Geodesics on an ellipsoid

4105: 4093: 5553: 4192: 9066: 9041: 9272: 9249: 948: 9394: 2463: 4528: 3872: 2630: 5522: 8748: 3929: 936: 3917: 7566: 3942: 443: 42: 7300: 8280: 574: 2451: 6939: 8743:{\displaystyle {\begin{aligned}\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 }}}}\\&\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{aligned}}} 6320: 8062: 6652: 7295:{\displaystyle {\begin{aligned}\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{aligned}}} 6046: 4991: 7752: 2289: 6362: 5261: 6897: 6315:{\displaystyle {\begin{aligned}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{aligned}}} 3817: 1659: 5620:

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

4680: 1209: 542:

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

8057:{\displaystyle {\begin{aligned}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{aligned}}} 9152:

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

3976:

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

5654:

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

8957: 2123: 5397:

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:

6647:{\displaystyle S_{12}=R_{2}^{2}(\alpha _{2}-\alpha _{1})+b^{2}\int _{\lambda _{1}}^{\lambda _{2}}\left({\frac {1}{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 ,} 2601:

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

2866: 3177: 1396: 3434: 9680:, 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 ( 5071: 2734: 6731: 1839: 1985: 9376:; 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 7538:

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,

3304: 7684: 9119:, 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. 4986:{\displaystyle {\begin{aligned}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{aligned}}} 3059: 5836: 2440: 3635: 1035: 2968: 3652: 1534: 5930: 8795: 10899:(1891). "Über die Curve, welche alle von einem Punkte ausgehenden geodätischen Linien eines Rotationsellipsoides berührt" [The envelope of geodesic lines emanating from a single point on an ellipsoid]. In 2284:{\displaystyle {\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 }}.} 1523: 906:

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.

543:

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.

513:

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

9507:), 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. 8285: 4669: 4522: 9332:; 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 7757: 6944: 6051: 4685: 1539: 5988: 2102: 2769: 2040: 9692:). 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: 3070: 3493: 1290: 9901:

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

5729:, 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 3848:

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.

2347: 7466:

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

5256:{\displaystyle K={\frac {1}{\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}}}.} 8239:

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 (

6892:{\displaystyle \tan {\frac {E_{12}}{2}}={\frac {\sin {\tfrac {1}{2}}(\beta _{2}+\beta _{1})}{\cos {\tfrac {1}{2}}(\beta _{2}-\beta _{1})}}\tan {\frac {\omega _{12}}{2}}.} 2655: 5648:

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

3338: 9366:

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

9473:

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

1752: 3218: 7579: 3875:

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).

922:

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.

2979: 11748: 534:). 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 1926: 1204:{\displaystyle 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}}.} 5767: 5675:, 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 3812:{\displaystyle \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 ',} 2358: 1654:{\displaystyle {\begin{aligned}ds&={\sqrt {\rho ^{2}\varphi '^{2}+R^{2}}}\,d\lambda \\&\equiv L(\varphi ,\varphi ')\,d\lambda ,\end{aligned}}} 3564: 8773:

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

2885: 8952:{\displaystyle \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 ,} 5867: 1679: 401: 4291:. 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 9534:

are stable (a geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse

9166:. Two examples are given in Figs. 18 and 19. Figure 18 shows practically the same behavior as for an oblate ellipsoid of revolution (because 898:

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

11449: 10528: 9659: 11372:(1850) . "Sur les lignes de courbure de la surface de l'ellipsoïde" [On the lines of curvature on the surface of the ellipsoid]. In 9390:

closed, but fill the area bounded by the limiting lines of latitude (in the case of Figs. 18–19) or longitude (in the case of Figs. 20–21).

6930: 4423: 4386:

Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments (

1444: 482:

on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry (

223: 9552:

and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.)

391: 359: 4556:. As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions 9186:

are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at

7513:

Solving the geodesic problem for an ellipsoid of revolution is mathematically straightforward: because of symmetry, geodesics have a

10800:"Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution" 829: 9822: 5739:). Once this area is known, the area of a polygon may be computed by summing the contributions from all the edges of the polygon. 369: 17: 4390:), determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by 5290:

As we see from Fig. 14 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by

9091:

On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by

6919:

Solving the geodesic problems entails mapping the geodesic onto the auxiliary sphere and solving the corresponding problem in

4562: 11298: 11104: 10857: 10557: 10300: 10151: 10077: 4431: 505:

showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the

349: 11671: 10486: 9943:

This notation for the semi-axes is incompatible with that used in the previous section on ellipsoids of revolution in which

9613:

The direct and inverse geodesic problems no longer play the central role in geodesy that they once did. Instead of solving

10123: 6908: 824:

for the inverse problem, and its two adjacent sides. For a sphere the solutions to these problems are simple exercises in

10532: 9517:

The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic.

9325:

and the geodesic encircles the ellipsoid in a "transpolar" sense. The geodesic oscillates east and west of the ellipse

5941: 2861:{\displaystyle {\frac {1}{a}}{\frac {ds}{d\sigma }}={\frac {d\lambda }{d\omega }}={\frac {\sin \beta }{\sin \varphi }}.} 2062: 866:

During the 18th century geodesics were typically referred to as "shortest lines". The term "geodesic line" (actually, a

329: 10920: 7505:, addendum) extends the method to use elliptic integrals which can be applied to ellipsoids with arbitrary flattening. 4188:), so that the geodesic completes 2 (resp. 3) complete oscillations about the equator on one circuit of the ellipsoid. 130: 9176:); compare to Fig. 9. However, if the starting point is at a higher latitude (Fig. 18) the distortions resulting from 859:). The full solution for the direct problem (complete with computational tables and a worked out example) is given by 10718: 9621:

as a two-dimensional problem in spheroidal trigonometry, these problems are now solved by three-dimensional methods (

9112:. To survey the other geodesics, it is convenient to consider geodesics that intersect the middle principal section, 429: 3172:{\displaystyle {\frac {1}{a}}{\frac {ds}{d\sigma }}={\frac {d\lambda }{d\omega }}={\sqrt {1-e^{2}\cos ^{2}\beta }}.} 9902: 8200: 7560: 2008: 1391:{\displaystyle \cos \alpha \,ds=\rho \,d\varphi =-{\frac {dR}{\sin \varphi }},\quad \sin \alpha \,ds=R\,d\lambda ,} 8112:

coordinate system: the grid lines intersect at right angles. The principal sections of the ellipsoid, defined by

11856: 11846: 9648: 2114: 9386:

In Figs. 18–21, the geodesics are (very nearly) closed. As noted above, in the typical case, the geodesics are

4104: 4092: 11851: 9630: 4359:. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to 215: 11341:. Popular Lectures in Mathematics. Vol. 13. Translated by Collins, P.; Brown, R. B. New York: Macmillan. 5552: 3522: 2301: 231: 8192:

They are "lines of curvature" on the ellipsoid: they are parallel to the directions of principal curvature (

7535:, et al.), there was a complete understanding of the properties of geodesics on an ellipsoid of revolution. 4378:, and this means that meridional geodesics stop being shortest paths before the antipodal point is reached. 2614:, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for 5609:

if continued past the cut locus form an envelope illustrated in Fig. 15. Here the geodesics for which

5062: 4053:, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. 9). 3851:

There are also several ways of approximating geodesics on a terrestrial ellipsoid (with small flattening) (

3448: 886:

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

2476: 10802:[The geodesic on an ellipsoid and various applications of a remarkable analytical substitution]. 11179:, Dept. of Geodesy and Geomatics Engineering, Lecture Notes, Fredericton, N.B.: Univ. of New Brunswick, 4191: 11413: 11397: 11393: 10949: 9614: 1014: 10627: 9504: 9393: 9065: 9040: 11409: 11405: 10924: 9677: 9271: 9248: 3980:

All other geodesics are typified by Figs. 8 and 9 which show a geodesic starting on the equator with

2314: 947: 194: 11314:"Sur quelques cas particuliers où les équations du mouvement d'un point matériel peuvent s'intégrer" 3514: 593:

It is possible to reduce the various geodesic problems into one of two types. Consider two points:

11784: 11560: 11421: 10896: 10841: 10795: 10758: 10662: 10395:[Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini]. 7520: 7444:, the integrals (3) and (4) can be found by numerical quadrature or by expressing them in terms of 5858: 161: 108: 11765: 10520: 8208: 6691:

over its edges. This result holds provided that the polygon does not include a pole; if it does,

4527: 2729:{\displaystyle \cos \alpha \,d\sigma =d\beta ,\quad \sin \alpha \,d\sigma =\cos \beta \,d\omega .} 2462: 11704:"Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations" 11438: 11316:[On special cases where the equations of motion of a point particle can be integrated]. 10376: 10372: 9797: 9689: 9208: 8219: 6920: 4249:

up to the point at which they cease to be shortest paths. (The flattening has been increased to

3871: 2629: 833: 531: 299: 259: 11826: 4020:. The geodesic completes one full oscillation in latitude before the longitude has increased by 3429:{\displaystyle {\frac {s}{b}}=\int _{0}^{\sigma }{\sqrt {1+k^{2}\sin ^{2}\sigma '}}\,d\sigma ',} 11633: 11466: 11223: 10388: 10322: 9839: 8214:

Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points (

5521: 2049: 1913: 891: 839:

For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by

825: 498: 279: 11793:, a set of 500000 geodesics for the WGS84 ellipsoid, computed using high-precision arithmetic. 11454: 11377: 11249: 10828: 10799: 10507: 10392: 8156:

are equal. Here and in the other figures in this section the parameters of the ellipsoid are

4259:

in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant

3928: 3859:. However, these are typically comparable in complexity to the method for the exact solution ( 11506: 11486: 11313: 11261: 11245: 10904: 10762: 10645: 10508:"Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits" 10066: 9955:

stood for the equatorial radius and polar semi-axis. Thus the corresponding inequalities are

9792: 8227: 7524: 3856: 3186:

as the independent parameter in both of these differential equations and thereby to express

935: 562:

and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry (

555: 459: 422: 339: 89: 11679:(Technical report) (4th ed.). Monaco: International Hydrographic Bureau. Archived from 11326: 9722:

methods for solving systems of differential equations by a change of independent variables (

9562:

of a geodesic is not an umbilical point, its envelope is an astroid with two cusps lying on

8993: 7543:

discovered a constant of the motion allowing this problem to be reduced to quadrature also (

11718: 11569: 11401: 11346: 11219: 11191: 11180: 11142: 11114: 11019: 10961: 10867: 10832: 10742:

A Course of Mathematics in Three Volumes Composed for the Use of the Royal Military Academy

10608: 10364: 10351: 10184: 7556: 6031: 1834:{\displaystyle s_{12}=\int _{\lambda _{1}}^{\lambda _{2}}L(\varphi ,\varphi ')\,d\lambda ,} 510: 239: 189: 115: 11673:

A Manual on Technical Aspects of the United Nations Convention on the Law of the Sea, 1982

11265: 11195: 10887: 10631: 10510:[Elements of spheroidal trigonometry taken from the method of maxima and minima]. 10354:[Geodesic lines and the lines of curvature of the surfaces of the second degree]. 10220: 9931: 8188:

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

3299:{\displaystyle \sin \beta =\sin \beta (\sigma ;\alpha _{0})=\cos \alpha _{0}\sin \sigma ,} 1435: 8: 11092: 11038: 10702: 10425: 9787: 9782: 9755: 9637: 8153: 8082: 7679:{\displaystyle h={\frac {X^{2}}{a^{2}}}+{\frac {Y^{2}}{b^{2}}}+{\frac {Z^{2}}{c^{2}}}=1,} 4340:, this relation is exact and as a consequence the equator is only a shortest geodesic if 559: 551: 463: 151: 99: 11722: 11573: 11184: 11146: 11023: 10965: 10881: 10707: 10455: 10188: 5688:. All the geodesics are tangent to the envelope which is shown in green in the figure. 960:

Here the equations for a geodesic are developed; the derivation closely follows that of

11585: 11546: 11267:

Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures

11203: 11158: 11068: 11009: 10819: 10782: 10641: 10393:"Détermination géometrique de la perpendiculaire à la méridienne tracée par M. Cassini" 10228: 10200: 10174: 9701: 7565: 7515: 7445: 5854: 4547: 3941: 3916: 535: 515: 442: 71: 41: 11796: 10740: 10612: 10429: 10249: 10232: 9917:, §17) uses the term "coefficient of convergence of ordinates" for the geodesic scale. 9625:). Nevertheless, terrestrial geodesics still play an important role in several areas: 8247:

The day before yesterday, I reduced to quadrature the problem of geodesic lines on an

4278:

showed that, on any surface, geodesics and geodesic circle intersect at right angles.

3054:{\displaystyle {\frac {\sin \beta }{\sin \varphi }}={\sqrt {1-e^{2}\cos ^{2}\beta }},} 1409: 11744: 11699: 11589: 11383: 11350: 11294: 11275: 11233: 11207: 11162: 11118: 11100: 10977: 10910: 10871: 10853: 10823: 10786: 10746: 10724: 10714: 10678: 10618: 10589: 10563: 10553: 10490: 10465: 10439: 10306: 10296: 10268: 10204: 10147: 10109: 10083: 10073: 10043:, there are other simple closed geodesics similar to those shown in Figs. 11 and 12 ( 9618: 7705:

are Cartesian coordinates centered on the ellipsoid and, without loss of generality,

7474:

ellipsoid; however the inverse method fails to converge for nearly antipodal points.

5725: 4285:, the locus of points which have multiple (two in this case) shortest geodesics from 1917: 171: 156: 11646:

Sjöberg, L. E. (2006). "Determination of areas on the plane, sphere and ellipsoid".

11522: 10580:. Translated by Willis, J. M. St. Louis: Aeronautical Chart and Information Center. 10210: 7501:

and improves the solution of the inverse problem so that it converges in all cases.

11841: 11761: 11726: 11655: 11618: 11602: 11577: 11538: 11373: 11334: 11309: 11150: 11078: 11027: 10969: 10937: 10900: 10811: 10774: 10670: 10581: 10413: 10352:"Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré" 10334: 10244: 10192: 10162: 10139: 10101: 9730: 8252: 7532: 7481: 6705: 1980:{\displaystyle L-\varphi '{\frac {\partial L}{\partial \varphi '}}={\text{const.}}} 573: 468: 415: 166: 11130: 10765:[The theory of the calculus of variations and of differential equations]. 10540:(Technical report). Lausanne, Switzerland: Fédération Aéronautique Internationale. 10485:] (Technical report). Reihe B: Angewandte Geodäsie, Heft Nr. 292 (in German). 10233:"Jacobi's condition for problems of the calculus of variations in parametric form" 3988:. The geodesic oscillates about the equator. The equatorial crossings are called 11861: 11680: 11342: 11288: 11110: 10863: 10688: 10165:(2010) . "The calculation of longitude and latitude from geodesic measurements". 10061: 8149: 5831:{\displaystyle T=\int dT=\int {\frac {1}{K}}\cos \varphi \,d\varphi \,d\lambda ,} 5335: 11703: 7305:

The mapping of the geodesic involves evaluating the integrals for the distance,

6704:

must be added to the sum. If the edges are specified by their vertices, then a

4024:. Thus, on each successive northward crossing of the equator (see Fig. 8), 2435:{\displaystyle \sin \alpha _{1}\cos \beta _{1}=\sin \alpha _{2}\cos \beta _{2}.} 11730: 11659: 11502: 11482: 11462: 11083: 11056: 10845: 10736: 10545: 10503: 10417: 10347: 10281: 10262: 9817: 8988:, the familiar Clairaut relation. A derivation of Jacobi's result is given by 7528: 5714: 5661:

as Fig. 13 and is drawn in the same projection.) The two shorter geodesics are

3630:{\displaystyle d\omega ={\frac {\sin \alpha _{0}}{\cos ^{2}\beta }}\,d\sigma ,} 2450: 539: 379: 146: 120: 11750:

Application of three-dimensional geodesy to adjustments of horizontal networks

11032: 10997: 10338: 10143: 2963:{\displaystyle \tan \beta ={\sqrt {1-e^{2}}}\tan \varphi =(1-f)\tan \varphi ,} 11835: 11369: 11279: 11047: 10914: 10875: 10815: 10728: 10698: 10567: 10494: 10469: 10451: 10318: 10310: 10131: 10072:. Translated by Vogtmann, K.; Weinstein, A. (2nd ed.). Springer-Verlag. 9802: 6326: 494: 479: 11237: 10981: 10960:. Vol. 3. Translated by Carta, M. W. Washington, DC: Army Map Service. 10778: 10750: 10682: 10285: 8259: 6914: 5925:{\displaystyle \Gamma =\int K\,dT=\int \cos \varphi \,d\varphi \,d\lambda ,} 4046:

completes more that a full circuit; see Fig. 10). For nearly all values of

478:

is the shortest path between two points on a curved surface, analogous to a

11641:(Technical report). Washington, D.C.: U.S. Federal Aviation Administration. 11434: 11425: 11387: 11354: 11211: 11122: 10622: 10443: 10272: 10196: 10113: 10087: 9829: 7398: 104: 11790: 10593: 7376:

is given; this cannot be easily related to the equivalent spherical angle

4147:, §3.5.19). Two such geodesics are illustrated in Figs. 11 and 12. Here 10973: 10953: 10674: 10617:. Translated by Morehead, J. C.; Hiltebeitel, A. M. Princeton Univ. Lib. 10585: 10258: 10105: 8255:, which become the well known elliptic integrals if 2 axes are set equal. 2052:, using a geometrical construction; a similar derivation is presented by 691:

at the two endpoints. The two geodesic problems usually considered are:

61: 11816: 5269:, Eq. (6.5.1.)) solved the Gauss-Jacobi equation for this case enabling 788:

As can be seen from Fig. 1, these problems involve solving the triangle

11581: 11550: 11171: 11154: 9834: 8109: 7480:

continues the expansions to sixth order which suffices to provide full

5627:; points on the envelope may be computed by finding the point at which 1214:(In most applications in geodesy, the ellipsoid is taken to be oblate, 11622: 11606: 10941: 10852:. Translated by Balagangadharan, K. New Delhi: Hindustan Book Agency. 10763:"Zur Theorie der Variations-Rechnung und der Differential-Gleichungen" 9514:

Through any point on the ellipsoid, there are two umbilical geodesics.

3922:

Fig. 8. Following the geodesic on the ellipsoid for about 5 circuits.

10669:. Vol. 1. St. Louis: Aeronautical Chart and Information Center. 8777:. However, for geodesics that start at an umbilical points, we have 8761:. These two functions are just Abelian integrals..." Two constants 7519:, given by Clairaut's relation allowing the problem to be reduced to 4292: 4282: 4225: 612: 546:

By the end of the 18th century, an ellipsoid of revolution (the term

140: 66: 11542: 11526: 10690:

Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie

8171:, and it is viewed in an orthographic projection from a point above 1518:{\displaystyle ds^{2}=\rho ^{2}\,d\varphi ^{2}+R^{2}\,d\lambda ^{2}} 1224:; however, the theory applies without change to prolate ellipsoids, 11073: 9807: 5702: 600: 547: 474: 319: 136: 125: 11806: 11770:

Monatsberichte der Königlichen Akademie der Wissenschaft zu Berlin

11014: 10179: 9926:

This section is adapted from the documentation for GeographicLib (

518:(with three distinct semi-axes), only three geodesics are closed. 9812: 9740:

investigations into the number and stability of periodic orbits (

6335:

has been substituted. Applying this formula to the quadrilateral

5692: 5642: 1256:

Let an elementary segment of a path on the ellipsoid have length

843:. A systematic solution for the paths of geodesics was given by 675: 506: 455: 199: 56: 33: 11382:(in French) (5th ed.). Paris: Bachelier. pp. 139–160. 10646:"Geometrical demonstration of some properties of geodesic lines" 2468:

Fig. 5. The elementary geodesic problem on the auxiliary sphere.

27:

Shortest paths on a bounded deformed sphere-like quadric surface

11667: 8230:

with the ends of the string pinned to the two umbilical points.

1892:. The shortest path or geodesic entails finding that function 490: 309: 269: 207: 11635:

Order 8260.54A, The United States Standard for Area Navigation

11232:] (in French). Vol. 2. Paris: Crapelet. p. 112. 8785:

determines the direction at the umbilical point. The constant

6902: 5359:, to be a shortest path it must satisfy the Jacobi condition ( 4028:

falls short of a full circuit of the equator by approximately

11250:"Analyse des triangles tracées sur la surface d'un sphéroïde" 10173:(8). Translated by Karney, C. F. F.; Deakin, R. E.: 852–861. 9670: 7471: 7414:

is small, the integrals are typically evaluated as a series (

6665:). The integral can be expressed as a series valid for small 5723:

is a polygon whose sides are geodesics. It is analogous to a

2479:

of spherical trigonometry relating two sides of the triangle

867: 497:(all of which are closed) and the problems reduce to ones in 94: 11558:

Rainsford, H. F. (1955). "Long geodesics on the ellipsoid".

10381:

Abhandlungen Königlichen Akademie der Wissenschaft zu Berlin

7390:

is unknown. Thus, the solution of the problem requires that

5655:

increasing length. (This figure uses the same position for

953:

Fig. 3. Differential element of a geodesic on an ellipsoid.

11810: 11766:"Über die geodätischen Linien auf dem dreiaxigen Ellipsoid" 11629: 11445:. Translated by Motte, A. New York: Adee. pp. 405–409. 5058:. Their basic definitions are illustrated in Fig. 14. 4885: 4736: 3064:

so that the differential equations for the geodesic become

995:

Consider an ellipsoid of revolution with equatorial radius

396: 289: 10909:(in German). Vol. 7. Berlin: Reimer. pp. 72–87. 10614:

General Investigations of Curved Surfaces of 1827 and 1825

4059:

Two additional closed geodesics for the oblate ellipsoid,

521: 10512:

Mémoires de l'Académie Royale des Sciences de Berlin 1753

10438:] (in French). Vol. 3. Paris: Gauthier-Villars. 9510:

Umbilical geodesic enjoy several interesting properties.

6909:

Geographical distance § Ellipsoidal-surface formulae

4531:

Fig. 14. Definition of reduced length and geodesic scale.

4314:. The longitudinal extent of cut locus is approximately 3992:

and the points of maximum or minimum latitude are called

2456:

Fig. 4. Geodesic problem mapped to the auxiliary sphere.

10404:

Danielsen, J. S. (1989). "The Area under the Geodesic".

10397:

Mémoires de l'Académie Royale des Sciences de Paris 1733

9754:, geodesics on a triaxial ellipsoid reduce to a case of 9488:

so that asymptotically the geodesic lies on the ellipse

7337:

Handling the direct problem is straightforward, because

5641:

calls this star-like figure produced by the envelope an

5338:

to the starting point. In order for a geodesic between

4664:{\displaystyle t(s_{2})=Cm(s_{1},s_{2})+DM(s_{1},s_{2})} 4143:, another class of simple closed geodesics is possible ( 4042:(for a prolate ellipsoid, this quantity is negative and 3996:; the parametric latitudes of the vertices are given by 8966:

is the angle the geodesic makes with lines of constant

7550: 6923:. When solving the "elementary" spherical triangle for 5393:. If this condition is not satisfied, then there is a 4517:{\displaystyle {\frac {d^{2}t(s)}{ds^{2}}}+K(s)t(s)=0,} 4381: 2633:

Fig. 6. Differential element of a geodesic on a sphere.

550:

is also used) was a well-accepted approximation to the

9545:. If it is perturbed, it will swing out of the plane 8201:

confocal systems of hyperboloids of one and two sheets

8108:(in green) are given in Fig. 17. These constitute an 6820: 6771: 6657:

where the integral is over the geodesic line (so that

4180:, for the green (resp. blue) geodesic is chosen to be 3194:

as integrals. Applying the sine rule to the vertices

8798: 8283: 7755: 7582: 7344:

can be determined directly from the given quantities

6942: 6734: 6365: 6049: 5944: 5870: 5770: 5761:. The area of any closed region of the ellipsoid is 5074: 4683: 4565: 4434: 3855:, §6); some of these are described in the article on 3655: 3567: 3451: 3341: 3221: 3073: 2982: 2888: 2772: 2658: 2361: 2317: 2126: 2065: 2011: 1929: 1755: 1537: 1447: 1293: 1262:. From Figs. 2 and 3, we see that if its azimuth is 1038: 10667:

Mathematical and Physical Theories of Higher Geodesy

8152:(two of which are visible in this figure) where the 7508: 11057:"Geodesics on an arbitrary ellipsoid of revolution" 8089:is consistent with the previous sections. However, 6682:The area of a geodesic polygon is given by summing 3934:

Fig. 9. The same geodesic after about 70 circuits.

3825:and the limits on the integrals are chosen so that 941:

Fig. 2. Differential element of a meridian ellipse.

11768:[Geodesic lines on a triaxial ellipsoid]. 11527:"Sur les lignes géodésiques des surfaces convexes" 10706: 10697: 10065: 9685: 9204: 8951: 8742: 8215: 8199:They are also intersections of the ellipsoid with 8085:for an oblate ellipsoid, so the use of the symbol 8056: 7678: 7294: 6891: 6646: 6314: 5983:{\displaystyle \Gamma =2\pi -\sum _{j}\theta _{j}} 5982: 5924: 5830: 5255: 4985: 4663: 4516: 4195:Fig. 13. Geodesics (blue) from a single point for 3811: 3629: 3498:and the limits on the integral are chosen so that 3487: 3428: 3298: 3171: 3053: 2962: 2860: 2728: 2434: 2341: 2283: 2097:{\displaystyle d\alpha =\sin \varphi \,d\lambda .} 2096: 2034: 1979: 1833: 1653: 1517: 1390: 1203: 647:(see Fig. 1). The connecting geodesic (from 527: 11787:of books and articles on geodesics on ellipsoids. 11531:Transactions of the American Mathematical Society 11443:The Mathematical Principles of Natural Philosophy 10633:Disquisitiones generales circa superficies curvas 10237:Transactions of the American Mathematical Society 9761:extensions to an arbitrary number of dimensions ( 5527:Fig. 15. The envelope of geodesics from a point 5063:Gaussian curvature for an ellipsoid of revolution 4232:Fig. 13 shows geodesics (in blue) emanating 2564:In order to find the relation for the third side 11833: 11287:Leick, A.; Rapoport, L.; Tatarnikov, D. (2015). 11169: 10379:[General theory of geodesic triangles]. 9541:, which goes through all 4 umbilical points, is 8251:. They are the simplest formulas in the world, 8126:are shown in red. The third principal section, 977: 11743: 11509:[Elements of spheroidal trigonometry]. 11489:[Elements of spheroidal trigonometry]. 11469:[Elements of spheroidal trigonometry]. 11455:Liber Tertius, Prop. XIX. Prob. II. pp. 422–424 10804:Journal für die Reine und Angewandte Mathematik 10767:Journal für die Reine und Angewandte Mathematik 10745:. Vol. 3. London: F. C. and J. Rivington. 10713:. Translated by Nemenyi, P. New York: Chelsea. 10479:Methoden der ellipsoidischen Dreiecksberechnung 9622: 9003: 7523:. By the early 19th century (with the work of 4117:If the ellipsoid is sufficiently oblate, i.e., 1438:. The elementary segment is therefore given by 11529:[Geodesics lines on convex surfaces]. 11206:Boston: Hillard, Gray, Little, & Wilkins. 10377:"Allgemeine Theorie der geodätischen Dreiecke" 8226:in Fig. 17 can be generated with the familiar 4224:; geodesic circles are shown in green and the 526:There are several ways of defining geodesics ( 11507:"Elementi di trigonometria sferoidica, Pt. 3" 11487:"Elementi di trigonometria sferoidica, Pt. 2" 11467:"Elementi di trigonometria sferoidica, Pt. 1" 10323:"On the geodesic lines on an oblate spheroid" 9520:Whereas the closed geodesics on the ellipses 9356:, all meridians are geodesics; the effect of 8909: 8879: 8837: 8807: 8701: 8671: 8478: 8448: 6520: 6481: 6245: 6205: 6034:, and subtracting this from the equation for 5708: 5316:. On a closed surface such as an ellipsoid, 5236: 5196: 5144: 5104: 2056:, §10). Differentiating this relation gives 2035:{\displaystyle R\sin \alpha ={\text{const.}}} 1916:and the minimizing condition is given by the 992:also provide derivations of these equations. 423: 11318:Journal de Mathématiques Pures et Appliquées 11286: 11252:[Analysis of spheroidal triangles]. 10947: 10574: 10356:Journal de Mathématiques Pures et Appliquées 10008:playing the role of the parametric latitude. 8097:from the spherical longitude defined above. 973: 965: 878:Nous désignerons cette ligne sous le nom de 567: 558:entailed reducing all the measurements to a 11760: 11450:Philosophiæ Naturalis Principia Mathematica 11091: 10431:Leçons sur la théorie générale des surfaces 10371: 10279: 10068:Mathematical Methods of Classical Mechanics 10044: 9716: 9705: 7544: 6903:Solution of the direct and inverse problems 5325:oscillates about zero. The point at which 4144: 1710:. The length of an arbitrary path between 989: 925: 11332: 10138:. Translated by Senechal, L. J. Springer. 9905:for a particle on a surface of revolution. 9711:the development of differential geometry ( 7569:Fig. 17. Triaxial ellipsoidal coordinates. 4268:, which are the geodesic circles centered 3945:Fig. 10. Geodesic on a prolate ellipsoid ( 3322:. Substituting this into the equation for 2053: 430: 416: 397:Spatial Reference System Identifier (SRID) 392:International Terrestrial Reference System 11557: 11426:"Geodesic Utilities: Inverse and Forward" 11307: 11254:Mémoires de l'Institut National de France 11170:Krakiwsky, E. J.; Thomson, D. B. (1974), 11082: 11072: 11031: 11013: 10990:, 8th edition (Metzler, Stuttgart, 1941). 10650:Cambridge and Dublin Mathematical Journal 10436:Lessons on the general theory of surfaces 10403: 10248: 10178: 10094: 9914: 9658:the method of measuring distances in the 8997: 8753:As Jacobi notes "a function of the angle 8586: 8363: 7431: 6672: 6634: 6298: 6291: 6175: 6143: 6136: 6079: 5912: 5905: 5883: 5818: 5811: 5736: 4396:, and a second geodesic a small distance 3794: 3617: 3411: 2716: 2697: 2668: 2084: 1821: 1637: 1597: 1501: 1474: 1378: 1365: 1316: 1303: 969: 830:formulas for solving a spherical triangle 11800: 11697: 11521: 11511:Memorie dell'Istituto Nazionale Italiano 11491:Memorie dell'Istituto Nazionale Italiano 11471:Memorie dell'Istituto Nazionale Italiano 11260: 11244: 11218: 11190: 10387: 9898: 9823:Map projection of the triaxial ellipsoid 9741: 9697: 9681: 9392: 7564: 7463: 7449: 7415: 5558:Fig. 16. The four geodesics connecting 5477: 4526: 4190: 3940: 3870: 3866: 3550:. In order to express the equation for 3510: 2649:infinitesimally (see Fig. 6), we obtain 2628: 2595:, it is useful to consider the triangle 2045: 1426:is the radius of the circle of latitude 871: 844: 840: 572: 441: 11645: 11379:Application de l'Analyse à la Géometrie 11129: 10660: 10544: 10424: 10345: 10257: 9762: 9696:the development of elliptic integrals ( 9200: 8989: 8222:). For example, the lines of constant 7427: 6931:Napier's rules for quadrantal triangles 5758: 5375:), that there is no point conjugate to 5368: 5266: 563: 522:Geodesics on an ellipsoid of revolution 14: 11834: 11820: 11666: 11501: 11481: 11461: 11432: 11133:(1980). "Geodesics on the ellipsoid". 11054: 11045: 10994: 10934:Geometric Reference Systems in Geodesy 10931: 10894: 10839: 10793: 10757: 10735: 10476: 10317: 10160: 10130: 10060: 9927: 9769: 9734: 9723: 9641: 9500: 8271: 8267: 8240: 7721: 7540: 7502: 7477: 7457: 7453: 7423: 7419: 7402: 7361: 6676: 5701:. Likewise, the geodesic circles are 5638: 5603:The geodesics from a particular point 5364: 5360: 4387: 3860: 985: 961: 887: 860: 856: 852: 848: 502: 11756:(Technical report). NOAA. NOS NGS-13. 11367: 11361:Кратчайшие Линии: Вариационные Задачи 10606: 10501: 10483:Methods for ellipsoidal triangulation 10450: 10227: 9712: 9629:for measuring distances and areas in 8204: 8193: 5861:applied to a geodesic polygon states 5372: 4408: 4275: 3646:and Clairaut's relation. This yields 3488:{\displaystyle k=e'\cos \alpha _{0},} 2352:and Clairaut's relation then becomes 483: 11628: 11612: 11596: 11339:Shortest Paths: Variational Problems 11274:] (in French). Paris: Courcier. 11256:(in French) (1st semester): 130–161. 10640: 10464:] (in French). Paris: Courcier. 9652: 9496: 8234: 7551:Triaxial ellipsoid coordinate system 7435: 7367:In the case of the inverse problem, 5695:of the geodesic circles centered at 4382:Differential properties of geodesics 3852: 3521:is the same as the equation for the 3517:) pointed out that the equation for 981: 11439:"Book 3, Proposition 19, Problem 3" 11420: 10527: 9663: 8769:appear in the solution. Typically 6352:, and performing the integral over 402:Universal Transverse Mercator (UTM) 364:European Terrestrial Ref. Sys. 1989 24: 11813:utility for geodesic calculations. 10693:, Vol. 1 (Teubner, Leipzig, 1880). 10122:(Moscow, 1962) by U.S. Air Force ( 9878:azimuth instead; this is given by 8996:); he gives the solution found by 7726:(triaxial) ellipsoidal coordinates 7573:Consider the ellipsoid defined by 7327:and these depend on the parameter 6176: 6080: 5945: 5871: 5847:is an element of surface area and 1955: 1947: 472:, a slightly flattened sphere. A 458:specifically with the solution of 274:Ordnance Survey Great Britain 1936 240:Discrete Global Grid and Geocoding 131:Horizontal position representation 25: 11873: 11827:Drawing geodesics on Google Maps. 11778: 10293:Algorithms for Global Positioning 10250:10.1090/S0002-9947-1916-1501037-4 9932:Geodesics on a triaxial ellipsoid 9462:(the geodesic leaves the ellipse 9397:Fig. 22. An umbilical geodesic, 8249:ellipsoid with three unequal axes 7509:Geodesics on a triaxial ellipsoid 3882:Geodesic on an oblate ellipsoid ( 3332:and integrating the result gives 2754:gives differential equations for 446:A geodesic on an oblate ellipsoid 9903:conservation of angular momentum 9270: 9247: 9064: 9039: 9000:for general quadratic surfaces. 8228:string construction for ellipses 7561:Ellipsoidal-harmonic coordinates 7408:In geodetic applications, where 7360:; for a sample calculation, see 6008:. Multiplying the equation for 6002:is the exterior angle at vertex 5742:Here an expression for the area 5551: 5520: 4103: 4091: 3927: 3915: 2461: 2449: 946: 934: 911:In its adoption by other fields 581:on an ellipsoid of revolution. 190:Global Nav. Sat. Systems (GNSSs) 40: 11230:Treatise on Celestial Mechanics 11200:Treatise on Celestial Mechanics 10578:Geometry of the Earth Ellipsoid 10487:Deutsche Geodätische Kommission 10286:"11, Geometry of the Ellipsoid" 10004:gives a prolate ellipsoid with 9649:Federal Aviation Administration 9608: 8757:equals a function of the angle 8517: 5485:Geodesics from a single point ( 4882: 4733: 2687: 2342:{\displaystyle R=a\cos \beta ,} 2227: 2171: 2115:ordinary differential equations 1355: 1133: 1066: 354:N. American Vertical Datum 1988 11272:Exercises in Integral Calculus 11173:Geodetic position computations 10011: 9988: 9937: 9920: 9908: 9892: 9851: 9686:Hilbert & Cohn-Vossen 1952 9631:geographic information systems 9205:Hilbert & Cohn-Vossen 1952 8266:The solution given by Jacobi ( 8216:Hilbert & Cohn-Vossen 1952 7734:triaxial ellipsoidal longitude 6857: 6831: 6808: 6782: 6565: 6550: 6420: 6394: 5691:The astroid is the (exterior) 5287:to be expressed as integrals. 4922: 4896: 4866: 4840: 4773: 4747: 4717: 4691: 4658: 4632: 4620: 4594: 4582: 4569: 4502: 4496: 4490: 4484: 4457: 4451: 3747: 3735: 3262: 3243: 2945: 2933: 1912:. This is an exercise in the 1818: 1801: 1634: 1617: 1410:meridional radius of curvature 1125: 1113: 1022:, and the second eccentricity 528:Hilbert & Cohn-Vossen 1952 384:Internet link to a point 2010 314:Geodetic Reference System 1980 232:Quasi-Zenith Sat. Sys. (QZSS) 13: 1: 11202:. Vol. 1. Translated by 10988:Handbuch der Vermessungskunde 10636:(Dieterich, Göttingen, 1828). 10295:. Wellesley-Cambridge Press. 10053: 9874:. Some authors calculate the 7730:triaxial ellipsoidal latitude 5449:, the supplemental condition 978:Krakiwsky & Thomson (1974 828:, whose solution is given by 489:If the Earth is treated as a 374:Chinese obfuscated datum 2002 11817:GeographicLib implementation 11785:Online geodesic bibliography 11359:Translation from Russian of 10709:Geometry and the Imagination 10600:Геометрия земного эллипсоида 10598:Translation from Russian of 10120:Курс сфероидической геодезии 10118:Translation from Russian of 10098:Course in Spheroidal Geodesy 9970:for an oblate ellipsoid and 9768:geodesic flow on a surface ( 9341:ω = 180° − ω 9004:Survey of triaxial geodesics 8154:principal radii of curvature 8148:. These lines meet at four 6661:is implicitly a function of 4173:and the equatorial azimuth, 3972:. Compare with Fig. 8. 324:Geographic point coord. 1983 7: 11225:Traité de Mécanique Céleste 10457:Développements de Géométrie 9776: 9623:Vincenty & Bowring 1978 9592:is the portion of the line 7323: 7317: 5993:is the geodesic excess and 5709:Area of a geodesic polygon 3819: 3642: 3436: 2750: 2744: 2736: 2109: 1998: 1398: 801:for the direct problem and 466:is well approximated by an 284:Systema Koordinat 1942 goda 10: 11878: 11731:10.1179/sre.1975.23.176.88 11660:10.1179/003962606780732100 11615:Geometric geodesy, part II 11084:10.1007/s00190-023-01813-2 10998:"Algorithms for geodesics" 10418:10.1179/003962689791474267 10096:Bagratuni, G. V. (1967) . 9454:(an umbilical point), and 8133:, is covered by the lines 7554: 6912: 6906: 5712: 5334:becomes zero is the point 3206:in the spherical triangle 2537:and their opposite angles 1436:normal radius of curvature 966:Jordan & Eggert (1941) 915:, frequently shortened to 851:(and subsequent papers in 344:World Geodetic System 1984 11747:; Bowring, B. R. (1978). 11599:Geometric geodesy, part I 11055:Karney, C. F. F. (2024). 11046:Karney, C. F. F. (2015). 11033:10.1007/s00190-012-0578-z 10996:Karney, C. F. F. (2013). 10906:Jacobi's Gesammelte Werke 10630:. English translation of 10576:Gan'shin, V. V. (1969) . 10552:. Cambridge Univ. Press. 10531:(2018). "Section 8.2.3". 10339:10.1080/14786447008640411 10167:Astronomische Nachrichten 10144:10.1007/978-3-540-70997-8 9678:principle of least action 9436:If the starting point is 9295:If the starting point is 9130:∈ (−90°, 90°) 9122:If the starting point is 6915:Azimuth § In geodesy 5419:for a prolate ellipsoid, 5402:for an oblate ellipsoid, 3838:at the equator crossing, 2107:This, together with Eqs. 1007:. Define the flattening 990:Borre & Strang (2012) 454:arose in connection with 452:geodesics on an ellipsoid 334:North American Datum 1983 304:South American Datum 1969 11735:Addendum: Survey Review 11422:National Geodetic Survey 11135:Inventiones Mathematicae 10919:Op. post., completed by 10886:(Reimer, Berlin, 1866). 10883:Vorlesungen über Dynamik 10816:10.1515/crll.1839.19.309 10462:Developments in geometry 9985:for a prolate ellipsoid. 9845: 9669:help Muslims find their 7724:, §§26–27) employed the 6708:for the geodesic excess 3182:The last step is to use 926:Equations for a geodesic 743:inverse geodesic problem 195:Global Pos. System (GPS) 162:Spatial reference system 18:Inverse geodetic problem 11632:(2007). "Appendix 2.". 11447:English translation of 11293:(4th ed.). Wiley. 11290:GPS Satellite Surveying 10986:English translation of 10880:English translation of 10779:10.1515/crll.1837.17.68 10687:English translation of 10209:English translation of 9798:Great-circle navigation 9469:at right angles), then 9011:Circumpolar geodesics, 8100:Grid lines of constant 6921:great-circle navigation 5757:is developed following 2113:, leads to a system of 834:great-circle navigation 747:second geodesic problem 697:direct geodesic problem 538:—i.e., the analogue of 11857:Calculus of variations 11847:Geodesic (mathematics) 11791:Test set for geodesics 10550:Calculus of Variations 10327:Philosophical Magazine 10197:10.1002/asna.201011352 9671:direction toward Mecca 9556:If the starting point 9543:exponentially unstable 9433: 9217:Transpolar geodesics, 8953: 8744: 8264: 8058: 7680: 7570: 7397:be found iteratively ( 7296: 6893: 6648: 6316: 5984: 5926: 5832: 5257: 4987: 4665: 4532: 4518: 4229: 3973: 3876: 3813: 3631: 3489: 3430: 3300: 3173: 3055: 2964: 2862: 2730: 2643:is extended by moving 2634: 2436: 2343: 2285: 2098: 2036: 1981: 1914:calculus of variations 1835: 1655: 1519: 1392: 1205: 909: 884: 832:. (See the article on 826:spherical trigonometry 701:first geodesic problem 590: 585:is the north pole and 556:triangulation networks 499:spherical trigonometry 460:triangulation networks 447: 11852:Differential geometry 11772:(in German): 986–997. 11762:Weierstrass, K. T. W. 11093:Klingenberg, W. P. A. 10399:(in French): 406–416. 10383:(in German): 119–176. 10267:. Oxford: Clarendon. 9793:Geographical distance 9651:for area navigation ( 9586:. The cut locus for 9575:and the other two on 9396: 8954: 8745: 8245: 8243:, Letter to Bessel), 8059: 7681: 7568: 7311:, and the longitude, 7297: 6907:Further information: 6894: 6706:convenient expression 6649: 6317: 5985: 5927: 5833: 5478:Envelope of geodesics 5258: 4988: 4666: 4530: 4519: 4424:Gauss-Jacobi equation 4330:lies on the equator, 4194: 3944: 3874: 3867:Behavior of geodesics 3857:geographical distance 3814: 3632: 3490: 3431: 3301: 3174: 3056: 2965: 2871:The relation between 2863: 2731: 2632: 2581:, and included angle 2437: 2344: 2286: 2099: 2037: 1982: 1836: 1656: 1520: 1393: 1206: 896: 876: 576: 554:. The adjustment of 445: 90:Geographical distance 11617:, Ohio State Univ., 11613:Rapp, R. H. (1993), 11601:, Ohio State Univ., 11597:Rapp, R. H. (1991), 10974:10.5281/zenodo.35316 10936:, Ohio State Univ., 10901:K. T. W. Weierstrass 10850:Lectures on Dynamics 10675:10.5281/zenodo.32050 10586:10.5281/zenodo.32854 10106:10.5281/zenodo.32371 9647:in the rules of the 8796: 8789:may be expressed as 8281: 7753: 7580: 7557:Geodetic coordinates 6940: 6732: 6679:, §6 and addendum). 6363: 6047: 5942: 5868: 5859:Gauss–Bonnet theorem 5768: 5072: 4681: 4563: 4432: 4281:The red line is the 4098:Fig. 11. Side view. 3653: 3565: 3449: 3339: 3219: 3071: 2980: 2886: 2770: 2656: 2579:spherical arc length 2359: 2315: 2124: 2063: 2009: 1927: 1753: 1680:Lagrangian function 1535: 1445: 1291: 1036: 1001:and polar semi-axis 493:, the geodesics are 264:Sea Level Datum 1929 116:Geodetic coordinates 11809:, man page for the 11723:1975SurRv..23...88V 11574:1955BGeod..29...12R 11561:Bulletin Géodésique 11185:1974gpc..book.....K 11147:1980InMat..59..119K 11097:Riemannian Geometry 11024:2013JGeod..87...43K 10966:1962hage.book.....J 10958:Handbook of Geodesy 10932:Jekeli, C. (2012), 10477:Ehlert, D. (1993). 10189:2010AN....331..852K 9840:Vincenty's formulae 9788:Figure of the Earth 9783:Earth section paths 9756:dynamical billiards 9638:maritime boundaries 9605:between the cusps. 8083:parametric latitude 6607: 6464: 6393: 6276: 6174: 6121: 6078: 4110:Fig. 12. Top view. 3714: 3640:which follows from 3369: 2593:spherical longitude 2485:(see Fig. 4), 2302:parametric latitude 1797: 589:lie on the equator. 577:Fig. 1. A geodesic 560:reference ellipsoid 552:figure of the Earth 464:figure of the Earth 294:European Datum 1950 252:Standards (history) 152:Reference ellipsoid 100:Figure of the Earth 11739:(180): 294 (1976). 11582:10.1007/BF02527187 11155:10.1007/BF01390041 11061:Journal of Geodesy 11002:Journal of Geodesy 10833:French translation 10373:Christoffel, E. B. 9702:elliptic functions 9636:the definition of 9480:becomes closer to 9434: 9311:∈ (0°, 180°) 8980:, this reduces to 8949: 8740: 8738: 8054: 8052: 7676: 7571: 7516:constant of motion 7446:elliptic integrals 7438:). For arbitrary 7292: 7290: 6889: 6829: 6780: 6644: 6593: 6436: 6379: 6312: 6310: 6262: 6160: 6107: 6064: 5980: 5969: 5922: 5855:Gaussian curvature 5828: 5253: 4983: 4981: 4661: 4548:Gaussian curvature 4533: 4514: 4230: 3974: 3877: 3809: 3700: 3627: 3485: 3426: 3355: 3316:is the azimuth at 3296: 3169: 3051: 2960: 2858: 2726: 2635: 2432: 2339: 2281: 2094: 2032: 1977: 1831: 1769: 1651: 1649: 1515: 1388: 1201: 591: 536:geodesic curvature 516:triaxial ellipsoid 448: 172:Vertical positions 11300:978-1-119-01828-5 11106:978-3-11-008673-7 10921:F. H. A. Wangerin 10859:978-81-85931-91-3 10831:, Dec. 28, 1838. 10559:978-1-107-64083-2 10534:FAI Sporting Code 10302:978-0-9802327-3-8 10153:978-3-540-70996-1 10136:Geometry Revealed 10079:978-0-387-96890-2 9619:geodetic networks 8731: 8728: 8665: 8584: 8508: 8505: 8442: 8361: 8262:, 28th Dec. '38. 8253:Abelian integrals 8235:Jacobi's solution 8045: 8044: 8019: 7879: 7878: 7853: 7665: 7638: 7611: 6933:can be employed, 6884: 6861: 6828: 6779: 6756: 6618: 6586: 6526: 6257: 6102: 5960: 5800: 5726:spherical polygon 5616:is a multiple of 5248: 5166: 5094: 5023:is the so-called 4941: 4792: 4476: 3792: 3789: 3615: 3537:′ cosα 3523:arc on an ellipse 3409: 3350: 3164: 3125: 3102: 3082: 3046: 3007: 2919: 2853: 2824: 2801: 2781: 2276: 2246: 2222: 2190: 2166: 2145: 2030: 1990:Substituting for 1975: 1967: 1918:Beltrami identity 1848:is a function of 1595: 1350: 1196: 1175: 1171: 1128: 1103: 1099: 1061: 919:, was preferred. 794:given one angle, 568:Leick et al. 2015 440: 439: 388: 387: 167:Spatial relations 157:Satellite geodesy 112: 16:(Redirected from 11869: 11773: 11757: 11755: 11740: 11734: 11708: 11694: 11692: 11691: 11685: 11678: 11663: 11654:(301): 583–593. 11642: 11640: 11625: 11609: 11593: 11554: 11518: 11498: 11478: 11458: 11446: 11429: 11417: 11391: 11364: 11358: 11329: 11325: 11304: 11283: 11257: 11241: 11215: 11187: 11178: 11166: 11126: 11088: 11086: 11076: 11051: 11042: 11037: 11035: 11017: 10991: 10985: 10944: 10928: 10918: 10897:Jacobi, C. G. J. 10891: 10879: 10842:Jacobi, C. G. J. 10836: 10829:Letter to Bessel 10827: 10796:Jacobi, C. G. J. 10790: 10759:Jacobi, C. G. J. 10754: 10732: 10712: 10694: 10686: 10657: 10637: 10626: 10603: 10597: 10571: 10541: 10539: 10524: 10519: 10498: 10473: 10447: 10421: 10400: 10384: 10368: 10363: 10342: 10333:(268): 329–340. 10314: 10290: 10276: 10254: 10252: 10224: 10217:, 241–254 (1825) 10208: 10182: 10157: 10127: 10117: 10091: 10071: 10048: 10045:Klingenberg 1982 10042: 10041: 10040: 10036: 10031: 10030: 10024: 10015: 10009: 10007: 10003: 9992: 9986: 9984: 9969: 9954: 9948: 9941: 9935: 9924: 9918: 9912: 9906: 9896: 9890: 9888: 9873: 9863: 9855: 9753: 9717:Christoffel 1869 9706:Weierstrass 1861 9604: 9591: 9585: 9574: 9561: 9551: 9540: 9533: 9526: 9494: 9487: 9483: 9479: 9472: 9468: 9461: 9453: 9445: 9431: 9430: 9420: 9419: 9409: 9408: 9382: 9375: 9365: 9355: 9345: 9338: 9331: 9324: 9320: 9312: 9304: 9287: 9286: 9274: 9264: 9263: 9251: 9240: 9239: 9229: 9228: 9198: 9185: 9175: 9165: 9151: 9147: 9139: 9131: 9118: 9111: 9104: 9097: 9083: 9082: 9068: 9058: 9057: 9043: 9032: 9031: 9021: 9020: 8998:Liouville (1846) 8987: 8979: 8969: 8965: 8958: 8956: 8955: 8950: 8939: 8938: 8923: 8922: 8913: 8912: 8906: 8905: 8893: 8892: 8883: 8882: 8867: 8866: 8851: 8850: 8841: 8840: 8834: 8833: 8821: 8820: 8811: 8810: 8788: 8784: 8780: 8776: 8772: 8768: 8764: 8760: 8756: 8749: 8747: 8746: 8741: 8739: 8732: 8730: 8729: 8715: 8714: 8705: 8704: 8698: 8697: 8685: 8684: 8675: 8674: 8668: 8666: 8664: 8663: 8645: 8644: 8635: 8634: 8616: 8615: 8606: 8605: 8596: 8593: 8585: 8577: 8576: 8567: 8566: 8548: 8547: 8538: 8537: 8528: 8525: 8513: 8509: 8507: 8506: 8492: 8491: 8482: 8481: 8475: 8474: 8462: 8461: 8452: 8451: 8445: 8443: 8435: 8434: 8425: 8424: 8406: 8405: 8396: 8395: 8383: 8382: 8373: 8370: 8362: 8354: 8353: 8344: 8343: 8325: 8324: 8315: 8314: 8305: 8302: 8225: 8184: 8177: 8170: 8150:umbilical points 8147: 8143: 8139: 8132: 8125: 8118: 8107: 8103: 8092: 8088: 8080: 8076: 8063: 8061: 8060: 8055: 8053: 8046: 8043: 8042: 8030: 8029: 8020: 8018: 8017: 7999: 7998: 7989: 7988: 7970: 7969: 7960: 7959: 7950: 7949: 7880: 7877: 7876: 7864: 7863: 7854: 7846: 7845: 7836: 7835: 7817: 7816: 7807: 7806: 7794: 7793: 7784: 7783: 7745: 7719: 7704: 7685: 7683: 7682: 7677: 7666: 7664: 7663: 7654: 7653: 7644: 7639: 7637: 7636: 7627: 7626: 7617: 7612: 7610: 7609: 7600: 7599: 7590: 7545:Klingenberg 1982 7500: 7499: 7498: 7494: 7482:double precision 7469: 7443: 7413: 7396: 7389: 7382: 7375: 7359: 7352: 7343: 7333: 7314: 7310: 7301: 7299: 7298: 7293: 7291: 7284: 7283: 7185: 7184: 7125: 7124: 7068: 7067: 6962: 6961: 6929:in Fig. 5, 6928: 6898: 6896: 6895: 6890: 6885: 6880: 6879: 6870: 6862: 6860: 6856: 6855: 6843: 6842: 6830: 6821: 6811: 6807: 6806: 6794: 6793: 6781: 6772: 6762: 6757: 6752: 6751: 6742: 6724: 6703: 6690: 6670: 6664: 6660: 6653: 6651: 6650: 6645: 6624: 6620: 6619: 6617: 6616: 6606: 6601: 6592: 6587: 6585: 6568: 6546: 6545: 6532: 6527: 6525: 6524: 6523: 6511: 6510: 6501: 6500: 6485: 6484: 6471: 6463: 6462: 6461: 6451: 6450: 6449: 6435: 6434: 6419: 6418: 6406: 6405: 6392: 6387: 6375: 6374: 6355: 6351: 6340: 6333:for an ellipsoid 6332: 6321: 6319: 6318: 6313: 6311: 6281: 6277: 6275: 6270: 6258: 6256: 6255: 6254: 6249: 6248: 6235: 6234: 6225: 6224: 6209: 6208: 6201: 6200: 6191: 6173: 6168: 6153: 6126: 6122: 6120: 6115: 6103: 6095: 6077: 6072: 6039: 6029: 6020: 6011: 6007: 6001: 5989: 5987: 5986: 5981: 5979: 5978: 5968: 5931: 5929: 5928: 5923: 5852: 5846: 5837: 5835: 5834: 5829: 5801: 5793: 5756: 5750: 5734: 5721:geodesic polygon 5705:of the astroid. 5700: 5687: 5674: 5660: 5653: 5636: 5626: 5619: 5615: 5608: 5595: 5594: 5582: 5581: 5569: 5563: 5555: 5545: 5544: 5532: 5524: 5513: 5512: 5500: 5499: 5498: 5494: 5472: 5458: 5448: 5440: 5432: 5415: 5392: 5386: 5380: 5358: 5349: 5343: 5333: 5324: 5315: 5299: 5286: 5277: 5262: 5260: 5259: 5254: 5249: 5247: 5246: 5245: 5240: 5239: 5226: 5225: 5216: 5215: 5200: 5199: 5193: 5192: 5182: 5181: 5172: 5167: 5165: 5164: 5155: 5154: 5153: 5148: 5147: 5134: 5133: 5124: 5123: 5108: 5107: 5100: 5095: 5093: 5082: 5053: 5022: 4992: 4990: 4989: 4984: 4982: 4972: 4971: 4970: 4969: 4957: 4956: 4946: 4942: 4940: 4939: 4938: 4925: 4921: 4920: 4908: 4907: 4888: 4865: 4864: 4852: 4851: 4823: 4822: 4821: 4820: 4808: 4807: 4797: 4793: 4791: 4790: 4789: 4776: 4772: 4771: 4759: 4758: 4739: 4716: 4715: 4703: 4702: 4670: 4668: 4667: 4662: 4657: 4656: 4644: 4643: 4619: 4618: 4606: 4605: 4581: 4580: 4555: 4545: 4523: 4521: 4520: 4515: 4477: 4475: 4474: 4473: 4460: 4447: 4446: 4436: 4421: 4406: 4395: 4377: 4364: 4358: 4339: 4329: 4323: 4313: 4300: 4290: 4273: 4267: 4258: 4257: 4253: 4248: 4244: 4237: 4223: 4222: 4210: 4209: 4208: 4204: 4187: 4183: 4179: 4172: 4171: 4170: 4166: 4161: 4160: 4154: 4145:Klingenberg 1982 4142: 4141: 4140: 4136: 4131: 4130: 4124: 4107: 4095: 4084: 4083: 4082: 4078: 4073: 4072: 4066: 4052: 4045: 4041: 4027: 4023: 4019: 4010: 4009: 4005: 3987: 3971: 3970: 3960: 3959: 3958: 3954: 3931: 3919: 3908: 3907: 3897: 3896: 3895: 3891: 3844: 3837: 3821: 3818: 3816: 3815: 3810: 3805: 3793: 3791: 3790: 3788: 3777: 3776: 3767: 3766: 3751: 3727: 3716: 3713: 3708: 3699: 3698: 3671: 3670: 3636: 3634: 3633: 3628: 3616: 3614: 3607: 3606: 3596: 3595: 3594: 3578: 3557: 3553: 3549: 3543: 3542: 3541: 3520: 3508: 3494: 3492: 3491: 3486: 3481: 3480: 3465: 3438: 3435: 3433: 3432: 3427: 3422: 3410: 3408: 3397: 3396: 3387: 3386: 3371: 3368: 3363: 3351: 3343: 3331: 3321: 3315: 3305: 3303: 3302: 3297: 3283: 3282: 3261: 3260: 3212:in Fig. 5 gives 3211: 3205: 3199: 3193: 3189: 3185: 3178: 3176: 3175: 3170: 3165: 3157: 3156: 3147: 3146: 3131: 3126: 3124: 3116: 3108: 3103: 3101: 3093: 3085: 3083: 3075: 3060: 3058: 3057: 3052: 3047: 3039: 3038: 3029: 3028: 3013: 3008: 3006: 2995: 2984: 2969: 2967: 2966: 2961: 2920: 2918: 2917: 2902: 2878: 2874: 2867: 2865: 2864: 2859: 2854: 2852: 2841: 2830: 2825: 2823: 2815: 2807: 2802: 2800: 2792: 2784: 2782: 2774: 2763: 2759: 2738: 2735: 2733: 2732: 2727: 2648: 2642: 2625: 2621: 2617: 2613: 2607: 2600: 2590: 2576: 2560: 2550: 2536: 2525: 2524: 2520: 2510: 2499: 2498: 2494: 2484: 2465: 2453: 2441: 2439: 2438: 2433: 2428: 2427: 2412: 2411: 2393: 2392: 2377: 2376: 2348: 2346: 2345: 2340: 2307: 2300:in terms of the 2299: 2290: 2288: 2287: 2282: 2277: 2272: 2252: 2247: 2245: 2237: 2229: 2223: 2221: 2207: 2196: 2191: 2189: 2181: 2173: 2167: 2162: 2151: 2146: 2144: 2136: 2128: 2103: 2101: 2100: 2095: 2054:Lyusternik (1964 2041: 2039: 2038: 2033: 2031: 2028: 1995: 1986: 1984: 1983: 1978: 1976: 1973: 1968: 1966: 1965: 1953: 1945: 1943: 1911: 1903:which minimizes 1902: 1891: 1871: 1851: 1847: 1840: 1838: 1837: 1832: 1817: 1796: 1795: 1794: 1784: 1783: 1782: 1765: 1764: 1745: 1727: 1709: 1698: 1690: 1685: 1677: 1660: 1658: 1657: 1652: 1650: 1633: 1607: 1596: 1594: 1593: 1581: 1580: 1579: 1566: 1565: 1556: 1524: 1522: 1521: 1516: 1514: 1513: 1500: 1499: 1487: 1486: 1473: 1472: 1460: 1459: 1433: 1429: 1425: 1407: 1400: 1397: 1395: 1394: 1389: 1351: 1349: 1338: 1330: 1283: 1277: 1271: 1265: 1261: 1252: 1245: 1239: 1234:, in which case 1233: 1223: 1210: 1208: 1207: 1202: 1197: 1195: 1181: 1176: 1170: 1169: 1157: 1156: 1147: 1146: 1141: 1129: 1109: 1104: 1098: 1097: 1085: 1084: 1075: 1074: 1062: 1057: 1046: 1028: 1021: 1012: 1006: 1000: 950: 938: 880:ligne géodésique 870:) was coined by 823: 800: 793: 783: 776: 769: 760: 754: 737: 730: 724: 715: 708: 690: 683: 673: 664: 658: 652: 646: 637: 628: 622: 610: 598: 469:oblate ellipsoid 432: 425: 418: 256: 255: 235: 227: 219: 211: 203: 143: 102: 44: 30: 29: 21: 11877: 11876: 11872: 11871: 11870: 11868: 11867: 11866: 11832: 11831: 11801:Vincenty (1975) 11781: 11776: 11753: 11706: 11698: 11689: 11687: 11683: 11676: 11638: 11543:10.2307/1986219 11433: 11368: 11363:(Moscow, 1955). 11333: 11308: 11301: 11262:Legendre, A. M. 11246:Legendre, A. M. 11176: 11107: 11050:. Version 1.44. 11048:"GeographicLib" 10995: 10948: 10895: 10860: 10840: 10810:(19): 309–313. 10794: 10721: 10703:Cohn-Vossen, S. 10661: 10607: 10602:(Moscow, 1967). 10575: 10560: 10537: 10502: 10452:Dupin, P. C. F. 10389:Clairaut, A. C. 10346: 10303: 10288: 10161: 10154: 10095: 10080: 10056: 10051: 10038: 10034: 10033: 10026: 10020: 10019: 10018: 10016: 10012: 10005: 9995: 9993: 9989: 9971: 9956: 9950: 9944: 9942: 9938: 9925: 9921: 9915:Bagratuni (1962 9913: 9909: 9899:Laplace (1799a) 9897: 9893: 9883: 9879: 9869: 9862: 9858: 9856: 9852: 9848: 9779: 9748: 9662:Sporting Code ( 9611: 9603: 9593: 9587: 9580: 9577:ω = ω 9576: 9573: 9563: 9557: 9546: 9535: 9528: 9521: 9489: 9485: 9481: 9474: 9470: 9463: 9459: 9455: 9451: 9447: 9443: 9437: 9428: 9426: 9422: 9417: 9415: 9411: 9406: 9404: 9398: 9381: 9378:ω = ω 9377: 9367: 9357: 9347: 9344: 9340: 9337: 9334:ω = ω 9333: 9326: 9322: 9318: 9314: 9310: 9306: 9302: 9296: 9293: 9292: 9291: 9290: 9289: 9284: 9282: 9278: 9275: 9267: 9266: 9261: 9259: 9255: 9252: 9243: 9242: 9237: 9235: 9231: 9226: 9224: 9218: 9197: 9187: 9177: 9167: 9164: 9154: 9153:latitude lines 9149: 9145: 9141: 9137: 9133: 9129: 9123: 9113: 9106: 9099: 9092: 9089: 9088: 9087: 9086: 9085: 9080: 9078: 9072: 9069: 9061: 9060: 9055: 9053: 9047: 9044: 9035: 9034: 9029: 9027: 9023: 9018: 9016: 9012: 9006: 8981: 8971: 8970:. In the limit 8967: 8963: 8934: 8930: 8918: 8914: 8908: 8907: 8901: 8897: 8888: 8884: 8878: 8877: 8862: 8858: 8846: 8842: 8836: 8835: 8829: 8825: 8816: 8812: 8806: 8805: 8797: 8794: 8793: 8786: 8782: 8778: 8774: 8770: 8766: 8762: 8758: 8754: 8737: 8736: 8710: 8706: 8700: 8699: 8693: 8689: 8680: 8676: 8670: 8669: 8667: 8659: 8655: 8640: 8636: 8630: 8626: 8611: 8607: 8601: 8597: 8595: 8594: 8572: 8568: 8562: 8558: 8543: 8539: 8533: 8529: 8527: 8526: 8524: 8511: 8510: 8487: 8483: 8477: 8476: 8470: 8466: 8457: 8453: 8447: 8446: 8444: 8430: 8426: 8420: 8416: 8401: 8397: 8391: 8387: 8378: 8374: 8372: 8371: 8349: 8345: 8339: 8335: 8320: 8316: 8310: 8306: 8304: 8303: 8301: 8291: 8284: 8282: 8279: 8278: 8256: 8237: 8223: 8179: 8172: 8157: 8145: 8141: 8134: 8127: 8120: 8113: 8105: 8101: 8090: 8086: 8078: 8068: 8051: 8050: 8038: 8034: 8025: 8021: 8013: 8009: 7994: 7990: 7984: 7980: 7965: 7961: 7955: 7951: 7948: 7929: 7923: 7922: 7891: 7885: 7884: 7872: 7868: 7859: 7855: 7841: 7837: 7831: 7827: 7812: 7808: 7802: 7798: 7789: 7785: 7782: 7763: 7756: 7754: 7751: 7750: 7737: 7706: 7690: 7659: 7655: 7649: 7645: 7643: 7632: 7628: 7622: 7618: 7616: 7605: 7601: 7595: 7591: 7589: 7581: 7578: 7577: 7563: 7553: 7511: 7496: 7492: 7491: 7485: 7467: 7464:Vincenty (1975) 7439: 7409: 7395: 7391: 7388: 7384: 7381: 7377: 7374: 7368: 7358: 7354: 7351: 7345: 7342: 7338: 7332: 7328: 7312: 7306: 7289: 7288: 7279: 7275: 7232: 7220: 7219: 7180: 7176: 7163: 7151: 7150: 7120: 7116: 7094: 7082: 7081: 7063: 7059: 7025: 7013: 7012: 6963: 6957: 6953: 6943: 6941: 6938: 6937: 6924: 6917: 6911: 6905: 6875: 6871: 6869: 6851: 6847: 6838: 6834: 6819: 6812: 6802: 6798: 6789: 6785: 6770: 6763: 6761: 6747: 6743: 6741: 6733: 6730: 6729: 6723: 6719: 6715: 6709: 6702: 6692: 6689: 6683: 6666: 6662: 6658: 6612: 6608: 6602: 6597: 6591: 6569: 6538: 6534: 6533: 6531: 6519: 6518: 6506: 6502: 6496: 6492: 6480: 6479: 6475: 6470: 6469: 6465: 6457: 6453: 6452: 6445: 6441: 6440: 6430: 6426: 6414: 6410: 6401: 6397: 6388: 6383: 6370: 6366: 6364: 6361: 6360: 6353: 6350: 6346: 6342: 6336: 6328: 6309: 6308: 6271: 6266: 6250: 6244: 6243: 6242: 6230: 6226: 6220: 6216: 6204: 6203: 6202: 6196: 6192: 6190: 6189: 6185: 6169: 6164: 6151: 6150: 6116: 6111: 6094: 6093: 6089: 6073: 6068: 6057: 6050: 6048: 6045: 6044: 6035: 6032:authalic radius 6028: 6022: 6019: 6013: 6009: 6003: 6000: 5994: 5974: 5970: 5964: 5943: 5940: 5939: 5869: 5866: 5865: 5848: 5842: 5792: 5769: 5766: 5765: 5752: 5749: 5743: 5730: 5717: 5711: 5696: 5682: 5676: 5672: 5666: 5656: 5649: 5637:on a geodesic. 5634: 5628: 5622: 5617: 5614: 5610: 5604: 5601: 5600: 5599: 5598: 5597: 5592: 5590: 5584: 5579: 5577: 5571: 5565: 5559: 5556: 5548: 5547: 5542: 5540: 5534: 5528: 5525: 5516: 5515: 5510: 5508: 5502: 5496: 5492: 5491: 5486: 5480: 5467: 5460: 5459:is required if 5456: 5450: 5446: 5442: 5438: 5434: 5427: 5420: 5410: 5403: 5388: 5382: 5376: 5357: 5351: 5345: 5339: 5332: 5326: 5323: 5317: 5314: 5307: 5301: 5298: 5291: 5285: 5279: 5276: 5270: 5241: 5235: 5234: 5233: 5221: 5217: 5211: 5207: 5195: 5194: 5188: 5184: 5183: 5177: 5173: 5171: 5160: 5156: 5149: 5143: 5142: 5141: 5129: 5125: 5119: 5115: 5103: 5102: 5101: 5099: 5086: 5081: 5073: 5070: 5069: 5052: 5045: 5038: 5028: 5021: 5014: 5007: 4997: 4980: 4979: 4965: 4961: 4952: 4948: 4947: 4934: 4930: 4926: 4916: 4912: 4903: 4899: 4889: 4887: 4884: 4883: 4869: 4860: 4856: 4847: 4843: 4834: 4833: 4816: 4812: 4803: 4799: 4798: 4785: 4781: 4777: 4767: 4763: 4754: 4750: 4740: 4738: 4735: 4734: 4720: 4711: 4707: 4698: 4694: 4684: 4682: 4679: 4678: 4652: 4648: 4639: 4635: 4614: 4610: 4601: 4597: 4576: 4572: 4564: 4561: 4560: 4551: 4536: 4469: 4465: 4461: 4442: 4438: 4437: 4435: 4433: 4430: 4429: 4412: 4407:away from it. 4397: 4391: 4384: 4372: 4366: 4360: 4348: 4341: 4337: 4331: 4325: 4321: 4315: 4312: 4302: 4296: 4286: 4269: 4266: 4260: 4255: 4251: 4250: 4246: 4243: 4239: 4233: 4220: 4218: 4212: 4206: 4202: 4201: 4196: 4185: 4181: 4178: 4174: 4168: 4164: 4163: 4156: 4150: 4149: 4148: 4138: 4134: 4133: 4126: 4120: 4119: 4118: 4115: 4114: 4113: 4112: 4111: 4108: 4100: 4099: 4096: 4087: 4086: 4080: 4076: 4075: 4068: 4062: 4061: 4060: 4051: 4047: 4043: 4040: 4029: 4025: 4021: 4017: 4014:− |α 4007: 4003: 4002: 3997: 3985: 3981: 3968: 3966: 3962: 3956: 3952: 3951: 3946: 3939: 3938: 3937: 3936: 3935: 3932: 3924: 3923: 3920: 3911: 3910: 3905: 3903: 3899: 3893: 3889: 3888: 3883: 3869: 3839: 3836: 3826: 3798: 3781: 3772: 3768: 3762: 3758: 3750: 3728: 3717: 3715: 3709: 3704: 3694: 3690: 3666: 3662: 3654: 3651: 3650: 3602: 3598: 3597: 3590: 3586: 3579: 3577: 3566: 3563: 3562: 3555: 3551: 3545: 3540: 3532: 3530: 3526: 3525:with semi-axes 3518: 3499: 3476: 3472: 3458: 3450: 3447: 3446: 3415: 3401: 3392: 3388: 3382: 3378: 3370: 3364: 3359: 3342: 3340: 3337: 3336: 3323: 3317: 3314: 3310: 3278: 3274: 3256: 3252: 3220: 3217: 3216: 3207: 3201: 3195: 3191: 3187: 3183: 3152: 3148: 3142: 3138: 3130: 3117: 3109: 3107: 3094: 3086: 3084: 3074: 3072: 3069: 3068: 3034: 3030: 3024: 3020: 3012: 2996: 2985: 2983: 2981: 2978: 2977: 2913: 2909: 2901: 2887: 2884: 2883: 2876: 2872: 2842: 2831: 2829: 2816: 2808: 2806: 2793: 2785: 2783: 2773: 2771: 2768: 2767: 2761: 2755: 2742:Combining Eqs. 2657: 2654: 2653: 2644: 2638: 2623: 2619: 2615: 2609: 2603: 2596: 2589: 2582: 2575: 2565: 2559: 2552: 2549: 2538: 2535: 2522: 2518: 2517: 2512: 2509: 2496: 2492: 2491: 2486: 2480: 2473: 2472: 2471: 2470: 2469: 2466: 2458: 2457: 2454: 2423: 2419: 2407: 2403: 2388: 2384: 2372: 2368: 2360: 2357: 2356: 2316: 2313: 2312: 2305: 2295: 2294:We can express 2253: 2251: 2238: 2230: 2228: 2208: 2197: 2195: 2182: 2174: 2172: 2152: 2150: 2137: 2129: 2127: 2125: 2122: 2121: 2117:for a geodesic 2064: 2061: 2060: 2046:Clairaut (1735) 2027: 2010: 2007: 2006: 1996:and using Eqs. 1991: 1972: 1958: 1954: 1946: 1944: 1936: 1928: 1925: 1924: 1910: 1904: 1893: 1890: 1883: 1873: 1870: 1863: 1853: 1849: 1845: 1810: 1790: 1786: 1785: 1778: 1774: 1773: 1760: 1756: 1754: 1751: 1750: 1743: 1736: 1729: 1725: 1718: 1711: 1700: 1692: 1688: 1681: 1665: 1648: 1647: 1626: 1605: 1604: 1589: 1585: 1575: 1571: 1567: 1561: 1557: 1555: 1548: 1538: 1536: 1533: 1532: 1509: 1505: 1495: 1491: 1482: 1478: 1468: 1464: 1455: 1451: 1446: 1443: 1442: 1431: 1427: 1413: 1405: 1339: 1331: 1329: 1292: 1289: 1288: 1279: 1273: 1267: 1263: 1257: 1253:are negative.) 1247: 1241: 1235: 1225: 1215: 1185: 1180: 1165: 1161: 1152: 1148: 1145: 1134: 1108: 1093: 1089: 1080: 1076: 1073: 1047: 1045: 1037: 1034: 1033: 1023: 1017: 1008: 1002: 996: 970:Bagratuni (1962 958: 957: 956: 955: 954: 951: 943: 942: 939: 928: 872:Laplace (1799b) 845:Legendre (1806) 841:Clairaut (1735) 822: 815: 808: 802: 799: 795: 789: 782: 778: 775: 771: 768: 762: 756: 750: 736: 732: 726: 723: 717: 714: 710: 704: 689: 685: 682: 678: 672: 666: 660: 654: 648: 645: 639: 636: 630: 624: 621: 615: 609: 603: 594: 524: 436: 407: 406: 253: 245: 244: 233: 225: 217: 209: 201: 185: 177: 176: 135: 85: 77: 76: 52: 28: 23: 22: 15: 12: 11: 5: 11875: 11865: 11864: 11859: 11854: 11849: 11844: 11830: 11829: 11824: 11814: 11804: 11794: 11788: 11780: 11779:External links 11777: 11775: 11774: 11758: 11741: 11717:(176): 88–93. 11695: 11664: 11643: 11626: 11610: 11594: 11555: 11537:(3): 237–274. 11519: 11513:(in Italian). 11499: 11493:(in Italian). 11479: 11473:(in Italian). 11459: 11430: 11428:. Version 3.0. 11418: 11365: 11335:Lyusternik, L. 11330: 11305: 11299: 11284: 11258: 11242: 11220:Laplace, P. S. 11216: 11192:Laplace, P. S. 11188: 11167: 11141:(2): 119–143. 11127: 11105: 11089: 11052: 11043: 10992: 10945: 10929: 10892: 10858: 10837: 10791: 10755: 10733: 10719: 10695: 10663:Helmert, F. R. 10658: 10638: 10604: 10572: 10558: 10546:Forsyth, A. R. 10542: 10525: 10499: 10474: 10448: 10426:Darboux, J. G. 10422: 10412:(232): 61–66. 10401: 10385: 10369: 10343: 10329:. 4th series. 10315: 10301: 10277: 10255: 10243:(2): 195–206. 10225: 10212:Astron. Nachr. 10158: 10152: 10128: 10092: 10078: 10057: 10055: 10052: 10050: 10049: 10010: 9987: 9936: 9919: 9907: 9891: 9881: 9860: 9849: 9847: 9844: 9843: 9842: 9837: 9832: 9827: 9826: 9825: 9818:Map projection 9815: 9810: 9805: 9800: 9795: 9790: 9785: 9778: 9775: 9774: 9773: 9766: 9759: 9745: 9738: 9727: 9720: 9709: 9674: 9673: 9667: 9656: 9645: 9634: 9610: 9607: 9601: 9578: 9571: 9554: 9553: 9518: 9515: 9457: 9449: 9441: 9424: 9413: 9402: 9379: 9342: 9335: 9316: 9308: 9300: 9280: 9276: 9269: 9268: 9257: 9253: 9246: 9245: 9244: 9233: 9222: 9216: 9215: 9214: 9213: 9195: 9162: 9143: 9135: 9127: 9076: 9070: 9063: 9062: 9051: 9045: 9038: 9037: 9036: 9025: 9014: 9010: 9009: 9008: 9007: 9005: 9002: 8960: 8959: 8948: 8945: 8942: 8937: 8933: 8929: 8926: 8921: 8917: 8911: 8904: 8900: 8896: 8891: 8887: 8881: 8876: 8873: 8870: 8865: 8861: 8857: 8854: 8849: 8845: 8839: 8832: 8828: 8824: 8819: 8815: 8809: 8804: 8801: 8751: 8750: 8735: 8727: 8724: 8721: 8718: 8713: 8709: 8703: 8696: 8692: 8688: 8683: 8679: 8673: 8662: 8658: 8654: 8651: 8648: 8643: 8639: 8633: 8629: 8625: 8622: 8619: 8614: 8610: 8604: 8600: 8592: 8589: 8583: 8580: 8575: 8571: 8565: 8561: 8557: 8554: 8551: 8546: 8542: 8536: 8532: 8523: 8520: 8516: 8514: 8512: 8504: 8501: 8498: 8495: 8490: 8486: 8480: 8473: 8469: 8465: 8460: 8456: 8450: 8441: 8438: 8433: 8429: 8423: 8419: 8415: 8412: 8409: 8404: 8400: 8394: 8390: 8386: 8381: 8377: 8369: 8366: 8360: 8357: 8352: 8348: 8342: 8338: 8334: 8331: 8328: 8323: 8319: 8313: 8309: 8300: 8297: 8294: 8292: 8290: 8287: 8286: 8236: 8233: 8232: 8231: 8212: 8197: 8104:(in blue) and 8065: 8064: 8049: 8041: 8037: 8033: 8028: 8024: 8016: 8012: 8008: 8005: 8002: 7997: 7993: 7987: 7983: 7979: 7976: 7973: 7968: 7964: 7958: 7954: 7947: 7944: 7941: 7938: 7935: 7932: 7930: 7928: 7925: 7924: 7921: 7918: 7915: 7912: 7909: 7906: 7903: 7900: 7897: 7894: 7892: 7890: 7887: 7886: 7883: 7875: 7871: 7867: 7862: 7858: 7852: 7849: 7844: 7840: 7834: 7830: 7826: 7823: 7820: 7815: 7811: 7805: 7801: 7797: 7792: 7788: 7781: 7778: 7775: 7772: 7769: 7766: 7764: 7762: 7759: 7758: 7687: 7686: 7675: 7672: 7669: 7662: 7658: 7652: 7648: 7642: 7635: 7631: 7625: 7621: 7615: 7608: 7604: 7598: 7594: 7588: 7585: 7552: 7549: 7510: 7507: 7432:Rainsford 1955 7393: 7386: 7379: 7372: 7356: 7349: 7340: 7330: 7303: 7302: 7287: 7282: 7278: 7274: 7271: 7268: 7265: 7262: 7259: 7256: 7253: 7250: 7247: 7244: 7241: 7238: 7235: 7233: 7231: 7228: 7225: 7222: 7221: 7218: 7215: 7212: 7209: 7206: 7203: 7200: 7197: 7194: 7191: 7188: 7183: 7179: 7175: 7172: 7169: 7166: 7164: 7162: 7159: 7156: 7153: 7152: 7149: 7146: 7143: 7140: 7137: 7134: 7131: 7128: 7123: 7119: 7115: 7112: 7109: 7106: 7103: 7100: 7097: 7095: 7093: 7090: 7087: 7084: 7083: 7080: 7077: 7074: 7071: 7066: 7062: 7058: 7055: 7052: 7049: 7046: 7043: 7040: 7037: 7034: 7031: 7028: 7026: 7024: 7021: 7018: 7015: 7014: 7011: 7008: 7005: 7002: 6999: 6996: 6993: 6990: 6987: 6984: 6981: 6978: 6975: 6972: 6969: 6966: 6964: 6960: 6956: 6952: 6949: 6946: 6945: 6904: 6901: 6900: 6899: 6888: 6883: 6878: 6874: 6868: 6865: 6859: 6854: 6850: 6846: 6841: 6837: 6833: 6827: 6824: 6818: 6815: 6810: 6805: 6801: 6797: 6792: 6788: 6784: 6778: 6775: 6769: 6766: 6760: 6755: 6750: 6746: 6740: 6737: 6721: 6720:− α 6717: 6713: 6700: 6687: 6673:Danielsen 1989 6655: 6654: 6643: 6640: 6637: 6633: 6630: 6627: 6623: 6615: 6611: 6605: 6600: 6596: 6590: 6584: 6581: 6578: 6575: 6572: 6567: 6564: 6561: 6558: 6555: 6552: 6549: 6544: 6541: 6537: 6530: 6522: 6517: 6514: 6509: 6505: 6499: 6495: 6491: 6488: 6483: 6478: 6474: 6468: 6460: 6456: 6448: 6444: 6439: 6433: 6429: 6425: 6422: 6417: 6413: 6409: 6404: 6400: 6396: 6391: 6386: 6382: 6378: 6373: 6369: 6348: 6347:− α 6344: 6341:, noting that 6323: 6322: 6307: 6304: 6301: 6297: 6294: 6290: 6287: 6284: 6280: 6274: 6269: 6265: 6261: 6253: 6247: 6241: 6238: 6233: 6229: 6223: 6219: 6215: 6212: 6207: 6199: 6195: 6188: 6184: 6181: 6178: 6172: 6167: 6163: 6159: 6156: 6154: 6152: 6149: 6146: 6142: 6139: 6135: 6132: 6129: 6125: 6119: 6114: 6110: 6106: 6101: 6098: 6092: 6088: 6085: 6082: 6076: 6071: 6067: 6063: 6060: 6058: 6056: 6053: 6052: 6026: 6017: 5996: 5991: 5990: 5977: 5973: 5967: 5963: 5959: 5956: 5953: 5950: 5947: 5933: 5932: 5921: 5918: 5915: 5911: 5908: 5904: 5901: 5898: 5895: 5892: 5889: 5886: 5882: 5879: 5876: 5873: 5839: 5838: 5827: 5824: 5821: 5817: 5814: 5810: 5807: 5804: 5799: 5796: 5791: 5788: 5785: 5782: 5779: 5776: 5773: 5759:Sjöberg (2006) 5747: 5737:Danielsen 1989 5715:Equal-area map 5710: 5707: 5680: 5670: 5632: 5612: 5588: 5575: 5557: 5550: 5549: 5538: 5526: 5519: 5518: 5517: 5506: 5484: 5483: 5482: 5481: 5479: 5476: 5475: 5474: 5465: 5454: 5444: 5436: 5425: 5417: 5408: 5355: 5330: 5321: 5312: 5305: 5296: 5283: 5274: 5264: 5263: 5252: 5244: 5238: 5232: 5229: 5224: 5220: 5214: 5210: 5206: 5203: 5198: 5191: 5187: 5180: 5176: 5170: 5163: 5159: 5152: 5146: 5140: 5137: 5132: 5128: 5122: 5118: 5114: 5111: 5106: 5098: 5092: 5089: 5085: 5080: 5077: 5056:geodesic scale 5050: 5043: 5036: 5025:reduced length 5019: 5012: 5005: 4994: 4993: 4978: 4975: 4968: 4964: 4960: 4955: 4951: 4945: 4937: 4933: 4929: 4924: 4919: 4915: 4911: 4906: 4902: 4898: 4895: 4892: 4886: 4881: 4878: 4875: 4872: 4870: 4868: 4863: 4859: 4855: 4850: 4846: 4842: 4839: 4836: 4835: 4832: 4829: 4826: 4819: 4815: 4811: 4806: 4802: 4796: 4788: 4784: 4780: 4775: 4770: 4766: 4762: 4757: 4753: 4749: 4746: 4743: 4737: 4732: 4729: 4726: 4723: 4721: 4719: 4714: 4710: 4706: 4701: 4697: 4693: 4690: 4687: 4686: 4672: 4671: 4660: 4655: 4651: 4647: 4642: 4638: 4634: 4631: 4628: 4625: 4622: 4617: 4613: 4609: 4604: 4600: 4596: 4593: 4590: 4587: 4584: 4579: 4575: 4571: 4568: 4525: 4524: 4513: 4510: 4507: 4504: 4501: 4498: 4495: 4492: 4489: 4486: 4483: 4480: 4472: 4468: 4464: 4459: 4456: 4453: 4450: 4445: 4441: 4383: 4380: 4370: 4346: 4335: 4319: 4310: 4264: 4245:a multiple of 4241: 4216: 4176: 4109: 4102: 4101: 4097: 4090: 4089: 4088: 4058: 4057: 4056: 4055: 4049: 4038: 4015: 3983: 3964: 3933: 3926: 3925: 3921: 3914: 3913: 3912: 3901: 3881: 3880: 3879: 3878: 3868: 3865: 3834: 3823: 3822: 3808: 3804: 3801: 3797: 3787: 3784: 3780: 3775: 3771: 3765: 3761: 3757: 3754: 3749: 3746: 3743: 3740: 3737: 3734: 3731: 3726: 3723: 3720: 3712: 3707: 3703: 3697: 3693: 3689: 3686: 3683: 3680: 3677: 3674: 3669: 3665: 3661: 3658: 3638: 3637: 3626: 3623: 3620: 3613: 3610: 3605: 3601: 3593: 3589: 3585: 3582: 3576: 3573: 3570: 3538: 3511:Legendre (1811 3496: 3495: 3484: 3479: 3475: 3471: 3468: 3464: 3461: 3457: 3454: 3440: 3439: 3425: 3421: 3418: 3414: 3407: 3404: 3400: 3395: 3391: 3385: 3381: 3377: 3374: 3367: 3362: 3358: 3354: 3349: 3346: 3312: 3307: 3306: 3295: 3292: 3289: 3286: 3281: 3277: 3273: 3270: 3267: 3264: 3259: 3255: 3251: 3248: 3245: 3242: 3239: 3236: 3233: 3230: 3227: 3224: 3180: 3179: 3168: 3163: 3160: 3155: 3151: 3145: 3141: 3137: 3134: 3129: 3123: 3120: 3115: 3112: 3106: 3100: 3097: 3092: 3089: 3081: 3078: 3062: 3061: 3050: 3045: 3042: 3037: 3033: 3027: 3023: 3019: 3016: 3011: 3005: 3002: 2999: 2994: 2991: 2988: 2971: 2970: 2959: 2956: 2953: 2950: 2947: 2944: 2941: 2938: 2935: 2932: 2929: 2926: 2923: 2916: 2912: 2908: 2905: 2900: 2897: 2894: 2891: 2869: 2868: 2857: 2851: 2848: 2845: 2840: 2837: 2834: 2828: 2822: 2819: 2814: 2811: 2805: 2799: 2796: 2791: 2788: 2780: 2777: 2740: 2739: 2725: 2722: 2719: 2715: 2712: 2709: 2706: 2703: 2700: 2696: 2693: 2690: 2686: 2683: 2680: 2677: 2674: 2671: 2667: 2664: 2661: 2587: 2573: 2557: 2547: 2546:− α 2533: 2507: 2467: 2460: 2459: 2455: 2448: 2447: 2446: 2445: 2444: 2443: 2442: 2431: 2426: 2422: 2418: 2415: 2410: 2406: 2402: 2399: 2396: 2391: 2387: 2383: 2380: 2375: 2371: 2367: 2364: 2350: 2349: 2338: 2335: 2332: 2329: 2326: 2323: 2320: 2292: 2291: 2280: 2275: 2271: 2268: 2265: 2262: 2259: 2256: 2250: 2244: 2241: 2236: 2233: 2226: 2220: 2217: 2214: 2211: 2206: 2203: 2200: 2194: 2188: 2185: 2180: 2177: 2170: 2165: 2161: 2158: 2155: 2149: 2143: 2140: 2135: 2132: 2105: 2104: 2093: 2090: 2087: 2083: 2080: 2077: 2074: 2071: 2068: 2043: 2042: 2026: 2023: 2020: 2017: 2014: 1988: 1987: 1971: 1964: 1961: 1957: 1952: 1949: 1942: 1939: 1935: 1932: 1908: 1888: 1881: 1868: 1861: 1842: 1841: 1830: 1827: 1824: 1820: 1816: 1813: 1809: 1806: 1803: 1800: 1793: 1789: 1781: 1777: 1772: 1768: 1763: 1759: 1741: 1734: 1723: 1716: 1662: 1661: 1646: 1643: 1640: 1636: 1632: 1629: 1625: 1622: 1619: 1616: 1613: 1610: 1608: 1606: 1603: 1600: 1592: 1588: 1584: 1578: 1574: 1570: 1564: 1560: 1554: 1551: 1549: 1547: 1544: 1541: 1540: 1526: 1525: 1512: 1508: 1504: 1498: 1494: 1490: 1485: 1481: 1477: 1471: 1467: 1463: 1458: 1454: 1450: 1402: 1401: 1387: 1384: 1381: 1377: 1374: 1371: 1368: 1364: 1361: 1358: 1354: 1348: 1345: 1342: 1337: 1334: 1328: 1325: 1322: 1319: 1315: 1312: 1309: 1306: 1302: 1299: 1296: 1272:is related to 1212: 1211: 1200: 1194: 1191: 1188: 1184: 1179: 1174: 1168: 1164: 1160: 1155: 1151: 1144: 1140: 1137: 1132: 1127: 1124: 1121: 1118: 1115: 1112: 1107: 1102: 1096: 1092: 1088: 1083: 1079: 1072: 1069: 1065: 1060: 1056: 1053: 1050: 1044: 1041: 974:Gan'shin (1967 952: 945: 944: 940: 933: 932: 931: 930: 929: 927: 924: 904:geodesic line: 820: 813: 806: 797: 786: 785: 780: 773: 766: 739: 734: 721: 712: 687: 680: 670: 643: 638:and longitude 634: 619: 607: 540:straight lines 523: 520: 438: 437: 435: 434: 427: 420: 412: 409: 408: 405: 404: 399: 394: 386: 385: 382: 376: 375: 372: 366: 365: 362: 356: 355: 352: 346: 345: 342: 336: 335: 332: 326: 325: 322: 316: 315: 312: 306: 305: 302: 296: 295: 292: 286: 285: 282: 276: 275: 272: 266: 265: 262: 254: 251: 250: 247: 246: 243: 242: 237: 229: 221: 213: 205: 197: 192: 186: 183: 182: 179: 178: 175: 174: 169: 164: 159: 154: 149: 147:Map projection 144: 133: 128: 123: 121:Geodetic datum 118: 113: 97: 92: 86: 83: 82: 79: 78: 75: 74: 69: 64: 59: 53: 50: 49: 46: 45: 37: 36: 26: 9: 6: 4: 3: 2: 11874: 11863: 11860: 11858: 11855: 11853: 11850: 11848: 11845: 11843: 11840: 11839: 11837: 11828: 11825: 11822: 11821:Karney (2013) 11818: 11815: 11812: 11808: 11805: 11802: 11799:implementing 11798: 11795: 11792: 11789: 11786: 11783: 11782: 11771: 11767: 11763: 11759: 11752: 11751: 11746: 11742: 11738: 11732: 11728: 11724: 11720: 11716: 11712: 11711:Survey Review 11705: 11701: 11696: 11686:on 2013-05-24 11682: 11675: 11674: 11669: 11665: 11661: 11657: 11653: 11649: 11648:Survey Review 11644: 11637: 11636: 11631: 11627: 11624: 11620: 11616: 11611: 11608: 11604: 11600: 11595: 11591: 11587: 11583: 11579: 11575: 11571: 11567: 11563: 11562: 11556: 11552: 11548: 11544: 11540: 11536: 11533:(in French). 11532: 11528: 11524: 11520: 11516: 11512: 11508: 11504: 11500: 11496: 11492: 11488: 11484: 11480: 11477:(1): 118–198. 11476: 11472: 11468: 11464: 11460: 11456: 11452: 11451: 11444: 11440: 11436: 11431: 11427: 11423: 11419: 11415: 11411: 11407: 11403: 11399: 11395: 11389: 11385: 11381: 11380: 11375: 11371: 11366: 11362: 11356: 11352: 11348: 11344: 11340: 11336: 11331: 11328: 11323: 11320:(in French). 11319: 11315: 11311: 11310:Liouville, J. 11306: 11302: 11296: 11292: 11291: 11285: 11281: 11277: 11273: 11269: 11268: 11263: 11259: 11255: 11251: 11247: 11243: 11239: 11235: 11231: 11227: 11226: 11221: 11217: 11213: 11209: 11205: 11201: 11197: 11196:"Book 1, §8." 11193: 11189: 11186: 11182: 11175: 11174: 11168: 11164: 11160: 11156: 11152: 11148: 11144: 11140: 11136: 11132: 11128: 11124: 11120: 11116: 11112: 11108: 11102: 11099:. de Gruyer. 11098: 11094: 11090: 11085: 11080: 11075: 11070: 11067:(1): 4:1–14. 11066: 11062: 11058: 11053: 11049: 11044: 11040: 11034: 11029: 11025: 11021: 11016: 11011: 11007: 11003: 10999: 10993: 10989: 10983: 10979: 10975: 10971: 10967: 10963: 10959: 10955: 10951: 10946: 10943: 10939: 10935: 10930: 10926: 10922: 10916: 10912: 10908: 10907: 10902: 10898: 10893: 10889: 10885: 10884: 10877: 10873: 10869: 10865: 10861: 10855: 10851: 10847: 10843: 10838: 10834: 10830: 10825: 10821: 10817: 10813: 10809: 10806:(in German). 10805: 10801: 10797: 10792: 10788: 10784: 10780: 10776: 10773:(17): 68–82. 10772: 10769:(in German). 10768: 10764: 10760: 10756: 10752: 10748: 10744: 10743: 10738: 10734: 10730: 10726: 10722: 10720:9780828400879 10716: 10711: 10710: 10704: 10700: 10696: 10692: 10691: 10684: 10680: 10676: 10672: 10668: 10664: 10659: 10655: 10651: 10647: 10643: 10639: 10635: 10634: 10629: 10624: 10620: 10616: 10615: 10610: 10605: 10601: 10595: 10591: 10587: 10583: 10579: 10573: 10569: 10565: 10561: 10555: 10551: 10547: 10543: 10536: 10535: 10530: 10526: 10522: 10517: 10514:(in French). 10513: 10509: 10505: 10500: 10496: 10492: 10488: 10484: 10480: 10475: 10471: 10467: 10463: 10459: 10458: 10453: 10449: 10445: 10441: 10437: 10433: 10432: 10427: 10423: 10419: 10415: 10411: 10407: 10406:Survey Review 10402: 10398: 10394: 10390: 10386: 10382: 10378: 10374: 10370: 10366: 10361: 10358:(in French). 10357: 10353: 10349: 10344: 10340: 10336: 10332: 10328: 10324: 10320: 10316: 10312: 10308: 10304: 10298: 10294: 10287: 10283: 10282:Strang, W. G. 10278: 10274: 10270: 10266: 10265: 10260: 10256: 10251: 10246: 10242: 10238: 10234: 10230: 10226: 10222: 10218: 10216: 10213: 10206: 10202: 10198: 10194: 10190: 10186: 10181: 10176: 10172: 10168: 10164: 10163:Bessel, F. W. 10159: 10155: 10149: 10145: 10141: 10137: 10133: 10129: 10125: 10124:FTD-MT-64-390 10121: 10115: 10111: 10107: 10103: 10099: 10093: 10089: 10085: 10081: 10075: 10070: 10069: 10063: 10062:Arnold, V. I. 10059: 10058: 10046: 10029: 10023: 10014: 10002: 9998: 9991: 9982: 9978: 9974: 9967: 9963: 9959: 9953: 9947: 9940: 9933: 9929: 9923: 9916: 9911: 9904: 9900: 9895: 9887: 9877: 9872: 9867: 9854: 9850: 9841: 9838: 9836: 9833: 9831: 9828: 9824: 9821: 9820: 9819: 9816: 9814: 9811: 9809: 9806: 9804: 9803:Great ellipse 9801: 9799: 9796: 9794: 9791: 9789: 9786: 9784: 9781: 9780: 9771: 9767: 9764: 9760: 9757: 9751: 9747:in the limit 9746: 9743: 9742:Poincaré 1905 9739: 9736: 9732: 9729:the study of 9728: 9725: 9721: 9718: 9714: 9710: 9707: 9703: 9699: 9698:Legendre 1811 9695: 9694: 9693: 9691: 9687: 9683: 9682:Laplace 1799a 9679: 9672: 9668: 9665: 9661: 9657: 9654: 9650: 9646: 9643: 9639: 9635: 9632: 9628: 9627: 9626: 9624: 9620: 9616: 9606: 9600: 9596: 9590: 9584: 9570: 9566: 9560: 9549: 9544: 9538: 9531: 9524: 9519: 9516: 9513: 9512: 9511: 9508: 9506: 9502: 9498: 9492: 9477: 9466: 9440: 9401: 9395: 9391: 9389: 9384: 9374: 9370: 9364: 9360: 9354: 9350: 9329: 9323:γ < 0 9299: 9273: 9250: 9221: 9212: 9210: 9206: 9202: 9194: 9190: 9184: 9180: 9174: 9170: 9161: 9157: 9150:γ > 0 9126: 9120: 9116: 9109: 9102: 9095: 9075: 9067: 9050: 9042: 9001: 8999: 8995: 8991: 8990:Darboux (1894 8985: 8982:sinα cos 8978: 8974: 8946: 8943: 8940: 8935: 8931: 8927: 8924: 8919: 8915: 8902: 8898: 8894: 8889: 8885: 8874: 8871: 8868: 8863: 8859: 8855: 8852: 8847: 8843: 8830: 8826: 8822: 8817: 8813: 8802: 8799: 8792: 8791: 8790: 8733: 8725: 8722: 8719: 8716: 8711: 8707: 8694: 8690: 8686: 8681: 8677: 8660: 8656: 8652: 8649: 8646: 8641: 8637: 8631: 8627: 8623: 8620: 8617: 8612: 8608: 8602: 8598: 8590: 8587: 8581: 8578: 8573: 8569: 8563: 8559: 8555: 8552: 8549: 8544: 8540: 8534: 8530: 8521: 8518: 8515: 8502: 8499: 8496: 8493: 8488: 8484: 8471: 8467: 8463: 8458: 8454: 8439: 8436: 8431: 8427: 8421: 8417: 8413: 8410: 8407: 8402: 8398: 8392: 8388: 8384: 8379: 8375: 8367: 8364: 8358: 8355: 8350: 8346: 8340: 8336: 8332: 8329: 8326: 8321: 8317: 8311: 8307: 8298: 8295: 8293: 8288: 8277: 8276: 8275: 8273: 8269: 8263: 8261: 8257: 8254: 8250: 8244: 8242: 8229: 8221: 8217: 8213: 8210: 8206: 8202: 8198: 8195: 8191: 8190: 8189: 8186: 8182: 8175: 8168: 8164: 8160: 8155: 8151: 8137: 8130: 8123: 8116: 8111: 8098: 8096: 8084: 8075: 8071: 8067:In the limit 8047: 8039: 8035: 8031: 8026: 8022: 8014: 8010: 8006: 8003: 8000: 7995: 7991: 7985: 7981: 7977: 7974: 7971: 7966: 7962: 7956: 7952: 7945: 7942: 7939: 7936: 7933: 7931: 7926: 7919: 7916: 7913: 7910: 7907: 7904: 7901: 7898: 7895: 7893: 7888: 7881: 7873: 7869: 7865: 7860: 7856: 7850: 7847: 7842: 7838: 7832: 7828: 7824: 7821: 7818: 7813: 7809: 7803: 7799: 7795: 7790: 7786: 7779: 7776: 7773: 7770: 7767: 7765: 7760: 7749: 7748: 7747: 7746:) defined by 7744: 7740: 7735: 7731: 7727: 7723: 7717: 7713: 7709: 7702: 7698: 7694: 7673: 7670: 7667: 7660: 7656: 7650: 7646: 7640: 7633: 7629: 7623: 7619: 7613: 7606: 7602: 7596: 7592: 7586: 7583: 7576: 7575: 7574: 7567: 7562: 7558: 7548: 7546: 7542: 7541:Jacobi (1839) 7536: 7534: 7530: 7526: 7522: 7518: 7517: 7506: 7504: 7489: 7484:accuracy for 7483: 7479: 7478:Karney (2013) 7475: 7473: 7465: 7461: 7459: 7455: 7451: 7450:Legendre 1806 7447: 7442: 7437: 7433: 7429: 7425: 7421: 7417: 7416:Legendre 1806 7412: 7406: 7405:for details. 7404: 7403:Karney (2013) 7400: 7371: 7365: 7363: 7362:Karney (2013) 7348: 7335: 7326: 7325: 7320: 7319: 7309: 7285: 7280: 7276: 7272: 7269: 7266: 7263: 7260: 7257: 7254: 7251: 7248: 7245: 7242: 7239: 7236: 7234: 7229: 7226: 7223: 7216: 7213: 7210: 7207: 7204: 7201: 7198: 7195: 7192: 7189: 7186: 7181: 7177: 7173: 7170: 7167: 7165: 7160: 7157: 7154: 7147: 7144: 7141: 7138: 7135: 7132: 7129: 7126: 7121: 7117: 7113: 7110: 7107: 7104: 7101: 7098: 7096: 7091: 7088: 7085: 7078: 7075: 7072: 7069: 7064: 7060: 7056: 7053: 7050: 7047: 7044: 7041: 7038: 7035: 7032: 7029: 7027: 7022: 7019: 7016: 7009: 7006: 7003: 7000: 6997: 6994: 6991: 6988: 6985: 6982: 6979: 6976: 6973: 6970: 6967: 6965: 6958: 6954: 6950: 6947: 6936: 6935: 6934: 6932: 6927: 6922: 6916: 6910: 6886: 6881: 6876: 6872: 6866: 6863: 6852: 6848: 6844: 6839: 6835: 6825: 6822: 6816: 6813: 6803: 6799: 6795: 6790: 6786: 6776: 6773: 6767: 6764: 6758: 6753: 6748: 6744: 6738: 6735: 6728: 6727: 6726: 6712: 6707: 6699: 6696: 6686: 6680: 6678: 6674: 6669: 6641: 6638: 6635: 6631: 6628: 6625: 6621: 6613: 6609: 6603: 6598: 6594: 6588: 6582: 6579: 6576: 6573: 6570: 6562: 6559: 6556: 6553: 6547: 6542: 6539: 6535: 6528: 6515: 6512: 6507: 6503: 6497: 6493: 6489: 6486: 6476: 6472: 6466: 6458: 6454: 6446: 6442: 6437: 6431: 6427: 6423: 6415: 6411: 6407: 6402: 6398: 6389: 6384: 6380: 6376: 6371: 6367: 6359: 6358: 6357: 6339: 6334: 6331: 6305: 6302: 6299: 6295: 6292: 6288: 6285: 6282: 6278: 6272: 6267: 6263: 6259: 6251: 6239: 6236: 6231: 6227: 6221: 6217: 6213: 6210: 6197: 6193: 6186: 6182: 6179: 6170: 6165: 6161: 6157: 6155: 6147: 6144: 6140: 6137: 6133: 6130: 6127: 6123: 6117: 6112: 6108: 6104: 6099: 6096: 6090: 6086: 6083: 6074: 6069: 6065: 6061: 6059: 6054: 6043: 6042: 6041: 6038: 6033: 6025: 6016: 6006: 5999: 5975: 5971: 5965: 5961: 5957: 5954: 5951: 5948: 5938: 5937: 5936: 5919: 5916: 5913: 5909: 5906: 5902: 5899: 5896: 5893: 5890: 5887: 5884: 5880: 5877: 5874: 5864: 5863: 5862: 5860: 5856: 5851: 5845: 5825: 5822: 5819: 5815: 5812: 5808: 5805: 5802: 5797: 5794: 5789: 5786: 5783: 5780: 5777: 5774: 5771: 5764: 5763: 5762: 5760: 5755: 5746: 5740: 5738: 5733: 5728: 5727: 5722: 5716: 5706: 5704: 5699: 5694: 5689: 5686: 5679: 5669: 5664: 5659: 5652: 5646: 5644: 5640: 5639:Jacobi (1891) 5631: 5625: 5607: 5587: 5574: 5568: 5562: 5554: 5537: 5531: 5523: 5505: 5489: 5471: 5464: 5453: 5431: 5424: 5418: 5414: 5407: 5401: 5400: 5399: 5396: 5391: 5385: 5379: 5374: 5370: 5366: 5362: 5354: 5348: 5342: 5337: 5329: 5320: 5310: 5304: 5294: 5288: 5282: 5273: 5268: 5267:Helmert (1880 5250: 5242: 5230: 5227: 5222: 5218: 5212: 5208: 5204: 5201: 5189: 5185: 5178: 5174: 5168: 5161: 5157: 5150: 5138: 5135: 5130: 5126: 5120: 5116: 5112: 5109: 5096: 5090: 5087: 5083: 5078: 5075: 5068: 5067: 5066: 5064: 5059: 5057: 5049: 5042: 5035: 5031: 5026: 5018: 5011: 5004: 5000: 4996:The quantity 4976: 4973: 4966: 4962: 4958: 4953: 4949: 4943: 4935: 4931: 4927: 4917: 4913: 4909: 4904: 4900: 4893: 4890: 4879: 4876: 4873: 4871: 4861: 4857: 4853: 4848: 4844: 4837: 4830: 4827: 4824: 4817: 4813: 4809: 4804: 4800: 4794: 4786: 4782: 4778: 4768: 4764: 4760: 4755: 4751: 4744: 4741: 4730: 4727: 4724: 4722: 4712: 4708: 4704: 4699: 4695: 4688: 4677: 4676: 4675: 4653: 4649: 4645: 4640: 4636: 4629: 4626: 4623: 4615: 4611: 4607: 4602: 4598: 4591: 4588: 4585: 4577: 4573: 4566: 4559: 4558: 4557: 4554: 4549: 4543: 4539: 4529: 4511: 4508: 4505: 4499: 4493: 4487: 4481: 4478: 4470: 4466: 4462: 4454: 4448: 4443: 4439: 4428: 4427: 4426: 4425: 4419: 4415: 4410: 4404: 4400: 4394: 4389: 4379: 4376: 4369: 4363: 4356: 4352: 4345: 4334: 4328: 4318: 4309: 4305: 4299: 4294: 4289: 4284: 4279: 4277: 4272: 4263: 4236: 4227: 4215: 4199: 4193: 4189: 4159: 4153: 4146: 4129: 4123: 4106: 4094: 4071: 4065: 4054: 4036: 4033: 4013: 4000: 3995: 3991: 3978: 3949: 3943: 3930: 3918: 3886: 3873: 3864: 3862: 3858: 3854: 3849: 3846: 3842: 3833: 3829: 3806: 3802: 3799: 3795: 3785: 3782: 3778: 3773: 3769: 3763: 3759: 3755: 3752: 3744: 3741: 3738: 3732: 3729: 3724: 3721: 3718: 3710: 3705: 3701: 3695: 3691: 3687: 3684: 3681: 3678: 3675: 3672: 3667: 3663: 3659: 3656: 3649: 3648: 3647: 3645: 3644: 3624: 3621: 3618: 3611: 3608: 3603: 3599: 3591: 3587: 3583: 3580: 3574: 3571: 3568: 3561: 3560: 3559: 3548: 3536: 3529: 3524: 3516: 3512: 3506: 3502: 3482: 3477: 3473: 3469: 3466: 3462: 3459: 3455: 3452: 3445: 3444: 3443: 3423: 3419: 3416: 3412: 3405: 3402: 3398: 3393: 3389: 3383: 3379: 3375: 3372: 3365: 3360: 3356: 3352: 3347: 3344: 3335: 3334: 3333: 3330: 3326: 3320: 3293: 3290: 3287: 3284: 3279: 3275: 3271: 3268: 3265: 3257: 3253: 3249: 3246: 3240: 3237: 3234: 3231: 3228: 3225: 3222: 3215: 3214: 3213: 3210: 3204: 3198: 3166: 3161: 3158: 3153: 3149: 3143: 3139: 3135: 3132: 3127: 3121: 3118: 3113: 3110: 3104: 3098: 3095: 3090: 3087: 3079: 3076: 3067: 3066: 3065: 3048: 3043: 3040: 3035: 3031: 3025: 3021: 3017: 3014: 3009: 3003: 3000: 2997: 2992: 2989: 2986: 2976: 2975: 2974: 2957: 2954: 2951: 2948: 2942: 2939: 2936: 2930: 2927: 2924: 2921: 2914: 2910: 2906: 2903: 2898: 2895: 2892: 2889: 2882: 2881: 2880: 2855: 2849: 2846: 2843: 2838: 2835: 2832: 2826: 2820: 2817: 2812: 2809: 2803: 2797: 2794: 2789: 2786: 2778: 2775: 2766: 2765: 2764: 2758: 2753: 2752: 2747: 2746: 2723: 2720: 2717: 2713: 2710: 2707: 2704: 2701: 2698: 2694: 2691: 2688: 2684: 2681: 2678: 2675: 2672: 2669: 2665: 2662: 2659: 2652: 2651: 2650: 2647: 2641: 2631: 2627: 2612: 2606: 2599: 2594: 2585: 2580: 2572: 2568: 2562: 2555: 2545: 2541: 2532: 2528: 2515: 2506: 2502: 2489: 2483: 2478: 2464: 2452: 2429: 2424: 2420: 2416: 2413: 2408: 2404: 2400: 2397: 2394: 2389: 2385: 2381: 2378: 2373: 2369: 2365: 2362: 2355: 2354: 2353: 2336: 2333: 2330: 2327: 2324: 2321: 2318: 2311: 2310: 2309: 2303: 2298: 2278: 2273: 2269: 2266: 2263: 2260: 2257: 2254: 2248: 2242: 2239: 2234: 2231: 2224: 2218: 2215: 2212: 2209: 2204: 2201: 2198: 2192: 2186: 2183: 2178: 2175: 2168: 2163: 2159: 2156: 2153: 2147: 2141: 2138: 2133: 2130: 2120: 2119: 2118: 2116: 2112: 2111: 2091: 2088: 2085: 2081: 2078: 2075: 2072: 2069: 2066: 2059: 2058: 2057: 2055: 2051: 2047: 2024: 2021: 2018: 2015: 2012: 2005: 2004: 2003: 2001: 2000: 1994: 1969: 1962: 1959: 1950: 1940: 1937: 1933: 1930: 1923: 1922: 1921: 1919: 1915: 1907: 1900: 1896: 1887: 1880: 1876: 1867: 1860: 1856: 1828: 1825: 1822: 1814: 1811: 1807: 1804: 1798: 1791: 1787: 1779: 1775: 1770: 1766: 1761: 1757: 1749: 1748: 1747: 1740: 1733: 1722: 1715: 1707: 1703: 1696: 1686: 1684: 1676: 1672: 1668: 1644: 1641: 1638: 1630: 1627: 1623: 1620: 1614: 1611: 1609: 1601: 1598: 1590: 1586: 1582: 1576: 1572: 1568: 1562: 1558: 1552: 1550: 1545: 1542: 1531: 1530: 1529: 1510: 1506: 1502: 1496: 1492: 1488: 1483: 1479: 1475: 1469: 1465: 1461: 1456: 1452: 1448: 1441: 1440: 1439: 1437: 1424: 1420: 1416: 1411: 1385: 1382: 1379: 1375: 1372: 1369: 1366: 1362: 1359: 1356: 1352: 1346: 1343: 1340: 1335: 1332: 1326: 1323: 1320: 1317: 1313: 1310: 1307: 1304: 1300: 1297: 1294: 1287: 1286: 1285: 1282: 1276: 1270: 1260: 1254: 1250: 1244: 1238: 1232: 1228: 1222: 1218: 1198: 1192: 1189: 1186: 1182: 1177: 1172: 1166: 1162: 1158: 1153: 1149: 1142: 1138: 1135: 1130: 1122: 1119: 1116: 1110: 1105: 1100: 1094: 1090: 1086: 1081: 1077: 1070: 1067: 1063: 1058: 1054: 1051: 1048: 1042: 1039: 1032: 1031: 1030: 1026: 1020: 1016: 1011: 1005: 999: 993: 991: 987: 986:Jekeli (2012) 983: 979: 975: 971: 967: 963: 962:Bessel (1825) 949: 937: 923: 920: 918: 914: 913:geodesic line 908: 905: 901: 895: 893: 889: 883: 881: 875: 873: 869: 864: 862: 861:Bessel (1825) 858: 854: 850: 849:Oriani (1806) 846: 842: 837: 835: 831: 827: 819: 812: 805: 792: 765: 759: 753: 748: 744: 740: 729: 720: 707: 702: 698: 694: 693: 692: 677: 669: 663: 657: 651: 642: 633: 627: 618: 614: 606: 602: 597: 588: 584: 580: 575: 571: 569: 565: 561: 557: 553: 549: 544: 541: 537: 533: 529: 519: 517: 512: 508: 504: 503:Newton (1687) 500: 496: 495:great circles 492: 487: 485: 481: 480:straight line 477: 476: 471: 470: 465: 461: 457: 453: 450:The study of 444: 433: 428: 426: 421: 419: 414: 413: 411: 410: 403: 400: 398: 395: 393: 390: 389: 383: 381: 378: 377: 373: 371: 368: 367: 363: 361: 358: 357: 353: 351: 348: 347: 343: 341: 338: 337: 333: 331: 328: 327: 323: 321: 318: 317: 313: 311: 308: 307: 303: 301: 298: 297: 293: 291: 288: 287: 283: 281: 278: 277: 273: 271: 268: 267: 263: 261: 258: 257: 249: 248: 241: 238: 236: 230: 228: 222: 220: 214: 212: 208:BeiDou (BDS) 206: 204: 198: 196: 193: 191: 188: 187: 181: 180: 173: 170: 168: 165: 163: 160: 158: 155: 153: 150: 148: 145: 142: 138: 134: 132: 129: 127: 124: 122: 119: 117: 114: 110: 109:circumference 106: 101: 98: 96: 93: 91: 88: 87: 81: 80: 73: 70: 68: 65: 63: 60: 58: 55: 54: 48: 47: 43: 39: 38: 35: 32: 31: 19: 11769: 11749: 11745:Vincenty, T. 11736: 11714: 11710: 11700:Vincenty, T. 11688:. Retrieved 11681:the original 11672: 11651: 11647: 11634: 11614: 11598: 11568:(1): 12–22. 11565: 11559: 11534: 11530: 11523:Poincaré, H. 11514: 11510: 11494: 11490: 11474: 11470: 11448: 11442: 11402:1796 edition 11378: 11374:J. Liouville 11360: 11338: 11321: 11317: 11289: 11271: 11266: 11253: 11229: 11224: 11204:Bowditch, N. 11199: 11172: 11138: 11134: 11096: 11064: 11060: 11008:(1): 43–55. 11005: 11001: 10987: 10957: 10933: 10905: 10882: 10849: 10807: 10803: 10770: 10766: 10741: 10708: 10689: 10666: 10653: 10649: 10632: 10613: 10609:Gauss, C. F. 10599: 10577: 10549: 10533: 10515: 10511: 10482: 10478: 10461: 10456: 10435: 10430: 10409: 10405: 10396: 10380: 10359: 10355: 10330: 10326: 10292: 10263: 10240: 10236: 10229:Bliss, G. A. 10214: 10211: 10170: 10166: 10135: 10119: 10097: 10067: 10027: 10021: 10013: 10000: 9996: 9990: 9980: 9976: 9972: 9965: 9961: 9957: 9951: 9945: 9939: 9922: 9910: 9894: 9885: 9875: 9870: 9865: 9853: 9830:Meridian arc 9772:, Chap. 12). 9763:Knörrer 1980 9749: 9675: 9612: 9609:Applications 9598: 9594: 9588: 9582: 9568: 9564: 9558: 9555: 9547: 9542: 9536: 9529: 9522: 9509: 9490: 9475: 9464: 9438: 9435: 9399: 9387: 9385: 9372: 9368: 9362: 9358: 9352: 9348: 9327: 9297: 9294: 9219: 9201:Chasles 1846 9192: 9188: 9182: 9178: 9172: 9168: 9159: 9155: 9124: 9121: 9114: 9107: 9100: 9093: 9090: 9073: 9048: 8983: 8976: 8972: 8961: 8752: 8265: 8258: 8248: 8246: 8238: 8187: 8180: 8173: 8169:= 1.01:1:0.8 8166: 8162: 8158: 8135: 8128: 8121: 8114: 8099: 8094: 8081:becomes the 8073: 8069: 8066: 7742: 7738: 7733: 7729: 7725: 7722:Jacobi (1866 7715: 7711: 7707: 7700: 7696: 7692: 7688: 7572: 7537: 7514: 7512: 7503:Karney (2013 7487: 7476: 7462: 7440: 7428:Helmert 1880 7410: 7407: 7399:root finding 7369: 7366: 7346: 7336: 7322: 7316: 7307: 7304: 6925: 6918: 6710: 6697: 6694: 6684: 6681: 6667: 6656: 6337: 6329: 6324: 6036: 6023: 6014: 6004: 5997: 5992: 5934: 5849: 5843: 5840: 5753: 5744: 5741: 5731: 5724: 5720: 5718: 5697: 5690: 5684: 5677: 5667: 5662: 5657: 5650: 5647: 5629: 5623: 5605: 5602: 5585: 5572: 5566: 5564:and a point 5560: 5535: 5529: 5503: 5487: 5469: 5462: 5451: 5429: 5422: 5412: 5405: 5394: 5389: 5383: 5377: 5371:, §§26–27) ( 5369:Forsyth 1927 5352: 5350:, of length 5346: 5340: 5327: 5318: 5308: 5302: 5292: 5289: 5280: 5271: 5265: 5060: 5055: 5047: 5040: 5033: 5029: 5024: 5016: 5009: 5002: 4998: 4995: 4673: 4552: 4541: 4537: 4534: 4417: 4413: 4411:showed that 4409:Gauss (1828) 4402: 4398: 4392: 4385: 4374: 4367: 4361: 4354: 4350: 4343: 4332: 4326: 4316: 4307: 4303: 4297: 4287: 4280: 4276:Gauss (1828) 4270: 4261: 4234: 4231: 4213: 4197: 4157: 4151: 4127: 4121: 4116: 4069: 4063: 4034: 4031: 4011: 3998: 3993: 3989: 3979: 3975: 3947: 3884: 3850: 3847: 3840: 3831: 3827: 3824: 3641: 3639: 3554:in terms of 3546: 3534: 3527: 3504: 3500: 3497: 3441: 3328: 3324: 3318: 3308: 3208: 3202: 3196: 3181: 3063: 2973:which gives 2972: 2870: 2756: 2749: 2743: 2741: 2645: 2639: 2637:If the side 2636: 2610: 2604: 2597: 2592: 2583: 2578: 2570: 2566: 2563: 2553: 2543: 2539: 2530: 2526: 2513: 2504: 2500: 2487: 2481: 2475:This is the 2474: 2351: 2296: 2293: 2108: 2106: 2044: 1997: 1992: 1989: 1905: 1898: 1894: 1885: 1878: 1874: 1865: 1858: 1854: 1843: 1746:is given by 1738: 1731: 1720: 1713: 1705: 1701: 1694: 1682: 1674: 1670: 1666: 1663: 1527: 1422: 1418: 1414: 1403: 1280: 1274: 1268: 1258: 1255: 1248: 1242: 1236: 1230: 1226: 1220: 1216: 1213: 1024: 1018: 1015:eccentricity 1009: 1003: 997: 994: 976:, Chap. 5), 959: 921: 916: 912: 910: 903: 899: 897: 885: 879: 877: 865: 838: 817: 810: 803: 790: 787: 763: 761:, determine 757: 751: 746: 742: 727: 725:, determine 718: 705: 700: 696: 674:, which has 667: 665:, of length 661: 655: 649: 640: 631: 629:at latitude 625: 616: 604: 595: 592: 586: 582: 578: 566:, Chap. 3) ( 564:Bomford 1952 545: 525: 501:. However, 488: 473: 467: 451: 449: 184:Technologies 139: / 51:Fundamentals 11414:PDF Figures 11131:Knörrer, H. 10699:Hilbert, D. 10642:Hart, A. S. 10348:Chasles, M. 10280:Borre, K.; 10259:Bomford, G. 10047:, §3.5.19). 9928:Karney 2015 9868:azimuth at 9770:Berger 2010 9735:Jacobi 1891 9724:Jacobi 1839 9642:UNCLOS 2006 9501:Arnold 1989 9209:pp. 223–224 8272:Jacobi 1866 8268:Jacobi 1839 8241:Jacobi 1839 8142:ω = 0° 8138:= ±90° 7468:0.1 mm 7458:Karney 2024 7454:Cayley 1870 7424:Bessel 1825 7420:Oriani 1806 6677:Karney 2013 5857:. Now the 5735:in Fig. 1 ( 5365:Jacobi 1866 5361:Jacobi 1837 4388:Ehlert 1993 4353:(1 − 3863:, §2.1.4). 3861:Jekeli 2012 3558:, we write 2048:found this 1852:satisfying 1687:depends on 888:Hutton 1811 532:pp. 220–221 62:Geodynamics 11836:Categories 11690:2013-08-15 11623:1811/24409 11607:1811/24333 11517:(2): 1–58. 11503:Oriani, B. 11497:(1): 1–58. 11483:Oriani, B. 11463:Oriani, B. 11435:Newton, I. 11324:: 345–378. 11074:2208.00492 10954:Eggert, O. 10950:Jordan, W. 10942:1811/51274 10846:A. Clebsch 10737:Hutton, C. 10518:: 258–293. 10319:Cayley, A. 10132:Berger, M. 10054:References 9994:The limit 9835:Rhumb line 9713:Gauss 1828 9615:adjustment 9471:γ = 0 8779:γ = 0 8274:, §28) is 8260:Königsberg 8205:Dupin 1813 8194:Monge 1796 8146:±180° 8110:orthogonal 7555:See also: 7521:quadrature 7490:| ≤ 6913:See also: 6343:Γ = α 6325:where the 5713:See also: 5428:| ≤ 5411:| ≤ 5373:Bliss 1916 4422:obeys the 4349:| ≤ 1669:′ = 982:Rapp (1993 484:Euler 1755 11590:122111614 11437:(1848) . 11370:Monge, G. 11280:312469983 11222:(1799b). 11194:(1829) . 11163:118792545 11015:1109.4448 10956:(1962) . 10915:630416023 10876:440645889 10844:(2009) . 10824:121670851 10787:119469290 10729:301610346 10665:(1964) . 10611:(1902) . 10568:250050479 10504:Euler, L. 10495:257615376 10470:560800801 10311:795014501 10205:118760590 10180:0908.1824 9752:→ 0 9653:RNAV 2007 9597:= − 9567:= − 9497:Hart 1849 9277:Fig. 21. 9254:Fig. 20. 9071:Fig. 19. 9046:Fig. 18. 8994:§§583–584 8944:α 8941:⁡ 8928:ω 8925:⁡ 8895:− 8875:− 8872:α 8869:⁡ 8856:β 8853:⁡ 8823:− 8800:γ 8726:γ 8720:ω 8717:⁡ 8687:− 8653:− 8650:ω 8647:⁡ 8621:ω 8618:⁡ 8591:ω 8582:ω 8579:⁡ 8553:ω 8550:⁡ 8522:∫ 8519:− 8503:γ 8500:− 8497:β 8494:⁡ 8464:− 8440:β 8437:⁡ 8414:− 8411:β 8408:⁡ 8385:− 8368:β 8359:β 8356:⁡ 8330:β 8327:⁡ 8299:∫ 8289:δ 8095:different 8032:− 8007:− 8004:ω 8001:⁡ 7975:ω 7972:⁡ 7946:β 7943:⁡ 7917:ω 7914:⁡ 7908:β 7905:⁡ 7866:− 7851:β 7848:⁡ 7825:− 7822:β 7819:⁡ 7796:− 7780:ω 7777:⁡ 7547:, §3.5). 7436:Rapp 1993 7277:α 7273:⁡ 7267:β 7264:⁡ 7255:α 7252:⁡ 7246:σ 7243:⁡ 7230:ω 7227:⁡ 7214:ω 7211:⁡ 7205:α 7202:⁡ 7193:σ 7190:⁡ 7178:α 7174:⁡ 7161:β 7158:⁡ 7145:β 7142:⁡ 7136:σ 7133:⁡ 7118:α 7114:⁡ 7108:ω 7105:⁡ 7092:α 7089:⁡ 7076:α 7073:⁡ 7061:α 7057:⁡ 7048:ω 7045:⁡ 7039:β 7036:⁡ 7023:σ 7020:⁡ 7007:σ 7004:⁡ 6998:ω 6995:⁡ 6986:β 6983:⁡ 6977:α 6974:⁡ 6955:α 6951:⁡ 6873:ω 6867:⁡ 6849:β 6845:− 6836:β 6817:⁡ 6800:β 6787:β 6768:⁡ 6739:⁡ 6639:λ 6632:φ 6629:⁡ 6589:− 6583:φ 6580:⁡ 6563:φ 6560:⁡ 6548:⁡ 6540:− 6516:φ 6513:⁡ 6490:− 6455:λ 6443:λ 6438:∫ 6412:α 6408:− 6399:α 6327:value of 6303:λ 6296:φ 6289:φ 6286:⁡ 6260:− 6240:φ 6237:⁡ 6214:− 6183:∫ 6177:Γ 6148:λ 6141:φ 6134:φ 6131:⁡ 6105:− 6087:∫ 6081:Γ 5972:θ 5962:∑ 5958:− 5955:π 5946:Γ 5917:λ 5910:φ 5903:φ 5900:⁡ 5894:∫ 5878:∫ 5872:Γ 5823:λ 5816:φ 5809:φ 5806:⁡ 5790:∫ 5778:∫ 5703:involutes 5457:≥ 0 5439:≠ 0 5336:conjugate 5231:β 5228:⁡ 5205:− 5139:φ 5136:⁡ 5113:− 5091:ν 5088:ρ 4306:= − 4293:antipodal 4283:cut locus 4226:cut locus 4037:sinα 4001:= ±( 3977:section. 3950:= − 3853:Rapp 1991 3800:σ 3783:σ 3779:⁡ 3742:− 3722:− 3711:σ 3702:∫ 3692:α 3688:⁡ 3679:− 3676:ω 3664:λ 3660:− 3657:λ 3622:σ 3612:β 3609:⁡ 3588:α 3584:⁡ 3572:ω 3474:α 3470:⁡ 3417:σ 3403:σ 3399:⁡ 3366:σ 3357:∫ 3291:σ 3288:⁡ 3276:α 3272:⁡ 3254:α 3247:σ 3241:β 3238:⁡ 3229:β 3226:⁡ 3162:β 3159:⁡ 3136:− 3122:ω 3114:λ 3099:σ 3044:β 3041:⁡ 3018:− 3004:φ 3001:⁡ 2993:β 2990:⁡ 2955:φ 2952:⁡ 2940:− 2928:φ 2925:⁡ 2907:− 2896:β 2893:⁡ 2850:φ 2847:⁡ 2839:β 2836:⁡ 2821:ω 2813:λ 2798:σ 2721:ω 2714:β 2711:⁡ 2702:σ 2695:α 2692:⁡ 2682:β 2673:σ 2666:α 2663:⁡ 2477:sine rule 2421:β 2417:⁡ 2405:α 2401:⁡ 2386:β 2382:⁡ 2370:α 2366:⁡ 2334:β 2331:⁡ 2274:ν 2270:α 2267:⁡ 2261:φ 2258:⁡ 2235:α 2219:φ 2216:⁡ 2210:ν 2205:α 2202:⁡ 2179:λ 2164:ρ 2160:α 2157:⁡ 2134:φ 2089:λ 2082:φ 2079:⁡ 2070:α 2022:α 2019:⁡ 1960:φ 1956:∂ 1948:∂ 1938:φ 1934:− 1826:λ 1812:φ 1805:φ 1788:λ 1776:λ 1771:∫ 1642:λ 1628:φ 1621:φ 1612:≡ 1602:λ 1569:φ 1559:ρ 1507:λ 1480:φ 1466:ρ 1383:λ 1363:α 1360:⁡ 1347:φ 1344:⁡ 1327:− 1321:φ 1314:ρ 1301:α 1298:⁡ 1190:− 1159:− 1120:− 1087:− 1052:− 984:, §1.2), 613:longitude 570:, §4.5). 511:meridians 141:Longitude 67:Geomatics 11797:NGS tool 11764:(1861). 11702:(1975). 11670:(2006). 11525:(1905). 11505:(1810). 11485:(1808). 11465:(1806). 11424:(2012). 11410:Fig. 3–4 11406:Fig. 1–2 11398:Fig. 3–4 11394:Fig. 1–2 11337:(1964). 11312:(1846). 11264:(1811). 11248:(1806). 11238:25448952 11095:(1982). 10982:34429043 10798:(1839). 10761:(1837). 10751:18031510 10739:(1811). 10705:(1952). 10683:17273288 10656:: 80–84. 10644:(1849). 10548:(1927). 10506:(1755). 10454:(1813). 10428:(1894). 10391:(1735). 10375:(1869). 10350:(1846). 10321:(1870). 10284:(2012). 10261:(1952). 10231:(1916). 10134:(2010). 10064:(1989). 9999:→ 9975:≥ 9964:≥ 9808:Geodesic 9777:See also 9731:caustics 9664:FAI 2018 9371:→ 9361:≠ 9181:≠ 9171:≈ 9158:= ± 8986:= const. 8975:→ 8072:→ 7714:≥ 7710:≥ 7525:Legendre 7470:for the 7383:because 6716:= α 6021:, where 5683:≤ 5665:, i.e., 5381:between 4322:∈ 3994:vertices 3803:′ 3786:′ 3507:= 0) = 0 3463:′ 3420:′ 3406:′ 2586:= ω 2556:= α 2529:− 2503:− 2308:, using 2050:relation 1963:′ 1941:′ 1815:′ 1691:through 1678:and the 1631:′ 1573:′ 1139:′ 972:, §15), 917:geodesic 900:geodetic 816:− 749:, given 703:, given 676:azimuths 601:latitude 548:spheroid 509:and the 475:geodesic 320:ISO 6709 218:(Europe) 216:Galileo 202:(Russia) 200:GLONASS 137:Latitude 126:Geodesic 84:Concepts 11842:Geodesy 11807:geod(1) 11719:Bibcode 11570:Bibcode 11551:1986219 11388:2829112 11376:(ed.). 11355:1048605 11347:0178386 11212:1294937 11181:Bibcode 11143:Bibcode 11123:8476832 11115:0666697 11039:Addenda 11020:Bibcode 10962:Bibcode 10903:(ed.). 10868:2569315 10848:(ed.). 10835:(1841). 10623:7824448 10521:Figures 10444:8566228 10362:: 5–20. 10273:1396190 10264:Geodesy 10185:Bibcode 10114:6150611 10088:4037141 10037:⁄ 10025:⁄ 9884:± 9866:forward 9864:is the 9813:Geodesy 9676:By the 9321:, then 9148:, then 7495:⁄ 7401:); see 7315:, Eqs. 6030:is the 5853:is the 5693:evolute 5643:astroid 5495:⁄ 5367:, §6) ( 5054:is the 4546:is the 4254:⁄ 4228:in red. 4205:⁄ 4186:75.192° 4184:(resp. 4182:53.175° 4167:⁄ 4155:⁄ 4137:⁄ 4125:⁄ 4079:⁄ 4067:⁄ 4006:⁄ 3961:) with 3955:⁄ 3898:) with 3892:⁄ 3531:√ 2521:⁄ 2495:⁄ 1693:ρ( 1434:is the 1408:is the 1266:, then 1251:′ 1027:′ 980:, §4), 507:equator 462:. The 456:geodesy 380:Geo URI 350:NAVD 88 260:NGVD 29 234:(Japan) 226:(India) 210:(China) 72:History 57:Geodesy 34:Geodesy 11862:Curves 11668:UNCLOS 11588: 11549: 11386: 11353: 11345: 11297: 11278: 11236: 11210: 11161: 11121: 11113: 11103: 10980: 10913: 10888:Errata 10874: 10866: 10856: 10822: 10785: 10749: 10727: 10717: 10681: 10621: 10594:493553 10592: 10566: 10556: 10493: 10468: 10442: 10309: 10299: 10271: 10221:Errata 10203: 10150: 10112: 10086: 10076: 10006:ω 9983:> 0 9968:> 0 9880:α 9859:α 9700:) and 9690:p. 222 9505:p. 265 9460:= 135° 9456:α 9448:ω 9423:α 9412:ω 9319:= 180° 9315:α 9313:, and 9307:ω 9285:9.966° 9279:ω 9256:ω 9232:α 9142:α 9140:, and 9134:ω 9105:, and 9081:87.48° 9024:α 9013:ω 8968:ω 8964:α 8962:where 8787:γ 8783:δ 8775:γ 8771:δ 8767:γ 8763:δ 8759:ω 8220:p. 188 8209:Part 5 8106:ω 8091:ω 7743:ω 7728:(with 7718:> 0 7689:where 7533:Bessel 7529:Oriani 7392:α 7385:α 7378:ω 7355:α 7339:α 7329:α 6356:gives 6040:gives 5995:θ 5935:where 5841:where 5673:> 0 5663:stable 5611:α 5443:α 5435:α 5395:nearby 5311:α 5295:α 5027:, and 4674:where 4535:where 4324:. If 4240:α 4175:α 4048:α 3982:α 3963:α 3900:α 3515:p. 180 3442:where 3311:α 3309:where 2591:, the 2577:, the 2511:, and 2029:const. 2002:gives 1974:const. 1844:where 1664:where 1432:ν 1430:, and 1419:ν 1406:ρ 1404:where 1264:α 1246:, and 1013:, the 988:, and 892:p. 115 796:α 779:α 777:, and 772:α 733:α 716:, and 711:α 686:α 679:α 491:sphere 370:GCJ-02 360:ETRS89 340:WGS 84 330:NAD 83 310:GRS 80 270:OSGB36 224:NAVIC 105:radius 11754:(PDF) 11707:(PDF) 11684:(PDF) 11677:(PDF) 11639:(PDF) 11586:S2CID 11547:JSTOR 11270:[ 11228:[ 11177:(PDF) 11159:S2CID 11069:arXiv 11010:arXiv 10820:S2CID 10783:S2CID 10538:(PDF) 10481:[ 10460:[ 10434:[ 10289:(PDF) 10201:S2CID 10175:arXiv 10032:< 9857:Here 9846:Notes 9444:= 90° 9346:. If 9303:= 90° 9262:39.9° 9146:= 90° 9056:45.1° 8183:= 30° 8176:= 40° 7472:WGS84 5441:; if 5433:, if 4238:with 4132:< 3990:nodes 3986:= 45° 3643:Eq. 2 1229:< 1219:> 868:curve 659:) is 300:SAD69 280:SK-42 95:Geoid 11811:PROJ 11630:RNAV 11384:OCLC 11351:OCLC 11295:ISBN 11276:OCLC 11234:OCLC 11208:OCLC 11119:OCLC 11101:ISBN 10978:OCLC 10911:OCLC 10872:OCLC 10854:ISBN 10808:1839 10771:1837 10747:OCLC 10725:OCLC 10715:ISBN 10679:OCLC 10619:OCLC 10590:OCLC 10564:OCLC 10554:ISBN 10491:OCLC 10466:OCLC 10440:OCLC 10307:OCLC 10297:ISBN 10269:OCLC 10148:ISBN 10110:OCLC 10084:OCLC 10074:ISBN 9949:and 9876:back 9527:and 9486:180° 9452:= 0° 9429:135° 9339:and 9238:180° 8781:and 8765:and 8140:and 8119:and 7732:and 7559:and 7353:and 7321:and 6536:tanh 6338:AFHB 5754:AFHB 5732:AFHB 5593:175° 5543:−30° 5511:−30° 5468:| = 5387:and 5344:and 5278:and 5061:The 5046:) = 5015:) = 4221:−30° 4022:360° 3544:and 3533:1 + 3200:and 3190:and 2875:and 2760:and 2748:and 2622:and 2551:and 1884:) = 1872:and 1864:) = 1728:and 1699:and 1278:and 857:1810 855:and 853:1808 847:and 755:and 741:the 731:and 695:the 684:and 623:and 611:and 290:ED50 107:and 11819:of 11727:doi 11656:doi 11619:hdl 11603:hdl 11578:doi 11539:doi 11412:). 11400:). 11327:PDF 11151:doi 11079:doi 11028:doi 10970:doi 10938:hdl 10925:PDF 10923:. 10812:doi 10775:doi 10671:doi 10628:PDF 10582:doi 10529:FAI 10414:doi 10365:PDF 10335:doi 10245:doi 10193:doi 10171:331 10140:doi 10102:doi 10017:If 9715:) ( 9684:) ( 9660:FAI 9617:of 9550:= 0 9539:= 0 9532:= 0 9525:= 0 9499:) ( 9493:= 0 9484:or 9478:= 0 9467:= 0 9407:90° 9388:not 9330:= 0 9227:90° 9211:). 9203:) ( 9138:= 0 9117:= 0 9110:= 0 9103:= 0 9096:= 0 9030:90° 8932:cos 8916:sin 8860:sin 8844:cos 8708:sin 8638:cos 8609:sin 8570:cos 8541:sin 8485:cos 8428:cos 8399:sin 8347:cos 8318:sin 8270:) ( 8144:or 8131:= 0 8124:= 0 8117:= 0 8093:is 7992:cos 7963:sin 7940:sin 7911:sin 7902:cos 7839:cos 7810:sin 7774:cos 7460:). 7456:) ( 7452:) ( 7434:) ( 7430:) ( 7426:) ( 7422:) ( 7418:) ( 7324:(4) 7318:(3) 7270:tan 7261:tan 7249:sin 7240:sin 7224:sin 7208:tan 7199:cot 7187:sin 7171:cos 7155:sin 7139:tan 7130:cot 7111:cos 7102:cos 7086:cos 7070:cot 7054:tan 7042:cos 7033:cos 7017:cos 7001:cot 6992:tan 6980:cos 6971:sin 6948:sin 6926:NEP 6864:tan 6814:cos 6765:sin 6736:tan 6725:is 6675:) ( 6626:sin 6577:sin 6557:sin 6504:sin 6283:cos 6228:sin 6128:cos 6012:by 5897:cos 5803:cos 5751:of 5635:= 0 5621:to 5580:26° 5533:at 5447:= 0 5363:) ( 5300:is 5219:cos 5127:sin 5065:is 4550:at 4338:= 0 4295:to 4247:15° 3969:45° 3906:45° 3843:= 0 3820:(4) 3770:sin 3685:sin 3600:cos 3581:sin 3509:. 3467:cos 3437:(3) 3390:sin 3329:d σ 3285:sin 3269:cos 3235:sin 3223:sin 3209:EGP 3150:cos 3032:cos 2998:sin 2987:sin 2949:tan 2922:tan 2890:tan 2879:is 2844:sin 2833:sin 2751:(2) 2745:(1) 2737:(2) 2708:cos 2689:sin 2660:cos 2598:NEP 2482:NAB 2414:cos 2398:sin 2379:cos 2363:sin 2328:cos 2264:sin 2255:tan 2213:cos 2199:sin 2154:cos 2110:(1) 2076:sin 2016:sin 1999:(1) 1528:or 1421:cos 1399:(1) 1357:sin 1341:sin 1295:cos 1284:by 902:or 894:), 836:.) 791:NAB 745:or 699:or 653:to 599:at 587:EFH 486:). 11838:: 11737:23 11725:. 11715:23 11713:. 11709:. 11652:38 11650:. 11584:. 11576:. 11566:37 11564:. 11545:. 11453:. 11441:. 11408:, 11396:, 11349:. 11343:MR 11322:11 11198:. 11157:. 11149:. 11139:59 11137:. 11117:. 11111:MR 11109:. 11077:. 11065:98 11063:. 11059:. 11026:. 11018:. 11006:87 11004:. 11000:. 10976:. 10968:. 10952:; 10870:. 10864:MR 10862:. 10818:. 10781:. 10723:. 10701:; 10677:. 10652:. 10648:. 10588:. 10562:. 10489:. 10410:30 10408:. 10360:11 10331:40 10325:. 10305:. 10291:. 10241:17 10239:. 10235:. 10219:. 10199:. 10191:. 10183:. 10169:. 10146:. 10108:. 10100:. 10082:. 9979:= 9960:= 9930:, 9765:); 9744:); 9737:); 9726:); 9719:); 9708:); 9688:, 9666:). 9655:); 9644:); 9581:+ 9503:, 9482:0° 9446:, 9427:= 9421:, 9418:0° 9416:= 9410:, 9405:= 9383:. 9351:= 9305:, 9283:= 9260:= 9236:= 9230:, 9225:= 9207:, 9191:= 9132:, 9098:, 9079:= 9054:= 9028:= 9022:, 9019:0° 9017:= 8992:, 8218:, 8211:). 8207:, 8196:). 8185:. 8178:, 8077:, 7741:, 7736:, 7720:. 7531:, 7527:, 7497:50 7380:12 7373:12 7364:. 7334:. 6877:12 6749:12 6714:12 6688:12 6372:12 5844:dT 5748:12 5719:A 5681:12 5671:12 5645:. 5633:12 5618:3° 5591:= 5589:12 5583:, 5578:= 5570:, 5541:= 5509:= 5501:, 5497:10 5490:= 5466:12 5455:12 5426:12 5409:12 5356:12 5331:12 5322:12 5306:12 5284:12 5275:12 5051:12 5039:, 5020:12 5008:, 4977:0. 4373:= 4371:12 4365:, 4347:12 4320:12 4301:, 4274:. 4265:12 4256:10 4219:= 4211:, 4207:10 4200:= 4162:= 4074:= 4018:|) 3967:= 3957:50 3904:= 3894:50 3887:= 3845:. 3830:= 3513:, 3325:ds 2640:EP 2626:. 2618:, 2608:; 2588:12 2574:12 2569:= 2567:AB 2561:. 2542:= 2516:= 2514:NB 2490:= 2488:NA 2304:, 1920:, 1909:12 1762:12 1737:, 1719:, 1675:dλ 1671:dφ 1417:= 1412:, 1281:dλ 1275:dφ 1269:ds 1259:ds 1240:, 1029:: 968:, 964:. 890:, 882:. 874:: 863:. 809:= 807:12 770:, 767:12 722:12 709:, 671:12 662:AB 579:AB 530:, 11823:. 11803:. 11733:. 11729:: 11721:: 11693:. 11662:. 11658:: 11621:: 11605:: 11592:. 11580:: 11572:: 11553:. 11541:: 11535:6 11515:2 11495:2 11475:1 11457:. 11416:. 11404:( 11392:( 11390:. 11357:. 11303:. 11282:. 11240:. 11214:. 11183:: 11165:. 11153:: 11145:: 11125:. 11087:. 11081:: 11071:: 11041:. 11036:. 11030:: 11022:: 11012:: 10984:. 10972:: 10964:: 10940:: 10927:. 10917:. 10890:. 10878:. 10826:. 10814:: 10789:. 10777:: 10753:. 10731:. 10685:. 10673:: 10654:4 10625:. 10596:. 10584:: 10570:. 10523:. 10516:9 10497:. 10472:. 10446:. 10420:. 10416:: 10367:. 10341:. 10337:: 10313:. 10275:. 10253:. 10247:: 10223:. 10215:4 10207:. 10195:: 10187:: 10177:: 10156:. 10142:: 10126:) 10116:. 10104:: 10090:. 10039:2 10035:1 10028:a 10022:c 10001:c 9997:b 9981:a 9977:a 9973:b 9966:b 9962:a 9958:a 9952:b 9946:a 9934:) 9889:. 9886:π 9882:2 9871:B 9861:2 9758:; 9750:c 9733:( 9704:( 9640:( 9633:; 9602:1 9599:β 9595:β 9589:A 9583:π 9579:1 9572:1 9569:β 9565:β 9559:A 9548:Y 9537:Y 9530:Z 9523:X 9495:( 9491:Y 9476:Y 9465:Y 9458:1 9450:1 9442:1 9439:β 9432:. 9425:1 9414:1 9403:1 9400:β 9380:1 9373:c 9369:b 9363:b 9359:a 9353:b 9349:a 9343:1 9336:1 9328:X 9317:1 9309:1 9301:1 9298:β 9288:. 9281:1 9265:. 9258:1 9241:. 9234:1 9223:1 9220:β 9199:( 9196:1 9193:β 9189:β 9183:b 9179:a 9173:b 9169:a 9163:1 9160:β 9156:β 9144:1 9136:1 9128:1 9125:β 9115:Y 9108:Z 9101:Y 9094:X 9084:. 9077:1 9074:β 9059:. 9052:1 9049:β 9033:. 9026:1 9015:1 8984:β 8977:a 8973:b 8947:, 8936:2 8920:2 8910:) 8903:2 8899:b 8890:2 8886:a 8880:( 8864:2 8848:2 8838:) 8831:2 8827:c 8818:2 8814:b 8808:( 8803:= 8755:β 8734:. 8723:+ 8712:2 8702:) 8695:2 8691:b 8682:2 8678:a 8672:( 8661:2 8657:c 8642:2 8632:2 8628:b 8624:+ 8613:2 8603:2 8599:a 8588:d 8574:2 8564:2 8560:b 8556:+ 8545:2 8535:2 8531:a 8489:2 8479:) 8472:2 8468:c 8459:2 8455:b 8449:( 8432:2 8422:2 8418:c 8403:2 8393:2 8389:b 8380:2 8376:a 8365:d 8351:2 8341:2 8337:c 8333:+ 8322:2 8312:2 8308:b 8296:= 8224:β 8203:( 8181:λ 8174:φ 8167:c 8165:: 8163:b 8161:: 8159:a 8136:β 8129:Y 8122:Z 8115:X 8102:β 8087:β 8079:β 8074:a 8070:b 8048:. 8040:2 8036:c 8027:2 8023:a 8015:2 8011:c 7996:2 7986:2 7982:b 7978:+ 7967:2 7957:2 7953:a 7937:c 7934:= 7927:Z 7920:, 7899:b 7896:= 7889:Y 7882:, 7874:2 7870:c 7861:2 7857:a 7843:2 7833:2 7829:c 7814:2 7804:2 7800:b 7791:2 7787:a 7771:a 7768:= 7761:X 7739:β 7716:c 7712:b 7708:a 7703:) 7701:Z 7699:, 7697:Y 7695:, 7693:X 7691:( 7674:, 7671:1 7668:= 7661:2 7657:c 7651:2 7647:Z 7641:+ 7634:2 7630:b 7624:2 7620:Y 7614:+ 7607:2 7603:a 7597:2 7593:X 7587:= 7584:h 7493:1 7488:f 7486:| 7448:( 7441:f 7411:f 7394:0 7387:0 7370:λ 7357:1 7350:1 7347:φ 7341:0 7331:0 7313:λ 7308:s 7286:. 7281:0 7258:= 7237:= 7217:, 7196:= 7182:0 7168:= 7148:, 7127:= 7122:0 7099:= 7079:, 7065:0 7051:= 7030:= 7010:, 6989:= 6968:= 6959:0 6887:. 6882:2 6858:) 6853:1 6840:2 6832:( 6826:2 6823:1 6809:) 6804:1 6796:+ 6791:2 6783:( 6777:2 6774:1 6759:= 6754:2 6745:E 6722:1 6718:2 6711:E 6701:2 6698:R 6695:π 6693:2 6685:S 6671:( 6668:f 6663:λ 6659:φ 6642:, 6636:d 6622:) 6614:2 6610:b 6604:2 6599:2 6595:R 6574:e 6571:2 6566:) 6554:e 6551:( 6543:1 6529:+ 6521:) 6508:2 6498:2 6494:e 6487:1 6482:( 6477:2 6473:1 6467:( 6459:2 6447:1 6432:2 6428:b 6424:+ 6421:) 6416:1 6403:2 6395:( 6390:2 6385:2 6381:R 6377:= 6368:S 6354:φ 6349:1 6345:2 6330:K 6306:, 6300:d 6293:d 6279:) 6273:2 6268:2 6264:R 6252:2 6246:) 6232:2 6222:2 6218:e 6211:1 6206:( 6198:2 6194:b 6187:( 6180:+ 6171:2 6166:2 6162:R 6158:= 6145:d 6138:d 6124:) 6118:2 6113:2 6109:R 6100:K 6097:1 6091:( 6084:+ 6075:2 6070:2 6066:R 6062:= 6055:T 6037:T 6027:2 6024:R 6018:2 6015:R 6010:Γ 6005:j 5998:j 5976:j 5966:j 5952:2 5949:= 5920:, 5914:d 5907:d 5891:= 5888:T 5885:d 5881:K 5875:= 5850:K 5826:, 5820:d 5813:d 5798:K 5795:1 5787:= 5784:T 5781:d 5775:= 5772:T 5745:S 5698:A 5685:π 5678:σ 5668:m 5658:A 5651:A 5630:m 5624:A 5613:1 5606:A 5596:. 5586:λ 5576:2 5573:φ 5567:B 5561:A 5546:. 5539:1 5536:φ 5530:A 5514:) 5507:1 5504:φ 5493:1 5488:f 5473:. 5470:π 5463:λ 5461:| 5452:m 5445:0 5437:0 5430:π 5423:λ 5421:| 5416:; 5413:π 5406:σ 5404:| 5390:B 5384:A 5378:A 5353:s 5347:B 5341:A 5328:m 5319:m 5313:1 5309:d 5303:m 5297:1 5293:d 5281:M 5272:m 5251:. 5243:2 5237:) 5223:2 5213:2 5209:e 5202:1 5197:( 5190:4 5186:a 5179:2 5175:b 5169:= 5162:2 5158:b 5151:2 5145:) 5131:2 5121:2 5117:e 5110:1 5105:( 5097:= 5084:1 5079:= 5076:K 5048:M 5044:2 5041:s 5037:1 5034:s 5032:( 5030:M 5017:m 5013:2 5010:s 5006:1 5003:s 5001:( 4999:m 4974:= 4967:1 4963:s 4959:= 4954:2 4950:s 4944:| 4936:2 4932:s 4928:d 4923:) 4918:2 4914:s 4910:, 4905:1 4901:s 4897:( 4894:M 4891:d 4880:, 4877:1 4874:= 4867:) 4862:1 4858:s 4854:, 4849:1 4845:s 4841:( 4838:M 4831:, 4828:1 4825:= 4818:1 4814:s 4810:= 4805:2 4801:s 4795:| 4787:2 4783:s 4779:d 4774:) 4769:2 4765:s 4761:, 4756:1 4752:s 4748:( 4745:m 4742:d 4731:, 4728:0 4725:= 4718:) 4713:1 4709:s 4705:, 4700:1 4696:s 4692:( 4689:m 4659:) 4654:2 4650:s 4646:, 4641:1 4637:s 4633:( 4630:M 4627:D 4624:+ 4621:) 4616:2 4612:s 4608:, 4603:1 4599:s 4595:( 4592:m 4589:C 4586:= 4583:) 4578:2 4574:s 4570:( 4567:t 4553:s 4544:) 4542:s 4540:( 4538:K 4512:, 4509:0 4506:= 4503:) 4500:s 4497:( 4494:t 4491:) 4488:s 4485:( 4482:K 4479:+ 4471:2 4467:s 4463:d 4458:) 4455:s 4452:( 4449:t 4444:2 4440:d 4420:) 4418:s 4416:( 4414:t 4405:) 4403:s 4401:( 4399:t 4393:s 4375:π 4368:λ 4362:A 4357:) 4355:f 4351:π 4344:λ 4342:| 4336:1 4333:φ 4327:A 4317:λ 4311:1 4308:φ 4304:φ 4298:A 4288:A 4271:A 4262:s 4252:1 4242:1 4235:A 4217:1 4214:φ 4203:1 4198:f 4177:0 4169:7 4165:2 4158:a 4152:b 4139:2 4135:1 4128:a 4122:b 4085:. 4081:7 4077:2 4070:a 4064:b 4050:0 4044:λ 4039:0 4035:f 4032:π 4030:2 4026:λ 4016:0 4012:π 4008:2 4004:1 3999:β 3984:0 3965:0 3953:1 3948:f 3909:. 3902:0 3890:1 3885:f 3841:σ 3835:0 3832:λ 3828:λ 3807:, 3796:d 3774:2 3764:2 3760:k 3756:+ 3753:1 3748:) 3745:f 3739:1 3736:( 3733:+ 3730:1 3725:f 3719:2 3706:0 3696:0 3682:f 3673:= 3668:0 3625:, 3619:d 3604:2 3592:0 3575:= 3569:d 3556:σ 3552:λ 3547:b 3539:0 3535:e 3528:b 3519:s 3505:σ 3503:( 3501:s 3483:, 3478:0 3460:e 3456:= 3453:k 3424:, 3413:d 3394:2 3384:2 3380:k 3376:+ 3373:1 3361:0 3353:= 3348:b 3345:s 3327:/ 3319:E 3313:0 3294:, 3280:0 3266:= 3263:) 3258:0 3250:; 3244:( 3232:= 3203:G 3197:E 3192:λ 3188:s 3184:σ 3167:. 3154:2 3144:2 3140:e 3133:1 3128:= 3119:d 3111:d 3105:= 3096:d 3091:s 3088:d 3080:a 3077:1 3049:, 3036:2 3026:2 3022:e 3015:1 3010:= 2958:, 2946:) 2943:f 2937:1 2934:( 2931:= 2915:2 2911:e 2904:1 2899:= 2877:φ 2873:β 2856:. 2827:= 2818:d 2810:d 2804:= 2795:d 2790:s 2787:d 2779:a 2776:1 2762:λ 2757:s 2724:. 2718:d 2705:= 2699:d 2685:, 2679:d 2676:= 2670:d 2646:P 2624:ω 2620:s 2616:σ 2611:E 2605:P 2584:N 2571:σ 2558:1 2554:A 2548:2 2544:π 2540:B 2534:2 2531:β 2527:π 2523:2 2519:1 2508:1 2505:β 2501:π 2497:2 2493:1 2430:. 2425:2 2409:2 2395:= 2390:1 2374:1 2337:, 2325:a 2322:= 2319:R 2306:β 2297:R 2279:. 2249:= 2243:s 2240:d 2232:d 2225:; 2193:= 2187:s 2184:d 2176:d 2169:; 2148:= 2142:s 2139:d 2131:d 2092:. 2086:d 2073:= 2067:d 2025:= 2013:R 1993:L 1970:= 1951:L 1931:L 1906:s 1901:) 1899:λ 1897:( 1895:φ 1889:2 1886:φ 1882:2 1879:λ 1877:( 1875:φ 1869:1 1866:φ 1862:1 1859:λ 1857:( 1855:φ 1850:λ 1846:φ 1829:, 1823:d 1819:) 1808:, 1802:( 1799:L 1792:2 1780:1 1767:= 1758:s 1744:) 1742:2 1739:λ 1735:2 1732:φ 1730:( 1726:) 1724:1 1721:λ 1717:1 1714:φ 1712:( 1708:) 1706:φ 1704:( 1702:R 1697:) 1695:φ 1689:φ 1683:L 1673:/ 1667:φ 1645:, 1639:d 1635:) 1624:, 1618:( 1615:L 1599:d 1591:2 1587:R 1583:+ 1577:2 1563:2 1553:= 1546:s 1543:d 1511:2 1503:d 1497:2 1493:R 1489:+ 1484:2 1476:d 1470:2 1462:= 1457:2 1453:s 1449:d 1428:φ 1423:φ 1415:R 1386:, 1380:d 1376:R 1373:= 1370:s 1367:d 1353:, 1336:R 1333:d 1324:= 1318:d 1311:= 1308:s 1305:d 1249:e 1243:e 1237:f 1231:b 1227:a 1221:b 1217:a 1199:. 1193:f 1187:1 1183:e 1178:= 1173:b 1167:2 1163:b 1154:2 1150:a 1143:= 1136:e 1131:, 1126:) 1123:f 1117:2 1114:( 1111:f 1106:= 1101:a 1095:2 1091:b 1082:2 1078:a 1071:= 1068:e 1064:, 1059:a 1055:b 1049:a 1043:= 1040:f 1025:e 1019:e 1010:f 1004:b 998:a 821:1 818:λ 814:2 811:λ 804:λ 798:1 784:. 781:2 774:1 764:s 758:B 752:A 738:; 735:2 728:B 719:s 713:1 706:A 688:2 681:1 668:s 656:B 650:A 644:2 641:λ 635:2 632:φ 626:B 620:1 617:λ 608:1 605:φ 596:A 583:N 431:e 424:t 417:v 111:) 103:( 20:)

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Knowledge

Geodesics on an ellipsoid

Index