 and α<sub>1</sub> = ±½π, the
| |
| equation for σ<sub>1</sub> is indeterminate and
| |
| σ<sub>1</sub> = 0 may be used.}} Use
| |
| the distance and longitude equations, Eqs. (4) and
| |
| (5), together with the known value of λ<sub>1</sub>, to
| |
| find ''s''<sub>1</sub> and λ<sub>0</sub>. Determine
| |
| ''s''<sub>2</sub> = ''s''<sub>1</sub> + ''s''<sub>12</sub> and invert the distance equation to find
| |
| σ<sub>2</sub>. Solve the triangle problem with ''P'' = ''B''
| |
| and α<sub>0</sub> and σ<sub>2</sub> given to find
| |
| β<sub>2</sub>, ω<sub>2</sub>, and α<sub>2</sub>.
| |
| Convert β<sub>2</sub> to φ<sub>2</sub> and substitute
| |
| σ<sub>2</sub> and ω<sub>2</sub> into the longitude
| |
| equation to give λ<sub>2</sub>.
| |
| | |
| 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=§5.9}}, and most subsequent
| |
| authors.
| |
| | |
| === Solution of the inverse problem ===
| |
| The ease with which the direct problem can be solved results from the
| |
| fact that given φ<sub>1</sub> and α<sub>1</sub>, we can
| |
| immediately find α<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 λ<sub>12</sub>,
| |
| but we cannot easily relate this to the equivalent spherical angle
| |
| ω<sub>12</sub> because α<sub>0</sub> is unknown.
| |
| Thus, the solution of the problem requires that α<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. λ<sub>12</sub> as a function of α<sub>1</sub> for φ<sub>1</sub> = −30° and φ<sub>2</sub> = 20°.
| |
| |header=Geodesics from a single point (''f'' = 1/10, φ<sub>1</sub> = −30°)
| |
| }}
| |
| | |
| Suppose point ''A'' in the inverse problem has
| |
| φ<sub>1</sub> = −30° and λ<sub>1</sub> = 0°. Fig.
| |
| 15 shows geodesics (in blue) emanating
| |
| ''A'' with α<sub>1</sub> a multiple of
| |
| 15° up to the point at which they cease to be shortest
| |
| paths. (The flattening has been increased to
| |
| 1/10 in order to accentuate the ellipsoidal effects.)
| |
| Also shown (in green) are curves of constant ''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 [[antipodes|antipodal]]
| |
| to ''A'', φ = −φ<sub>1</sub>. The longitudinal
| |
| extent of cut locus is approximately
| |
| λ<sub>12</sub> ∈ [π − ''f'' π cosφ<sub>1</sub>, π + ''f'' π cosφ<sub>1</sub>]. If
| |
| ''A'' lies on the equator, φ<sub>1</sub> = 0, this relation
| |
| is exact and as a consequence the equator is only a shortest geodesic if
| |
| |λ<sub>12</sub>| ≤ (1 − ''f'')π. For a prolate
| |
| ellipsoid, the cut locus is a segment of the anti-meridian centered on
| |
| the point antipodal to ''A'', λ<sub>12</sub> = π,
| |
| and this means that
| |
| meridional geodesics stop being shortest paths before the antipodal
| |
| point is reached.
| |
| | |
| The solution of the inverse problem involves determining, for a given
| |
| point ''B'' with latitude φ<sub>2</sub> and longitude
| |
| λ<sub>2</sub> which blue and green curves it lies on; this
| |
| determines α<sub>1</sub> and ''s''<sub>12</sub> respectively.
| |
| In Fig. 16, the ellipsoid has been "rolled out" onto a
| |
| [[plate carrée projection]]. Suppose φ<sub>2</sub> = 20°, the
| |
| green line in the figure. Then as α<sub>1</sub> is varied
| |
| between 0° and 180°, the longitude
| |
| at which the geodesic intersects φ = φ<sub>2</sub> varies between
| |
| 0° and 180° (see Fig. 17).
| |
| This behavior holds provided that
| |
| |φ<sub>2</sub>| ≤ |φ<sub>1</sub>| (otherwise the
| |
| geodesic does not reach φ<sub>2</sub> for some values of
| |
| α<sub>1</sub>). Thus, the inverse problem may be solved by
| |
| determining the value α<sub>1</sub> which results in the given
| |
| value of λ<sub>12</sub> when the geodesic intersects the circle
| |
| φ = φ<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 α<sub>1</sub>.
| |
| # Solve the so-called ''hybrid geodesic problem'', given φ<sub>1</sub>, φ<sub>2</sub>, and α<sub>1</sub> find λ<sub>12</sub>, ''s''<sub>12</sub>, and α<sub>2</sub>, corresponding to the ''first'' intersection of the geodesic with the circle φ = φ<sub>2</sub>.
| |
| # Compare the resulting λ<sub>12</sub> with the desired value and adjust α<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 λ<sub>12</sub> = 0 or
| |
| ±π. The shortest geodesic follows the equator if
| |
| φ<sub>1</sub> = φ<sub>2</sub> = 0 and
| |
| |λ<sub>12</sub>| ≤ (1 − ''f'')π. For a prolate
| |
| ellipsoid, the meridian is no longer the shortest geodesic if
| |
| λ<sub>12</sub> = ±π and the points are close to antipodal
| |
| (this will be discussed in the next section). There is no longitudinal
| |
| restriction on equatorial geodesics.
| |
| | |
| 2. In most cases a suitable starting value of α<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 ω<sub>12</sub> = λ<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'' = ''A'' to find α<sub>0</sub>,
| |
| σ<sub>1</sub>, ω<sub>1</sub>, and
| |
| λ<sub>0</sub>.{{refn|1=If φ<sub>1</sub> = φ<sub>2</sub> = 0, take
| |
| sinσ<sub>1</sub> = sinω<sub>1</sub> = −0, consistent with the relations
| |
| (8); this gives σ<sub>1</sub> = ω<sub>1</sub> = −π.}} Now find
| |
| α<sub>2</sub> from
| |
| sinα<sub>2</sub> = sinα<sub>0</sub>/cosβ<sub>2</sub>, taking
| |
| cosα<sub>2</sub> ≥ 0 (corresponding to the first, northward,
| |
| crossing of the circle φ = φ<sub>2</sub>). Next,
| |
| σ<sub>2</sub> is given by
| |
| tanσ<sub>2</sub> = tanβ<sub>2</sub>/cosα<sub>2</sub> and
| |
| ω<sub>2</sub> by
| |
| tanω<sub>2</sub> = tanσ<sub>2</sub>/sinα<sub>0</sub>.{{refn|name=atan}}
| |
| Finally, use the distance and longitude equations, Eqs. (4)
| |
| and (5), to find ''s''<sub>12</sub> and
| |
| λ<sub>12</sub>.{{refn|
| |
| The ordering in relations (8) automatically results in
| |
| σ<sub>12</sub> > 0.}}
| |
| | |
| 4. In order to discuss how α<sub>1</sub> is updated, let us define
| |
| the root-finding problem in more detail. The curve in Fig. 17
| |
| shows λ<sub>12</sub>(α<sub>1</sub>; φ<sub>1</sub>, φ<sub>2</sub>) where we regard
| |
| φ<sub>1</sub> and φ<sub>2</sub> as parameters and
| |
| α<sub>1</sub> as the independent variable. We seek the value of
| |
| α<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) ≤ 0 and ''g''(π) ≥ 0. In fact,
| |
| there is a unique root in the interval α<sub>1</sub> ∈ [0, π].
| |
| Any of a number of [[root-finding algorithms]] can be used to solve such
| |
| an equation. {{harvtxt|Karney|2013}} uses [[Newton's method]], which
| |
| requires a good starting guess; however it may be supplemented by a
| |
| [[fail-safe]] method, such as the [[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=§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
| |
| ω<sub>12</sub> = λ<sub>12</sub>, solving the resulting problem on
| |
| auxiliary sphere, and obtaining an updated estimate of
| |
| ω<sub>12</sub> using
| |
| :<math>
| |
| \omega_{12} = \lambda_{12} + f\sin\alpha_0 I(\sigma_1, \sigma_2; \alpha_0).
| |
| </math>
| |
| This process is repeated until convergence. {{harvtxt|Vincenty|1975a}}
| |
| uses this method 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.
| |
| | |
| 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:
| |
| * φ<sub>1</sub> = −φ<sub>2</sub> (with neither point at a pole). If α<sub>1</sub> = α<sub>2</sub>, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by interchanging α<sub>1</sub> and α<sub>2</sub>. (This occurs when λ<sub>12</sub> ≈ ±π for oblate ellipsoids.)
| |
| * λ<sub>12</sub> = ±π (with neither point at a pole). If α<sub>1</sub> = 0 or ±π, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by negating α<sub>1</sub> and α<sub>2</sub>. (This occurs when φ<sub>1</sub> + φ<sub>2</sub> ≈ 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 α<sub>1</sub> + α<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|[[Carl Friedrich Gauss|C. F. Gauss]]]]
| |
| [[File:Elwin Bruno Christoffel.JPG|thumb|150px|[[Elwin Christoffel|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 [[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>, ''s''<sub>2</sub>) = ''m''<sub>12</sub>, the so-called
| |
| ''reduced length'', and ''M''(''s''<sub>1</sub>, ''s''<sub>2</sub>) = ''M''<sub>12</sub>, the
| |
| ''geodesic scale''.{{refn|
| |
| {{harvtxt|Bagratuni|1962|loc=§17}} uses the term "coefficient of
| |
| convergence of ordinates" for the geodesic scale.}}
| |
| Their basic definitions are illustrated in
| |
| Fig. 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 [[Ellipsoid of revolution#Curvature|Gaussian curvature for an ellipsoid of revolution]]
| |
| is
| |
| :<math>
| |
| K = \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''α<sub>1</sub> is ''m''<sub>12</sub> ''d''α<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,
| |
| φ<sub>1</sub> = ½π, then the reduced length is the radius
| |
| of the circle of latitude,
| |
| ''m''<sub>12</sub> = ''a'' cosβ<sub>2</sub> = ''a'' sinσ<sub>12</sub>. Similarly, for a
| |
| meridional geodesic starting on the equator,
| |
| φ<sub>1</sub> = α<sub>1</sub> = 0, we have
| |
| ''M''<sub>12</sub> = cosσ<sub>12</sub>. In the typical case, these
| |
| quantities oscillate with a period of about 2π in
| |
| σ<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/√''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> > 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|Forsyth|1927|loc=§§26–27}}
| |
| {{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, |σ<sub>12</sub>| ≤ π;
| |
| * for a prolate ellipsoid, |λ<sub>12</sub>| ≤ π, if α<sub>0</sub> ≠ 0; if α<sub>0</sub> = 0, the supplemental condition ''m''<sub>12</sub> ≥ 0 is required if |λ<sub>12</sub>| = π.
| |
| The latter condition above can be used to determine whether the shortest
| |
| path is a meridian in the case of a prolate ellipsoid with
| |
| |λ<sub>12</sub>| = π. The derivative required to
| |
| solve the inverse method using Newton's method,
| |
| ∂λ<sub>12</sub>(α<sub>1</sub>; φ<sub>1</sub>, φ<sub>2</sub>) / ∂α<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'', θ) is centered on some point ''A''. The
| |
| coordinates of another point ''B'' are given by
| |
| ''r'' = ''s''<sub>12</sub> and θ = ½π − α<sub>1</sub> and
| |
| these coordinates are used to find the projected coordinates on a plane
| |
| map, ''x'' = ''r'' cosθ and ''y'' = ''r'' sinθ. The
| |
| result is the familiar [[azimuthal equidistant projection]]; in the
| |
| field of the [[differential geometry of surfaces]], it is called the
| |
| [[exponential map]]. Due to the basic properties of geodesics
| |
| {{harv|Gauss|1828}}, lines of constant ''r'' and lines of
| |
| constant θ 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'', ''y'') uses a reference
| |
| geodesic defined by ''A'' and α<sub>1</sub> as the
| |
| ''x'' axis. The point (''x'', ''y'') is found by traveling
| |
| a distance ''s''<sub>13</sub> = ''x'' from ''A'' along the reference
| |
| geodesic to an intermediate point ''C'' and then turning
| |
| ½π counter-clockwise and traveling along a
| |
| geodesic a distance ''s''<sub>32</sub> = ''y''. If ''A'' is on the
| |
| equator and α<sub>1</sub> = ½π, this gives the
| |
| [[equidistant cylindrical projection]]. If α<sub>1</sub> = 0,
| |
| this gives the [[Cassini projection|Cassini-Soldner projection]].
| |
| [[:fr:Carte de Cassini|Cassini's map of France]] placed ''A'' at
| |
| the [[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
| |
| [[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=§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 ρ = ''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'' = ''s''<sub>12</sub> and θ = ½π − α<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 φ<sub>1</sub> = −30°.
| |
| |image2=Four geodesics connecting two points on an oblate ellipsoid.svg
| |
| |caption2=Fig. 20. The four geodesics connecting ''A'' and a point ''B'', φ<sub>2</sub> = 26°, λ<sub>12</sub> = 175°.
| |
| |header=Geodesics from a single point (''f'' = 1/10, φ<sub>1</sub> = −30°)
| |
| }}
| |
| | |
| 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 α<sub>1</sub> is a multiple of
| |
| 3° are shown in light blue. (The geodesics are only
| |
| shown for their first passage close to the antipodal point, not for
| |
| subsequent ones.) Some geodesic circles are shown in green; these form
| |
| cusps on the envelope. The cut locus is shown in red. The envelope is
| |
| the locus of points which are conjugate to ''A''; points on the
| |
| envelope may be computed by finding the point at which
| |
| ''m''<sub>12</sub> = 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 [[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> > 0,
| |
| so that there is no nearby path connecting the two points which is
| |
| shorter; the other two are unstable. Only the shortest line (the first
| |
| one) has σ<sub>12</sub> ≤ π. 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|2012|loc=Table 1}}:
| |
| {| class="wikitable" style="text-align: right;"
| |
| |+ Geodesics for φ<sub>1</sub> = −30°, φ<sub>2</sub> = 29.9°, λ<sub>12</sub> = 179.8° (WGS84)
| |
| |-
| |
| !No.!!α<sub>1</sub> (°)!!α<sub>2</sub> (°)!!''s''<sub>12</sub> (m)!!σ<sub>12</sub> (°)!!''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||−22649.2935
| |
| |-
| |
| |4||−81.075605986||−99.282176388||20049364.2525||180.630976969||−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 γ is a parameter. (These are
| |
| generated by the rod of the [[trammel of Archimedes]].) This aids
| |
| in finding a good starting guess for α<sub>1</sub> for Newton's
| |
| method for in inverse problem in the case of nearly antipodal points
| |
| {{harv|Karney|2013|loc=§5}}.
| |
| | |
| The astroid is the (exterior) [[evolute]] of the geodesic circles
| |
| centered at ''A''. Likewise the geodesic circles are
| |
| [[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 [[Gaussian curvature]]. Now the
| |
| [[Gauss-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 θ<sub>''j''</sub> is the exterior angle at
| |
| vertex ''j''. Multiplying the equation for Γ
| |
| by ''R''<sub>2</sub><sup>2</sup>, where ''R''<sub>2</sub> is the
| |
| [[Earth radius#Authalic radius|authalic radius]], and subtracting this
| |
| from the equation for ''T'' gives{{refn|
| |
| {{harvtxt|Sjöberg|2006}} multiplies Γ by
| |
| ''b''<sup>2</sup> instead of ''R''<sub>2</sub><sup>2</sup>. However this leads to a
| |
| singular integrand {{harv|Karney|2012|loc=§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 [[Spheroid#Curvature|value of ''K'' for an ellipsoid]]
| |
| has been substituted.
| |
| Applying this formula to the quadrilateral ''AFHB'', noting
| |
| that Γ = α<sub>2</sub> − α<sub>1</sub>, and performing
| |
| the integral over φ gives
| |
| :<math>
| |
| \begin{align}
| |
| 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)}\\
| |
| &\qquad\qquad{}+
| |
| \frac{\tanh^{-1}(e \sin\phi)}{2e \sin\phi}
| |
| - \frac{R_2^2}{b^2}\biggr)\sin\phi
| |
| \,d\lambda,
| |
| \end{align}
| |
| </math>
| |
| where the integral is over the geodesic line (so that φ
| |
| is implicitly a function of λ). Converting this into
| |
| an integral over σ, we obtain
| |
| :<math>
| |
| \begin{align}
| |
| S_{12} &= R_2^2 E_{12} - e^2a^2\cos\alpha_0 \sin\alpha_0 \times\\
| |
| &\quad \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,
| |
| \end{align}
| |
| </math>
| |
| where
| |
| :<math>
| |
| t(x) = x + \sqrt{x^{-1} + 1}\,\sinh^{-1}\!\sqrt x,
| |
| </math>
| |
| and the notation ''E''<sub>12</sub> = α<sub>2</sub> − α<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=§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 2π ''R''<sub>2</sub><sup>2</sup> must be added to the
| |
| sum. If the edges are specified by their vertices, then a
| |
| [[Spherical excess|convenient expression]]
| |
| for ''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 [[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 0.5 mm for the WGS84 ellipsoid.
| |
| | |
| The algorithms given in {{harvtxt|Karney|2013}} are included in
| |
| GeographicLib {{harv|Karney|2013b}}. These are accurate to about
| |
| 15 nanometers for WGS84. Implementations in several
| |
| languages ([[C++]], [[C (programming language)|C]], Fortran,
| |
| [[Java (programming language)|Java]], [[JavaScript]],
| |
| [[Python (programming language)|Python]], [[Matlab]], and
| |
| [[Maxima (software)|Maxima]]) are included. 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>. A
| |
| [[command-line utility]], <code>GeodSolve</code>, for geodesic
| |
| calculations is included. As of version 4.9.0, the [[PROJ.4]] library
| |
| for cartographic projections includes the C implementation. This is
| |
| exposed in the two command-line utilities, <code>geod</code> and
| |
| <code>invgeod</code>, and in the library itself. These algorithms have
| |
| also been implemented in [[IDL (programming language)|IDL]] and
| |
| [[C Sharp (programming language)|C#]].
| |
| | |
| 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
| |
| 2–3 time slower than using series expansions; however it
| |
| provides accurate solutions for ellipsoids of revolution with
| |
| ''b''/''a'' ∈ [0.01, 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 [[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=§3.5}}.{{refn|
| |
| This section is adapted from the documentation for GeographicLib
| |
| {{harv|Karney|2013b|
| |
| 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 [[geoid]]. However geodesics on a surface of
| |
| the complexity of the geoid are partly [[Chaos theory|chaotic]]
| |
| {{harv|Waters|2011}}.}}
| |
| | |
| === Triaxial coordinate systems ===
| |
| [[File:Gaspard monge litho delpech.jpg|thumb|150px|[[Gaspard Monge]]]]
| |
| [[File:Charles Dupin.jpeg|thumb|150px|[[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'' ≥ ''b'' ≥ ''c'' > 0. A
| |
| point on the surface is specified by a latitude and longitude. The
| |
| ''geographical'' latitude and longitude (φ, λ) 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 (φ′, λ′)
| |
| 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>
| |
| Jacobi employed the ''ellipsoidal'' latitude and longitude
| |
| (β, ω) defined by
| |
| [[File:Triaxial ellipsoid coordinate system.svg|thumb|
| |
| Fig. 21.
| |
| [[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'' → ''a'', β
| |
| becomes the parametric latitude for an oblate ellipsoid, so the use of
| |
| the symbol β is consistent with the previous sections.
| |
| However ω is ''different'' from the spherical
| |
| longitude defined above.{{refn|
| |
| The limit ''b'' → ''c'' gives a prolate ellipsoid with
| |
| ω playing the role of the parametric latitude.}}
| |
| | |
| Grid lines of constant β (in blue) and
| |
| ω (in green) are given in Fig. 21. In contrast
| |
| to (φ, λ) and (φ′, λ′),
| |
| (β, ω) is an [[orthogonal]] coordinate system: the
| |
| grid lines intersect at right angles. The principal sections of the
| |
| ellipsoid, defined by ''X'' = 0 and ''Z'' = 0 are shown in
| |
| red. The third principal section, ''Y'' = 0, is covered by the
| |
| lines β = ±90° and ω = 0° or
| |
| ±180°. These lines meet at four
| |
| [[umbilical point]]s (two of which are visible in this figure) where the
| |
| [[principal curvature|principal radii of curvature]] are equal. Here
| |
| and in the other figures in this section the parameters of the ellipsoid
| |
| are ''a'':''b'':''c'' = 1.01:1:0.8, and it is viewed in an orthographic
| |
| projection from a point above φ = 40°,
| |
| λ = 30°.
| |
| | |
| The grid lines of the ellipsoidal coordinates may be interpreted in three
| |
| different ways
| |
| # They are "lines of curvature" on the ellipsoid, i.e., they are parallel to the directions of principal curvature {{harv|Monge|1796}}.
| |
| # They are also intersections of the ellipsoid with [[confocal ellipsoidal coordinates|confocal systems of hyperboloids of one and two sheets]] {{harv|Dupin|1813|loc = [http://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 β in Fig. 21 can be generated with the familiar [[Gardener's ellipse|string construction for ellipses]] with the ends of the string pinned to the two umbilical points.
| |
| | |
| Conversions between these three types of latitudes and longitudes
| |
| and the Cartesian coordinates are simple algebraic exercises.
| |
| | |
| The differential equations for a geodesic in ellipsoidal coordinates 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|[[C. G. J. Jacobi]]]]
| |
| [[File:Joseph liouville.jpeg|thumb|150px|[[Joseph Liouville]]]]
| |
| [[File:Darboux.jpg|thumb|150px|[[Jean-Gaston Darboux|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>
| |
| [[Königsberg]], 28th Dec. '38.
| |
| </blockquote>
| |
| | |
| The solution given by {{harvtxt|Jacobi|1839}} 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 "a function of the angle β equals
| |
| a function of the angle ω. These two functions are
| |
| just Abelian integrals..." Two constants δ and
| |
| γ appear in the solution. Typically
| |
| δ is zero if the lower limits of the integrals are
| |
| taken to be the starting point of the geodesic and the direction of the
| |
| geodesics is determined by γ. However for geodesics
| |
| that start at an umbilical points, we have γ = 0 and
| |
| δ determines the direction at the umbilical point.
| |
| The constant γ may be expressed as
| |
| :<math>
| |
| \gamma = (b^2-c^2)\cos^2\beta\sin^2\alpha-(a^2-b^2)\sin^2\omega\cos^2\alpha,
| |
| </math>
| |
| where α is the angle the geodesic makes with lines of
| |
| constant ω. In the limit ''b'' → ''a'',
| |
| this reduces to sinα cosβ = const., the
| |
| familiar Clairaut relation. A nice derivation of Jacobi's result is
| |
| given by {{harvtxt|Darboux|1894|loc=§§583–584}} 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, ω<sub>1</sub> = 0°, α<sub>1</sub> = 90°.
| |
| |caption1=Fig. 22. β<sub>1</sub> = 45.1°.
| |
| |caption2=Fig. 23. β<sub>1</sub> = 87.48°.
| |
| }}
| |
| On a triaxial ellipsoid, there are only 3 simple closed geodesics, the
| |
| three principal sections of the ellipsoid given by ''X'' = 0,
| |
| ''Y'' = 0, and ''Z'' = 0.{{refn|
| |
| If ''c''/''a'' < ½, there are other simple closed geodesics
| |
| similar to those shown in Figs. 13 and 14
| |
| {{harv|Klingenberg|1982|loc=§3.5.19}}.}}
| |
| To survey the other geodesics, it is convenient to consider geodesics
| |
| which intersect the middle principal section, ''Y'' = 0, at right
| |
| angles. Such geodesics are shown in Figs. 22–26,
| |
| which use the same ellipsoid parameters and the same viewing direction
| |
| as Fig. 21. In addition, the three principal ellipses are shown
| |
| in red in each of these figures.
| |
| | |
| If the starting point is β<sub>1</sub> ∈ (−90°, 90°),
| |
| ω<sub>1</sub> = 0, and α<sub>1</sub> = 90°, then the
| |
| geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic
| |
| oscillates north and south of the equator; on each oscillation it
| |
| completes slightly less that a full circuit around the ellipsoid
| |
| resulting, in the typical case, in the geodesic filling the area bounded
| |
| by the two latitude lines β = ±β<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'' ≈ ''b''); compare to Fig. 11.
| |
| However, if the starting point is at a higher latitude (Fig. 22)
| |
| the distortions resulting from ''a'' ≠ ''b'' are evident. All
| |
| tangents to a circumpolar geodesic touch the confocal single-sheeted
| |
| hyperboloid which intersects the ellipsoid at β = β<sub>1</sub>
| |
| {{harv|Chasles|1846}}
| |
| {{harv|Hilbert|Cohn-Vossen|1952|pp=223–224}}.
| |
| | |
| {{multiple image
| |
| |align=right
| |
| |direction=horizontal
| |
| |width=220
| |
| |image1=Transpolar geodesic on a triaxial ellipsoid case A.svg
| |
| |image2=Transpolar geodesic on a triaxial ellipsoid case B.svg
| |
| |header=Transpolar geodesics, β<sub>1</sub> = 90°, α<sub>1</sub> = 0°.
| |
| |caption1=Fig. 24. ω<sub>1</sub> = 39.9°.
| |
| |caption2=Fig. 25. ω<sub>1</sub> = 9.966°.
| |
| }}
| |
| If the starting point is β<sub>1</sub> = 90°,
| |
| ω<sub>1</sub> ∈ (0°, 180°), and
| |
| α<sub>1</sub> = 0°, then the geodesic encircles the ellipsoid
| |
| in a "transpolar" sense. The geodesic oscillates east and west of the
| |
| ellipse ''X'' = 0; on each oscillation it completes slightly more
| |
| that a full circuit around the ellipsoid resulting, in the typical case,
| |
| in the geodesic filling the area bounded by the two longitude lines
| |
| ω = ω<sub>1</sub> and ω = 180° − ω<sub>1</sub>.
| |
| If ''a'' = ''b'', all meridians are geodesics; the effect of
| |
| ''a'' ≠ ''b'' causes such geodesics to oscillate east and west.
| |
| Two examples are given in Figs. 24 and 25. The constriction
| |
| of the geodesic near the pole disappears in the limit
| |
| ''b'' → ''c''; in this case, the ellipsoid becomes a
| |
| prolate ellipsoid and Fig. 24 would resemble Fig. 12
| |
| (rotated on its side). All tangents to a transpolar geodesic touch the
| |
| confocal double-sheeted hyperboloid which intersects the ellipsoid at
| |
| ω = ω<sub>1</sub>.
| |
| | |
| [[File:Unstable umbilical geodesic on a triaxial ellipsoid.svg|thumb|
| |
| Fig. 26. An umbilical geodesic, β<sub>1</sub> = 90°,
| |
| ω<sub>1</sub> = 0°, α<sub>1</sub> = 45°.]]
| |
| If the starting point is β<sub>1</sub> = 90°,
| |
| ω<sub>1</sub> = 0° (an umbilical point), and
| |
| α<sub>1</sub> = 45° (the geodesic leaves the ellipse
| |
| ''Y'' = 0 at right angles), then the geodesic repeatedly
| |
| intersects the opposite umbilical point and returns to its starting
| |
| point. However on each circuit the angle at which it intersects
| |
| ''Y'' = 0 becomes closer to 0° or
| |
| 180° so that asymptotically the geodesic lies on the
| |
| ellipse ''Y'' = 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'' = 0 and ''Z'' = 0 are stable (an geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse ''Y'' = 0, which goes through all 4 umbilical points, is ''exponentially unstable''. If it is perturbed, it will swing out of the plane ''Y'' = 0 and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.)
| |
| | |
| If the starting point ''A'' of a geodesic is not an umbilical
| |
| point, then its envelope is an astroid with two cusps lying on
| |
| β = −β<sub>1</sub> and the other two on
| |
| ω = ω<sub>1</sub> + π {{harv|Sinclair|2003}}. The cut locus
| |
| for ''A'' is the portion
| |
| of the line β = −β<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|[[Karl Weierstrass]]]]
| |
| [[File:Henri Poincaré-2.jpg|thumb|150px|[[Henri Poincaré]]]]
| |
| The direct and inverse geodesic problems no longer play the central role
| |
| in geodesy that they once did. Instead of solving adjustment problems
| |
| 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 [[geographic information systems]];
| |
| * the definition of [[maritime boundaries]] {{harv|UNCLOS|2006}};
| |
| * in the rules of the [[Federal Aviation Administration]] for area navigation {{harv|RNAV|2007}};
| |
| * the method of measuring distances in the [[Fédération Aéronautique Internationale|FAI]] Sporting Code {{harv|FAI|2013}}.
| |
| | |
| By the [[principle of least action]], many problems in physics can be
| |
| formulated as a variational problem similar to that for geodesics. Indeed
| |
| the geodesic problem is equivalent to the motion of a particle
| |
| constrained to move on the surface, but otherwise subject to no forces
| |
| {{harv|Laplace|1799a}} {{harv|Hilbert|Cohn-Vossen|1952|p=222}}.
| |
| For this reason,
| |
| geodesics on simple surfaces such as ellipsoids of revolution or
| |
| triaxial ellipsoids are frequently used as "test cases" for exploring new
| |
| methods. Examples include:
| |
| * the development of elliptic integrals {{harv|Legendre|1811}} and [[elliptic functions]] {{harv|Weierstrass|1861}};
| |
| * the development of differential geometry {{harv|Gauss|1828}} {{harv|Christoffel|1869}};
| |
| * methods for solving systems of differential equations by a change of independent variables {{harv|Jacobi|1839}};
| |
| * the study of [[caustic (optics)|caustics]] {{harv|Jacobi|1891}};
| |
| * investigations into the number and stability of periodic orbits {{harv|Poincaré|1905}};
| |
| * in the limit ''c'' → 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]]
| |
| * [[Geodesics]]
| |
| * [[Geodesy]]
| |
| * [[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
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| |edition = 2nd
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| |publisher = Springer-Verlag
| |
| |others = Translated by K. Vogtmann & A. Weinstein
| |
| |oclc = 4037141
| |
| |isbn = 978-0-387-96890-2
| |
| }}
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| *{{cite book
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| |ref = {{harvid|Bagratuni|1962}} |year = 1967 |origyear = 1962
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| |last = Bagratuni |first = G. V.
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| |title = Course in Spheroidal Geodesy
| |
| |url = http://geographiclib.sf.net/geodesic-papers/bagratuni67.pdf
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| |postscript = . Translation of ''Курс сфероидической геодезии'' by U.S. Air Force ([http://www.dtic.mil/docs/citations/AD0650520 FTD-MT-64-390])
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| |oclc = 6150611
| |
| }}
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| *{{cite doi
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| |10.1007/BF03198517
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| |comment = Beltrami 1865
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| |noedit
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| }}
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| *{{cite book
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| |ref = harv |year = 2010
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| |last = Berger |first = M. |authorlink = Marcel Berger
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| |title = Geometry Revealed
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| |publisher = Springer
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| |others = Translated by L. J. Senechal
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| |isbn = 978-3-540-70996-1
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| |doi = 10.1007/978-3-540-70997-8
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| }}
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| *{{cite journal
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| |ref = {{harvid|Bessel|1825}} |year = 2010 |origyear = 1825
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| |last = Bessel |first = F. W. |authorlink = Friedrich Bessel
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| |doi = 10.1002/asna.201011352
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| |title = The calculation of longitude and latitude from geodesic measurements
| |
| |journal = Astronomische Nachrichten
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| |volume = 331 |issue = 8 |pages = 852–861
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| |arxiv = 0908.1824
| |
| |others = . Translated by C. F. F. Karney & R. E. Deakin
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| |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].
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| }}
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| *{{cite doi
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| |10.1002/asna.18270051202
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| |comment = Bessel 1827
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| |noedit
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| }}
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| *{{cite doi
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| |10.1090/S0002-9947-1916-1501037-4
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| |comment = Bliss 1916
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| |noedit
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| *{{cite book
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| |ref = harv |year = 1952
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| |last = Bomford |first = G.
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| |title = Geodesy
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| |publisher = Clarendon
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| |location = Oxford
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| |oclc = 1396190
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| }}
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| *{{cite doi
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| |10.1007/BF02198293
| |
| |comment = Carlson 1995
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| *{{cite journal
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| |ref = harv |year = 1991
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| |last1 = Casper |first1 = P. W.
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| |last2 = Bent |first2 = R. B.
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| |title = The effect of the Earth's oblate spheroid shape on the accuracy of a time-of-arrival lightning ground strike locating system
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| |url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910023392_1991023392.pdf
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| |postscript = . in Proceedings 1991 International Aerospace and Ground Conference on Lightning and Static Electricity, (Vol. 2).
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| }}
| |
| *{{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 = 389–405
| |
| |url = http://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA389
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1870
| |
| |last = Cayley |first = A. |authorlink = Arthur Cayley
| |
| |title = On the geodesic lines on an oblate spheroid
| |
| |journal = Philosophical Magazine (4th ser.)
| |
| |volume = 40
| |
| |pages = 329–340
| |
| |doi = 10.1080/14786447008640411
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| |doi_brokendate = August 5, 2013
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| |url = http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA329
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1846
| |
| |last = Chasles |first = M. |authorlink = Michel Chasles
| |
| |title = Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré
| |
| |language = French
| |
| |trans_title = Geodesic lines and the lines of curvature of the
| |
| surfaces of the second degree
| |
| |journal = Journal de Mathématiques Pures et Appliquées
| |
| |volume = 11
| |
| |pages = 5–20
| |
| |url = http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1846_1_11_A2_0
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1869
| |
| |last = Christoffel |first = E. B. |authorlink = Elwin Bruno Christoffel
| |
| |title = Allgemeine Theorie der geodätischen Dreiecke
| |
| |trans_title = General theory of geodesic triangles
| |
| |language = German
| |
| |journal = Abhandlungen Königlichen Akademie der Wissenschaft zu Berlin
| |
| |pages = 119–176
| |
| |url = http://books.google.com/books?id=EEtFAAAAcAAJ&pg=PA119
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1735
| |
| |last = Clairaut |first = A. C. |authorlink = Alexis Claude Clairaut
| |
| |title = Détermination géometrique de la perpendiculaire à la méridienne tracée par M. Cassini
| |
| |trans_title = Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini
| |
| |language = French
| |
| |journal = Mémoires de l'Académie Royale des Sciences de Paris 1733
| |
| |pages = 406–416
| |
| |url = http://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA406
| |
| }}
| |
| *{{cite doi
| |
| |10.1179/003962689791474267
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| |comment = Danielsen 1989
| |
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| }}
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| *{{cite book
| |
| |ref = harv |year = 1894
| |
| |last = Darboux |first = J. G. |authorlink = Jean Gaston Darboux
| |
| |title = Leçons sur la théorie générale des surfaces
| |
| |volume = 3
| |
| |language = French
| |
| |trans_title = Lessons on the general theory of surfaces
| |
| |publisher = Gauthier-Villars
| |
| |location = Paris
| |
| |url = http://books.google.com/books?id=hGMSAAAAIAAJ
| |
| |postscript = . [http://geographiclib.sourceforge.net/geodesic-papers/darboux94.pdf PDF].
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| |oclc = 8566228
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| }}
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| *{{cite book
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| |ref = harv |year = 2010
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| |author = DLMF |authorlink = Digital Library of Mathematical Functions
| |
| |editor1-last = Olver |editor1-first = F. W. J.
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| |editor1-link = Frank W. J. Olver
| |
| |editor2-last = Lozier |editor2-first = D. W.
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| |editor3-last = Boisvert |editor3-first = R. F.
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| |editor4-last = Clark |editor4-first = C. W.
| |
| |title = NIST Handbook of Mathematical Functions
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| |publisher = Cambridge Univ. Press
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| |url = http://dlmf.nist.gov
| |
| |isbn = 978-0-521-19225-5
| |
| |mr = 2723248
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| |displayeditors = 4
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| }}
| |
| *{{cite book
| |
| |ref = harv |year = 1813
| |
| |last = Dupin |first = P. C. F. |authorlink = Charles Dupin
| |
| |title = Développements de Géométrie
| |
| |language = French
| |
| |trans_title = Developments in geometry
| |
| |publisher = Courcier
| |
| |location = Paris
| |
| |url = http://books.google.com/books?id=j40AAAAAMAAJ
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| |oclc = 560800801
| |
| }}
| |
| *{{cite techreport
| |
| |ref = harv |year = 1993
| |
| |last = Ehlert |first = D.
| |
| |title = Methoden der ellipsoidischen Dreiecksberechnung
| |
| |trans_title = Methods for ellipsoidal triangulatioin
| |
| |language = German
| |
| |institution = [[:de:Deutsche Geodätische Kommission|Deutsche Geodätische Kommission]]
| |
| |series = Reihe B: Angewandte Geodäsie, Heft Nr. 292
| |
| |oclc = 257615376
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1755
| |
| |last = Euler |first = L. |authorlink = Leonhard Euler
| |
| |title = Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits
| |
| |trans_title = Elements of spheroidal trigonometry taken from the method of maxima and minima
| |
| |language = French
| |
| |journal = Mémoires de l'Académie Royale des Sciences de Berlin 1753
| |
| |pages = 258–293
| |
| |volume = 9
| |
| |url = http://books.google.com/books?id=QIIfAAAAYAAJ&pg=PA258
| |
| |postscript = . [http://books.google.com/books?id=QIIfAAAAYAAJ&pg=PA362-IA1 Figures].
| |
| }}
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| |10.1179/003962606780732100
| |
| |comment = Sjöberg 2006
| |
| |noedit
| |
| }}
| |
| *{{cite doi
| |
| |10.1061/(ASCE)SU.1943-5428.0000061
| |
| |comment = Sjöberg+Shirazian 2012
| |
| |noedit
| |
| }}
| |
| *{{cite doi
| |
| |10.1061/(ASCE)0733-9453(2005)131:1(20)
| |
| |comment = Thomas+Featherstone 2005
| |
| |noedit
| |
| }}
| |
| *{{cite techreport
| |
| |ref = harv |year = 1970
| |
| |last = Thomas |first = P. D.
| |
| |title = Spheroidal Geodesics, Reference Systems, & Local Geometry
| |
| |url = http://www.dtic.mil/docs/citations/AD0703541
| |
| |institution = U.S. Naval Oceanographic Office
| |
| |number = SP-138
| |
| }}
| |
| *{{cite techreport
| |
| |ref = harv |year = 2006
| |
| |last = UNCLOS |authorlink = UNCLOS
| |
| |title = A Manual on Technical Aspects of the United Nations Convention on the Law of the Sea, 1982
| |
| |institution = International Hydrographic Bureau
| |
| |edition = 4th
| |
| |location = Monaco
| |
| |url = http://www.iho.int/iho_pubs/CB/C-51_Ed4-EN.pdf
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1975a
| |
| |last = Vincenty |first = T. |authorlink = Thaddeus Vincenty
| |
| |title = Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations
| |
| |journal = Survey Review
| |
| |volume = 23
| |
| |number = 176
| |
| |pages = 88–93
| |
| |url = http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
| |
| |postscript = . Addendum: Survey Review '''23''' (180): 294 (1976).
| |
| }}
| |
| *{{cite techreport
| |
| |ref = harv |year = 1975b
| |
| |last = Vincenty |first = T. |authorlink = Thaddeus Vincenty
| |
| |title = Geodetic inverse solution between antipodal points
| |
| |institution = DMAAC Geodetic Survey Squadron
| |
| |url = http://geographiclib.sourceforge.net/geodesic-papers/vincenty75b.pdf
| |
| |accessdate = 2011-07-28
| |
| }}
| |
| *{{cite techreport
| |
| |ref = harv |year = 1978
| |
| |last1 = Vincenty |first1 = T. |authorlink1 = Thaddeus Vincenty
| |
| |last2 = Bowring |first2 = B. R.
| |
| |title = Application of three-dimensional geodesy to adjustments of horizontal networks
| |
| |institution = NOAA
| |
| |number = NOS NGS-13
| |
| |url = http://www.ngs.noaa.gov/PUBS_LIB/ApplicationOfThreeDimensionalGeodesyToAdjustmentsOfHorizontalNetworks_TM_NOS_NGS13.pdf
| |
| }}
| |
| *{{cite doi
| |
| |10.1016/j.physd.2011.11.010
| |
| |comment = Waters 2012
| |
| |noedit
| |
| }}
| |
| *{{cite journal
| |
| |ref = harv |year = 1861
| |
| |last = Weierstrass |first = K. T. W. |authorlink = Karl Weierstrass
| |
| |title = Über die geodätischen Linien auf dem dreiaxigen Ellipsoid
| |
| |trans_title = Geodesic lines on a triaxial ellipsoid
| |
| |language = German
| |
| |journal = Monatsbericht Königlichen Akademie der Wissenschaft zu Berlin
| |
| |pages = 986–997
| |
| |url = http://books.google.com/books?id=9O4GAAAAYAAJ&pg=PA257
| |
| |postscript = . [http://geographiclib.sourceforge.net/geodesic-papers/weierstrass-V1.pdf PDF].
| |
| }}
| |
| {{refend}}
| |
| | |
| == External links ==
| |
| * [http://geographiclib.sourceforge.net/geodesic-papers/biblio.html Online geodesic bibliography], approximately 150 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/cgi-bin/Inv_Fwd/forward2.prl NGS online solution of the direct problem].
| |
| ** [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|2013b}}:
| |
| ** [http://geographiclib.sourceforge.net/ GeographicLib web site] for downloading library and documentation.
| |
| ** [http://geographiclib.sourceforge.net/html/GeodSolve.1.html GeodSolve(1)], [[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://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/39108 Matlab implementation of the geodesic routines] (used for the figures for geodesics on ellipsoids of revolution in this article).
| |
| * 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:Geodesy]]
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| [[Category:Geodesic (mathematics)]]
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| [[Category:Differential geometry]]
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| [[Category:Calculus of variations]]
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