Английская Википедия:Geodesics on an ellipsoid

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Шаблон:Short description Шаблон:Geodesy

Файл:Long geodesic on an oblate ellipsoid.svg
A geodesic on an oblate ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry Шаблон:Harv.

If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Шаблон:Harvtxt showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the equator and the meridians are the only simple closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), only three geodesics are closed.

Geodesics on an ellipsoid of revolution

There are several ways of defining geodesics Шаблон:Harv. A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface that they start to return toward the starting point, so that other routes are more direct, and includes paths that intersect or re-trace themselves. Short enough segments of a geodesics are still the shortest route between their endpoints, but geodesics are not necessarily globally minimal (i.e. shortest among all possible paths). Every globally-shortest path is a geodesic, but not vice versa.

By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry Шаблон:Harv Шаблон:Harv.

Файл:Geodesic problem on an ellipsoid.svg
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: Шаблон:Math at latitude Шаблон:Math and longitude Шаблон:Math and Шаблон:Math at latitude Шаблон:Math and longitude Шаблон:Math (see Fig. 1). The connecting geodesic (from Шаблон:Math to Шаблон:Math) is Шаблон:Math, of length Шаблон:Math, which has azimuths Шаблон:Math and Шаблон:Math at the two endpoints.Шаблон:Refn The two geodesic problems usually considered are:

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

As can be seen from Fig. 1, these problems involve solving the triangle Шаблон:Math given one angle, Шаблон:Math for the direct problem and Шаблон:Math for the inverse problem, and its two adjacent sides. For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation.)

For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by Шаблон:Harvtxt. A systematic solution for the paths of geodesics was given by Шаблон:Harvtxt and Шаблон:Harvtxt (and subsequent papers in [[#Шаблон:Harvid|1808]] and [[#Шаблон:Harvid|1810]]). The full solution for the direct problem (complete with computational tables and a worked out example) is given by Шаблон:Harvtxt.

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

Nous désignerons cette ligne sous le nom de ligne géodésique [We will call this line the geodesic line].

This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example Шаблон:Harv,

A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the Earth; and it is therefore the proper itinerary measure of the distance between those two points.

In its adoption by other fields geodesic line, frequently shortened to geodesic, was preferred.

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

Equations for a geodesic

Шаблон:Multiple image

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

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

<math>

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

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

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

<math>

\cos\alpha\,ds = \rho\,d\varphi = - \frac{dR}{\sin\varphi}, \quad \sin\alpha\,ds = R\,d\lambda,</math> Шаблон:EquationRef where Шаблон:Mvar is the meridional radius of curvature, Шаблон:Math is the radius of the circle of latitude Шаблон:Mvar, and Шаблон:Mvar is the normal radius of curvature. The elementary segment is therefore given by

<math>ds^2 = \rho^2\,d\varphi^2 + R^2\,d\lambda^2</math>

or

<math>\begin{align}

ds &= \sqrt{\rho^2\varphi'^2 + R^2}\,d\lambda \\ &\equiv L(\varphi,\varphi')\,d\lambda, \end{align}</math> where Шаблон:Math and the [[Lagrangian mechanics#Euler–Lagrange equations and Hamilton's principle|Lagrangian function Шаблон:Math]] depends on Шаблон:Mvar through Шаблон:Math and Шаблон:Math. The length of an arbitrary path between Шаблон:Math and Шаблон:Math is given by

<math> s_{12} = \int_{\lambda_1}^{\lambda_2} L(\varphi, \varphi')\,d\lambda,</math>

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

<math>L - \varphi' \frac{\partial L}{\partial \varphi'} = \text{const.}</math>

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

<math>R\sin\alpha = \text{const.}</math>

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

<math>d\alpha=\sin\varphi\,d\lambda.</math>

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

<math>

\frac{d\varphi}{ds} = \frac{\cos\alpha}{\rho};\quad \frac{d\lambda}{ds} = \frac{\sin\alpha}{\nu\cos\varphi};\quad \frac{d\alpha}{ds} = \frac{\tan\varphi\sin\alpha}{\nu}.</math> We can express Шаблон:Math in terms of the parametric latitude, Шаблон:Mvar, using

<math>R = a\cos\beta,</math>

and Clairaut's relation then becomes

<math>\sin\alpha_1\cos\beta_1 = \sin\alpha_2\cos\beta_2.</math>

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

In order to find the relation for the third side Шаблон:Math, the spherical arc length, and included angle Шаблон:Math, the spherical longitude, it is useful to consider the triangle Шаблон:Math representing a geodesic starting at the equator; see Fig. 5. In this figure, the variables referred to the auxiliary sphere are shown with the corresponding quantities for the ellipsoid shown in parentheses. Quantities without subscripts refer to the arbitrary point Шаблон:Math; Шаблон:Math, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for Шаблон:Mvar, Шаблон:Mvar and Шаблон:Mvar.

Файл:Differential element of a geodesic on a sphere.svg
Fig. 6. Differential element of a geodesic on a sphere.

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

<math>

\cos\alpha\,d\sigma = d\beta, \quad \sin\alpha\,d\sigma = \cos\beta\,d\omega.</math> Шаблон:EquationRef Combining Eqs. Шаблон:EquationNote and Шаблон:EquationNote gives differential equations for Шаблон:Math and Шаблон:Mvar

<math>\frac 1 a \frac{ds}{d\sigma}

= \frac{d\lambda}{d\omega} = \frac{\sin\beta}{\sin\varphi}.</math>

The relation between Шаблон:Mvar and Шаблон:Mvar is

<math>\tan\beta = \sqrt{1-e^2} \tan\varphi = (1-f) \tan\varphi,</math>

which gives

<math>\frac{\sin\beta}{\sin\varphi} = \sqrt{1-e^2\cos^2\beta},</math>

so that the differential equations for the geodesic become

<math>\frac1a\frac{ds}{d\sigma} = \frac{d\lambda}{d\omega} = \sqrt{1-e^2\cos^2\beta}.</math>

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

<math>\sin\beta = \sin\beta(\sigma;\alpha_0) = \cos\alpha_0 \sin\sigma,</math>

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

<math>

\frac sb = \int_0^\sigma \sqrt{1 + k^2 \sin^2\sigma'}\,d\sigma', </math> Шаблон:EquationRef where

<math>k = e'\cos\alpha_0,</math>

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

<math>d\omega = \frac{\sin\alpha_0}{\cos^2\beta}\,d\sigma,</math>

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

<math>

\lambda - \lambda_0 = \omega - f\sin\alpha_0 \int_0^\sigma \frac{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}} \,d\sigma', </math> Шаблон:EquationRef and the limits on the integrals are chosen so that Шаблон:Math at the equator crossing, Шаблон:Math.

This completes the solution of the path of a geodesic using the auxiliary sphere. By this device a great circle can be mapped exactly to a geodesic on an ellipsoid of revolution.

There are also several ways of approximating geodesics on a terrestrial ellipsoid (with small flattening) Шаблон:Harv; some of these are described in the article on geographical distance. However, these are typically comparable in complexity to the method for the exact solution Шаблон:Harv.

Behavior of geodesics

Файл:Closed geodesics on an ellipsoid of revolution.svg
Fig. 7. Meridians and the equator are the only closed geodesics. (For the very flattened ellipsoids, there are other closed geodesics; see Figs. 11 and 12).

Шаблон:Multiple image

Файл:Long geodesic on a prolate ellipsoid.svg
Fig. 10. Geodesic on a prolate ellipsoid (Шаблон:Math) with Шаблон:Math. Compare with Fig. 8.

Fig. 7 shows the simple closed geodesics which consist of the meridians (green) and the equator (red). (Here the qualification "simple" means that the geodesic closes on itself without an intervening self-intersection.) This follows from the equations for the geodesics given in the previous section.

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

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

Файл:Geodesics and geodesic circles on an oblate ellipsoid.svg
Fig. 13. Geodesics (blue) from a single point for Шаблон:Math, Шаблон:Math; geodesic circles are shown in green and the cut locus in red.

Fig. 13 shows geodesics (in blue) emanating Шаблон:Math with Шаблон:Math a multiple of Шаблон:Val up to the point at which they cease to be shortest paths. (The flattening has been increased to Шаблон:Frac in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant Шаблон:Math, which are the geodesic circles centered Шаблон:Math. Шаблон:Harvtxt showed that, on any surface, geodesics and geodesic circle intersect at right angles. The red line is the cut locus, the locus of points which have multiple (two in this case) shortest geodesics from Шаблон:Math. On a sphere, the cut locus is a point. On an oblate ellipsoid (shown here), it is a segment of the circle of latitude centered on the point antipodal to Шаблон:Math, Шаблон:Math. The longitudinal extent of cut locus is approximately Шаблон:Math. If Шаблон:Math lies on the equator, Шаблон:Math, this relation is exact and as a consequence the equator is only a shortest geodesic if Шаблон:Math. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to Шаблон:Math, Шаблон:Math, and this means that meridional geodesics stop being shortest paths before the antipodal point is reached.

Differential properties of geodesics

Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments Шаблон:Harv, determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by Шаблон:Math, and a second geodesic a small distance Шаблон:Math away from it. Шаблон:Harvtxt showed that Шаблон:Math obeys the Gauss-Jacobi equation

<math>\frac{d^2t(s)}{ds^2} + K(s) t(s) = 0, </math>
Файл:Definition of reduced length and geodesic scale.svg
Fig. 14. Definition of reduced length and geodesic scale.

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

<math> t(s_2) = C m(s_1,s_2) + D M(s_1,s_2) </math>

where

<math>

\begin{align} m(s_1, s_1) &= 0, \quad \left.\frac{dm(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 1, \\ M(s_1, s_1) &= 1, \quad \left.\frac{dM(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 0. \end{align} </math> The quantity Шаблон:Math is the so-called reduced length, and Шаблон:Math is the geodesic scale.Шаблон:Refn Their basic definitions are illustrated in Fig. 14.

The Gaussian curvature for an ellipsoid of revolution is

<math>

K = \frac1{\rho\nu} = \frac{\bigl(1-e^2\sin^2\varphi\bigr)^2}{b^2} 

 = \frac{b^2}{a^4\bigl(1-e^2\cos^2\beta\bigr)^2}.

</math> Шаблон:Harvtxt solved the Gauss-Jacobi equation for this case enabling Шаблон:Math and Шаблон:Math to be expressed as integrals.

As we see from Fig. 14 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by Шаблон:Math is Шаблон:Math. On a closed surface such as an ellipsoid, Шаблон:Math oscillates about zero. The point at which Шаблон:Math becomes zero is the point conjugate to the starting point. In order for a geodesic between Шаблон:Math and Шаблон:Math, of length Шаблон:Math, to be a shortest path it must satisfy the Jacobi condition Шаблон:Harv Шаблон:Harv Шаблон:Harv Шаблон:Harv, that there is no point conjugate to Шаблон:Math between Шаблон:Math and Шаблон:Math. If this condition is not satisfied, then there is a nearby path (not necessarily a geodesic) which is shorter. Thus, the Jacobi condition is a local property of the geodesic and is only a necessary condition for the geodesic being a global shortest path. Necessary and sufficient conditions for a geodesic being the shortest path are:

Envelope of geodesics

Шаблон:Multiple image

The geodesics from a particular point Шаблон:Math if continued past the cut locus form an envelope illustrated in Fig. 15. Here the geodesics for which Шаблон:Math is a multiple of Шаблон:Val are shown in light blue. (The geodesics are only shown for their first passage close to the antipodal point, not for subsequent ones.) Some geodesic circles are shown in green; these form cusps on the envelope. The cut locus is shown in red. The envelope is the locus of points which are conjugate to Шаблон:Math; points on the envelope may be computed by finding the point at which Шаблон:Math on a geodesic. Шаблон:Harvtxt calls this star-like figure produced by the envelope an astroid.

Outside the astroid two geodesics intersect at each point; thus there are two geodesics (with a length approximately half the circumference of the ellipsoid) between Шаблон:Math and these points. This corresponds to the situation on the sphere where there are "short" and "long" routes on a great circle between two points. Inside the astroid four geodesics intersect at each point. Four such geodesics are shown in Fig. 16 where the geodesics are numbered in order of increasing length. (This figure uses the same position for Шаблон:Math as Fig. 13 and is drawn in the same projection.) The two shorter geodesics are stable, i.e., Шаблон:Math, so that there is no nearby path connecting the two points which is shorter; the other two are unstable. Only the shortest line (the first one) has Шаблон:Math. All the geodesics are tangent to the envelope which is shown in green in the figure.

The astroid is the (exterior) evolute of the geodesic circles centered at Шаблон:Math. Likewise, the geodesic circles are involutes of the astroid.

Area of a geodesic polygon Шаблон:Anchor

Шаблон:See also

A geodesic polygon is a polygon whose sides are geodesics. It is analogous to a spherical polygon, whose sides are great circles. The area of such a polygon may be found by first computing the area between a geodesic segment and the equator, i.e., the area of the quadrilateral Шаблон:Math in Fig. 1 Шаблон:Harv. Once this area is known, the area of a polygon may be computed by summing the contributions from all the edges of the polygon.

Here an expression for the area Шаблон:Math of Шаблон:Math is developed following Шаблон:Harvtxt. The area of any closed region of the ellipsoid is

<math> T = \int dT = \int \frac 1 K \cos\varphi\,d\varphi\,d\lambda,

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

<math>

\Gamma = \int K \,dT = \int \cos\varphi\,d\varphi\,d\lambda, </math> where

<math>

\Gamma = 2\pi - \sum_j \theta_j </math> is the geodesic excess and Шаблон:Math is the exterior angle at vertex Шаблон:Math. Multiplying the equation for Шаблон:Math by Шаблон:Math, where Шаблон:Math is the authalic radius, and subtracting this from the equation for Шаблон:Math gives

<math>

\begin{align} T &= R_2^2 \,\Gamma + \int \left(\frac 1 K - R_2^2\right)\cos\varphi\,d\varphi\,d\lambda \\ &=R_2^2 \,\Gamma + \int \left( \frac{b^2}{\bigl(1 - e^2\sin^2\varphi\bigr)^2} - R_2^2 \right)\cos\varphi\,d\varphi\,d\lambda, \end{align} </math> where the [[Spheroid#Curvature|value of Шаблон:Math for an ellipsoid]] has been substituted. Applying this formula to the quadrilateral Шаблон:Math, noting that Шаблон:Math, and performing the integral over Шаблон:Mvar gives

<math>

S_{12}=R_2^2 (\alpha_2-\alpha_1) + b^2 \int_{\lambda_1}^{\lambda_2} \left( \frac1{2\bigl(1 - e^2\sin^2\varphi\bigr)}+ \frac{\tanh^{-1}(e \sin\varphi)}{2e \sin\varphi} - \frac{R_2^2}{b^2}\right)\sin\varphi \,d\lambda, </math> where the integral is over the geodesic line (so that Шаблон:Mvar is implicitly a function of Шаблон:Mvar). The integral can be expressed as a series valid for small Шаблон:Math Шаблон:Harv Шаблон:Harv.

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

<math>

\tan\frac{E_{12}}2 = \frac{\sin\tfrac12 (\beta_2 + \beta_1)} {\cos\tfrac12 (\beta_2 - \beta_1)} \tan\frac{\omega_{12}}2. </math>

Solution of the direct and inverse problems

Шаблон:Further Шаблон:See also Solving the geodesic problems entails mapping the geodesic onto the auxiliary sphere and solving the corresponding problem in great-circle navigation. When solving the "elementary" spherical triangle for Шаблон:Math in Fig. 5, Napier's rules for quadrantal triangles can be employed,

<math>

\begin{align} \sin\alpha_0 &= \sin\alpha \cos\beta = \tan\omega \cot\sigma, \\ \cos\sigma &= \cos\beta \cos\omega = \tan\alpha_0 \cot\alpha, \\ \cos\alpha &= \cos\omega \cos\alpha_0 = \cot\sigma \tan\beta, \\ \sin\beta &= \cos\alpha_0 \sin\sigma = \cot\alpha \tan\omega, \\ \sin\omega &= \sin\sigma \sin\alpha = \tan\beta \tan\alpha_0. \end{align} </math> The mapping of the geodesic involves evaluating the integrals for the distance, Шаблон:Math, and the longitude, Шаблон:Mvar, Eqs. Шаблон:EquationNote and Шаблон:EquationNote and these depend on the parameter Шаблон:Math.

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

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

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

Шаблон:Harvtxt provides solutions for the direct and inverse problems; these are based on a series expansion carried out to third order in the flattening and provide an accuracy of about Шаблон:Val for the WGS84 ellipsoid; however the inverse method fails to converge for nearly antipodal points.

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

Geodesics on a triaxial ellipsoid

Solving the geodesic problem for an ellipsoid of revolution is mathematically straightforward: because of symmetry, geodesics have a constant of motion, given by Clairaut's relation allowing the problem to be reduced to quadrature. By the early 19th century (with the work of Legendre, Oriani, Bessel, et al.), there was a complete understanding of the properties of geodesics on an ellipsoid of revolution.

On the other hand, geodesics on a triaxial ellipsoid (with three unequal axes) have no obvious constant of the motion and thus represented a challenging unsolved problem in the first half of the 19th century. In a remarkable paper, Шаблон:Harvtxt discovered a constant of the motion allowing this problem to be reduced to quadrature also Шаблон:Harv.Шаблон:Refn

Triaxial ellipsoid coordinate system

Шаблон:See also

Файл:Triaxial ellipsoid coordinate system.svg
Fig. 17. Triaxial ellipsoidal coordinates.

Consider the ellipsoid defined by

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

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

<math>

\begin{align} 

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

\end{align} </math> In the limit Шаблон:Math, Шаблон:Mvar becomes the parametric latitude for an oblate ellipsoid, so the use of the symbol Шаблон:Mvar is consistent with the previous sections. However, Шаблон:Mvar is different from the spherical longitude defined above.Шаблон:Refn

Grid lines of constant Шаблон:Mvar (in blue) and Шаблон:Mvar (in green) are given in Fig. 17. These constitute an orthogonal coordinate system: the grid lines intersect at right angles. The principal sections of the ellipsoid, defined by Шаблон:Math and Шаблон:Math are shown in red. The third principal section, Шаблон:Math, is covered by the lines Шаблон:Math and Шаблон:Math or Шаблон:Math. These lines meet at four umbilical points (two of which are visible in this figure) where the principal radii of curvature are equal. Here and in the other figures in this section the parameters of the ellipsoid are Шаблон:Math, and it is viewed in an orthographic projection from a point above Шаблон:Math, Шаблон:Math.

The grid lines of the ellipsoidal coordinates may be interpreted in three different ways:

  1. They are "lines of curvature" on the ellipsoid: they are parallel to the directions of principal curvature Шаблон:Harv.
  2. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets Шаблон:Harv.
  3. Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points Шаблон:Harv. For example, the lines of constant Шаблон:Mvar in Fig. 17 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.

Jacobi's solution

Jacobi showed that the geodesic equations, expressed in ellipsoidal coordinates, are separable. Here is how he recounted his discovery to his friend and neighbor Bessel Шаблон:Harv,

The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal.

Königsberg, 28th Dec. '38.

The solution given by Jacobi Шаблон:Harv Шаблон:Harv is

<math>

\begin{align} \delta &= \int \frac {\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}\,d\beta} {\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta} 

\sqrt{\bigl(b^2-c^2\bigr)\cos^2\beta - \gamma}} \\[6pt]

&\quad - \int \frac {\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}\,d\omega} {\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2} 

\sqrt{\bigl(a^2-b^2\bigr)\sin^2\omega + \gamma}}.

\end{align} </math> As Jacobi notes "a function of the angle Шаблон:Mvar equals a function of the angle Шаблон:Mvar. These two functions are just Abelian integrals..." Two constants Шаблон:Math and Шаблон:Math appear in the solution. Typically Шаблон:Math is zero if the lower limits of the integrals are taken to be the starting point of the geodesic and the direction of the geodesics is determined by Шаблон:Math. However, for geodesics that start at an umbilical points, we have Шаблон:Math and Шаблон:Math determines the direction at the umbilical point. The constant Шаблон:Math may be expressed as

<math>

\gamma = \bigl(b^2-c^2\bigr)\cos^2\beta\sin^2\alpha- \bigl(a^2-b^2\bigl)\sin^2\omega\cos^2\alpha, </math> where Шаблон:Mvar is the angle the geodesic makes with lines of constant Шаблон:Mvar. In the limit Шаблон:Math, this reduces to Шаблон:Math, the familiar Clairaut relation. A derivation of Jacobi's result is given by Шаблон:Harvtxt; he gives the solution found by Шаблон:Harvtxt for general quadratic surfaces.

Survey of triaxial geodesics

Шаблон:Multiple image On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by Шаблон:Math, Шаблон:Math, and Шаблон:Math.Шаблон:Refn To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section, Шаблон:Math, at right angles. Such geodesics are shown in Figs. 18–22, which use the same ellipsoid parameters and the same viewing direction as Fig. 17. In addition, the three principal ellipses are shown in red in each of these figures.

If the starting point is Шаблон:Math, Шаблон:Math, and Шаблон:Math, then Шаблон:Math and the geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic oscillates north and south of the equator; on each oscillation it completes slightly less than a full circuit around the ellipsoid resulting, in the typical case, in the geodesic filling the area bounded by the two latitude lines Шаблон:Math. Two examples are given in Figs. 18 and 19. Figure 18 shows practically the same behavior as for an oblate ellipsoid of revolution (because Шаблон:Math); compare to Fig. 9. However, if the starting point is at a higher latitude (Fig. 18) the distortions resulting from Шаблон:Math are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at Шаблон:Math Шаблон:Harv Шаблон:Harv.

Шаблон:Multiple image If the starting point is Шаблон:Math, Шаблон:Math, and Шаблон:Math, then Шаблон:Math and the geodesic encircles the ellipsoid in a "transpolar" sense. The geodesic oscillates east and west of the ellipse Шаблон:Math; on each oscillation it completes slightly more than a full circuit around the ellipsoid. In the typical case, this results in the geodesic filling the area bounded by the two longitude lines Шаблон:Math and Шаблон:Math. If Шаблон:Math, all meridians are geodesics; the effect of Шаблон:Math causes such geodesics to oscillate east and west. Two examples are given in Figs. 20 and 21. The constriction of the geodesic near the pole disappears in the limit Шаблон:Math; in this case, the ellipsoid becomes a prolate ellipsoid and Fig. 20 would resemble Fig. 10 (rotated on its side). All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at Шаблон:Math.

Файл:Unstable umbilical geodesic on a triaxial ellipsoid.svg
Fig. 22. An umbilical geodesic, Шаблон:Math, Шаблон:Math, Шаблон:Math.

If the starting point is Шаблон:Math, Шаблон:Math (an umbilical point), and Шаблон:Math (the geodesic leaves the ellipse Шаблон:Math at right angles), then Шаблон:Math and the geodesic repeatedly intersects the opposite umbilical point and returns to its starting point. However, on each circuit the angle at which it intersects Шаблон:Math becomes closer to Шаблон:Val or Шаблон:Val so that asymptotically the geodesic lies on the ellipse Шаблон:Math Шаблон:Harv Шаблон:Harv, as shown in Fig. 22. A single geodesic does not fill an area on the ellipsoid. All tangents to umbilical geodesics touch the confocal hyperbola that intersects the ellipsoid at the umbilic points.

Umbilical geodesic enjoy several interesting properties.

  • Through any point on the ellipsoid, there are two umbilical geodesics.
  • The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic.
  • Whereas the closed geodesics on the ellipses Шаблон:Math and Шаблон:Math are stable (a geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse Шаблон:Math, which goes through all 4 umbilical points, is exponentially unstable. If it is perturbed, it will swing out of the plane Шаблон:Math and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.)

If the starting point Шаблон:Math of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on Шаблон:Math and the other two on Шаблон:Math. The cut locus for Шаблон:Math is the portion of the line Шаблон:Math between the cusps.

Applications

The direct and inverse geodesic problems no longer play the central role in geodesy that they once did. Instead of solving adjustment of geodetic networks as a two-dimensional problem in spheroidal trigonometry, these problems are now solved by three-dimensional methods Шаблон:Harv. Nevertheless, terrestrial geodesics still play an important role in several areas:

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 Шаблон:Harv Шаблон:Harv. For this reason, geodesics on simple surfaces such as ellipsoids of revolution or triaxial ellipsoids are frequently used as "test cases" for exploring new methods. Examples include:

See also

Notes

Шаблон:Reflist

References

Шаблон:Refbegin

Шаблон:Refend

External links

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