Shortest paths on a bounded deformed sphere-like quadric surface
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an "oblate ellipsoid", a slightly flattened sphere. A "geodesic" is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry .
If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid: in this case, the equator and the meridians are the only simple closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), only three geodesics are closed.
Geodesics on an ellipsoid of revolution.
There are several ways of defining geodesics . A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface that they start to return toward the starting point, so that other routes are more direct, and includes paths that intersect or re-trace themselves. Short enough segments of a geodesics are still the shortest route between their endpoints, but geodesics are not necessarily globally minimal (i.e. shortest among all possible paths). Every globally-shortest path is a geodesic, but not vice versa.
By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry .
It is possible to reduce the various geodesic problems into one of two types. Consider two points: "A" at latitude "φ"1 and longitude "λ"1 and "B" at latitude "φ"2 and longitude "λ"2 (see Fig. 1). The connecting geodesic (from "A" to "B") is "AB", of length "s"12, which has azimuths α1 and α2 at the two endpoints. The two geodesic problems usually considered are:
As can be seen from Fig. 1, these problems involve solving the triangle "NAB" given one angle, α1 for the direct problem and "λ"12 
 "λ"2 − "λ"1 for the inverse problem, and its two adjacent sides.
For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation.)
For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by . A systematic solution for the paths of geodesics was given by and (and subsequent papers in 1808 and 1810).
The full solution for the direct problem (complete with computational tables and a worked out example) is given by .
During the 18th century geodesics were typically referred to as "shortest lines".
The term "geodesic line" (actually, a curve) was coined by :
Nous désignerons cette ligne sous le nom de "ligne géodésique" [We will call this line the "geodesic line"].
This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example ,
A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a "geodetic" or "geodesic line:" it has the property of being the shortest which can be drawn between its two extremities on the surface of the Earth; and it is therefore the proper itinerary measure of the distance between those two points.
In its adoption by other fields "geodesic line", frequently shortened to "geodesic", was preferred.
This section treats the problem on an ellipsoid of revolution (both oblate and prolate). The problem on a triaxial ellipsoid is covered in the next section.
Equations for a geodesic.
Here the equations for a geodesic are developed; the derivation closely follows that of . , , , , , , and also provide derivations of these equations.
Consider an ellipsoid of revolution with equatorial radius "a" and polar semi-axis "b". Define the flattening "f", the eccentricity "e", and the second eccentricity "e"′:
formula_1
Let an elementary segment of a path on the ellipsoid have length "ds". From Figs. 2 and 3, we see that if its azimuth is α, then "ds" is related to "dφ" and "dλ" by
formula_2 (1)
where ρ is the meridional radius of curvature, "R" 
 "ν" cos"φ" is the radius of the circle of latitude φ, and ν is the normal radius of curvature.
The elementary segment is therefore given by
formula_3
or
formula_4
where "φ"′ 
 "dφ"/"dλ" and the Lagrangian function "L" depends on φ through ρ("φ") and "R"("φ"). The length of an arbitrary path between ("φ"1, "λ"1) and ("φ"2, "λ"2) is given by
formula_5
where φ is a function of λ satisfying "φ"("λ"1) 
 "φ"1 and "φ"("λ"2) 
 "φ"2. The shortest path or geodesic entails finding that function "φ"("λ") which minimizes "s"12. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity,
formula_6
Substituting for "L" and using Eqs. (1) gives
formula_7
 found this relation, using a geometrical construction; a similar derivation is presented by . Differentiating this relation gives
formula_8
This, together with Eqs. (1), leads to a system of ordinary differential equations for a geodesic
formula_9
We can express "R" in terms of the parametric latitude, β, using
formula_10
and Clairaut's relation then becomes
formula_11
This is the sine rule of spherical trigonometry relating two sides of the triangle "NAB" (see Fig. 4), "NA" 
 ;templatestyles src="Fraction/styles.css" / ;1⁄2"π" − "β"1, and "NB" 
 ;templatestyles src="Fraction/styles.css" / ;1⁄2"π" − "β"2 and their opposite angles "B" 
 "π" − α2 and "A" 
 α1.
In order to find the relation for the third side "AB" 
 "σ"12, the "spherical arc length", and included angle "N" 
 ω12, the "spherical longitude", it is useful to consider the triangle "NEP" representing a geodesic starting at the equator; see Fig. 5. In this figure, the variables referred to the auxiliary sphere are shown with the corresponding quantities for the ellipsoid shown in parentheses.
Quantities without subscripts refer to the arbitrary point "P"; "E", the point at which the geodesic crosses the equator in the northward direction, is used as the origin for σ, s and ω.
If the side "EP" is extended by moving "P" infinitesimally (see Fig. 6), we obtain
formula_12 (2)
Combining Eqs. (1) and (2) gives differential equations for "s" and λ
formula_13
The relation between β and φ is
formula_14
which gives
formula_15
so that the differential equations for the geodesic become
formula_16
The last step is to use σ as the independent parameter in both of these differential equations and thereby to express s and λ as integrals. Applying the sine rule to the vertices "E" and "G" in the spherical triangle "EGP" in Fig. 5 gives
formula_17
where α0 is the azimuth at "E".
Substituting this into the equation for "ds"/"d σ" and integrating the result gives
formula_18 (3)
where
formula_19
and the limits on the integral are chosen so that "s"("σ" = 0) = 0. pointed out that the equation for s is the same as the equation for the arc on an ellipse with semi-axes "b"√1 + "e"′2 cos2α0 and "b". In order to express the equation for λ in terms of σ, we write
formula_20
which follows from Eq. 2 and Clairaut's relation.
This yields
formula_21 (4)
and the limits on the integrals are chosen so that "λ" 
 "λ"0 at the equator crossing, "σ" 
 0.
This completes the solution of the path of a geodesic using the auxiliary sphere. By this device a great circle can be mapped exactly to a geodesic on an ellipsoid of revolution.
There are also several ways of approximating geodesics on a terrestrial ellipsoid (with small flattening) ; some of these are described in the article on geographical distance. However, these are typically comparable in complexity to the method for the exact solution .
Behavior of geodesics.
Fig. 7 shows the simple closed geodesics which consist of the meridians (green) and the equator (red). (Here the qualification "simple" means that the geodesic closes on itself without an intervening self-intersection.) This follows from the equations for the geodesics given in the previous section.
All other geodesics are typified by Figs. 8 and 9 which show a geodesic starting on the equator with α0 
 45°. The geodesic oscillates about the equator. The equatorial crossings are called "nodes" and the points of maximum or minimum latitude are called "vertices"; the parametric latitudes of the vertices are given by "β" 
 ±( ;templatestyles src="Fraction/styles.css" / ;1⁄2"π" − |α0|). The geodesic completes one full oscillation in latitude before the longitude has increased by . Thus, on each successive northward crossing of the equator (see Fig. 8), λ falls short of a full circuit of the equator by approximately 2"π" "f" sinα0 (for a prolate ellipsoid, this quantity is negative and λ completes more that a full circuit; see Fig. 10). For nearly all values of α0, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. 9).
If the ellipsoid is sufficiently oblate, i.e., , another class of simple closed geodesics is possible . Two such geodesics are illustrated in Figs. 11 and 12. Here and the equatorial azimuth, α0, for the green (resp. blue) geodesic is chosen to be (resp. ), so that the geodesic completes 2 (resp. 3) complete oscillations about the equator on one circuit of the ellipsoid.
Fig. 13 shows geodesics (in blue) emanating "A" with α1 a multiple of up to the point at which they cease to be shortest paths. (The flattening has been increased to ;templatestyles src="Fraction/styles.css" / ;1⁄10 in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant "s"12, which are the geodesic circles centered "A". showed that, on any surface, geodesics and geodesic circle intersect at right angles.
The red line is the cut locus, the locus of points which have multiple (two in this case) shortest geodesics from "A". On a sphere, the cut locus is a point. On an oblate ellipsoid (shown here), it is a segment of the circle of latitude centered on the point antipodal to "A", "φ" 
 −"φ"1. The longitudinal extent of cut locus is approximately "λ"12 ∈ ["π" (1 − "f" cos"φ"1), "π" (1 + "f" cos"φ"1)]. If
"A" lies on the equator, "φ"1 
 0, this relation is exact and as a consequence the equator is only a shortest geodesic if |"λ"12| ≤ "π" (1 − "f"). For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to "A", "λ"12 
 "π", and this means that meridional geodesics stop being shortest paths before the antipodal point is reached.
Differential properties of geodesics.
Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments , determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by "s", and a second geodesic a small distance "t"("s") away from it. showed that "t"("s") obeys the Gauss-Jacobi equation
formula_22
where "K"("s") is the Gaussian curvature at "s". As a second order, linear, homogeneous differential equation, its solution may be expressed as the sum of two independent solutions
formula_23
where
formula_24
The quantity "m"("s"1, "s"2) 
 "m"12 is the so-called "reduced length", and "M"("s"1, "s"2) 
 "M"12 is the "geodesic scale".
Their basic definitions are illustrated in Fig. 14.
The Gaussian curvature for an ellipsoid of revolution is
formula_25
 solved the Gauss-Jacobi equation for this case enabling "m"12 and "M"12 to be expressed as integrals.
As we see from Fig. 14 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by "d"α1 is "m"12 "d"α1. On a closed surface such as an ellipsoid, "m"12 oscillates about zero. The point at which "m"12 becomes zero is the point conjugate to the starting point. In order for a geodesic between "A" and "B", of length "s"12, to be a shortest path it must satisfy the Jacobi condition , that there is no point conjugate to "A" between "A" and "B". If this condition is not satisfied, then there is a