Geodesics on an ellipsoid


The study of [Spherical trigonometry|]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 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, 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 other distinct routes require less distance.
Locally, these geodesics are still identical to the shortest distance
between two points.
By the end of the 18th century, an ellipsoid of revolution 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: at latitude
and longitude and
at latitude and longitude
. The connecting geodesic
is, of length
, which has azimuths and
at the two endpoints. The two geodesic problems usually
considered are:
  1. the direct geodesic problem or first geodesic problem, given,, and, determine and ;
  2. the inverse geodesic problem or second geodesic problem, given and, determine,, and.
As can be seen from Fig. 1, these problems involve solving the triangle
given one angle, for the direct
problem and 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.
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
.
The full solution for the direct problem is given by.
During the 18th century geodesics were typically referred to as "shortest
lines".
The term "geodesic line" was coined by :

Nous désignerons cette ligne sous le nom de ligne géodésique .

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. 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
and polar semi-axis. Define the
flattening, the eccentricity
, and the second
eccentricity.
Let an elementary segment of a path on the ellipsoid have length
. From Figs. 2 and 3, we
see that if its azimuth is, then
is related to and by
where is the
meridional radius of curvature,
is the radius of the circle of latitude
, and is the
normal radius of curvature.
The elementary segment is therefore given by
or
where and the
Lagrangian function depends on
through and
. The length of an arbitrary path between
and is
given by
where is a function of satisfying
and
. The shortest path or geodesic
entails finding that function which minimizes
. This is an exercise in the
calculus of variations and the minimizing condition is given by the
Beltrami identity,
Substituting for and using Eqs. gives
found this relation,
using a geometrical construction; a similar derivation is presented by
. Differentiating this
relation gives
This, together with Eqs., leads to a system of
ordinary differential equations for a geodesic
We can express in terms of the
parametric latitude,
using
and Clairaut's
relation then becomes
This is the sine rule of spherical
trigonometry relating two sides of the triangle ,, and
and their opposite angles
and.
In order to find the relation for the third side
, the spherical arc length, and included
angle, the spherical longitude, it is
useful to consider the triangle 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
;, the point at which the geodesic crosses
the equator in the northward direction, is used as the origin for
, and.
If the side is extended by
moving infinitesimally, we
obtain
Combining Eqs. and gives differential
equations for and
The relation between and is
which gives
so that the differential equations for the geodesic become
The last step is to use as the independent
parameter in both of
these differential equations and thereby to express and
as integrals. Applying the sine rule to the vertices
and in the spherical triangle
in Fig. 5 gives
where is the azimuth at.
Substituting this into the equation for and
integrating the result gives
where
and the limits on the integral are chosen so that
. pointed out
that the equation for is the same as the equation for the
arc on an ellipse
with semi-axes and
. In order to express the equation for
in terms of, we write
which follows from Eq. and Clairaut's relation.
This yields
and the limits on the integrals are chosen
so that at the equator crossing,
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 ; 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 and the equator. 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
. 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
The geodesic completes one full oscillation in
latitude before the longitude has increased by.
Thus, on each successive northward crossing of the equator, falls short of a full circuit of
the equator by approximately . For nearly all
values of, the geodesic will fill that portion of
the ellipsoid between the two vertex latitudes.
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,
, for the green geodesic is chosen to
be , so that the
geodesic completes 2 complete oscillations about the equator
on one circuit of the ellipsoid.
Fig.
13 shows geodesics emanating
with a multiple of
up to the point at which they cease to be shortest
paths.
Also shown are curves of constant,
which are the geodesic circles centered.
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 shortest geodesics from. On a sphere, the cut
locus is a point. On an oblate ellipsoid, it is a segment
of the circle of latitude centered on the point antipodal
to,. The longitudinal
extent of cut locus is approximately
. If
lies on the equator,, this relation
is exact and as a consequence the equator is only a shortest geodesic if
. For a prolate
ellipsoid, the cut locus is a segment of the anti-meridian centered on
the point antipodal to,,
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,
and a second geodesic a small
distance away from it. showed that
obeys the
Gauss-Jacobi equation
where is the Gaussian curvature at.
As a second order, linear, homogeneous differential equation,
its solution may be expressed as the sum of two independent solutions
where
The quantity is the so-called
reduced length, and is the
geodesic scale.
Their basic definitions are illustrated in
Fig. 14.
The
Gaussian curvature for an ellipsoid of revolution
is
solved the Gauss-Jacobi
equation for this case enabling and
to be expressed as integrals.
As we see from Fig. 14, the separation of two
geodesics starting at the same point with azimuths differing by
is. On a closed
surface such as an ellipsoid, oscillates
about zero. The point at
which becomes zero is the point
conjugate to the starting point. In order
for a geodesic between and, of length
, to be a shortest path it must satisfy the
Jacobi condition
, that there is
no point conjugate to between and
. If this condition is not satisfied, then there is a
nearby path 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:
The geodesics from a particular point if continued
past the cut locus form an envelope illustrated in Fig. 15.
Here the geodesics for which is a multiple of
are shown in light blue. Some geodesic circles are shown in green; these form
cusps on the envelope. The cut locus is shown in red. The envelope is
the locus of points which are conjugate to ; points on the
envelope may be computed by finding the point at which
on a geodesic.
calls this star-like figure
produced by the envelope an astroid.
Outside the astroid two geodesics intersect at each point; thus there
are two geodesics between and these points.
This corresponds to the situation on the sphere where there are "short"
and "long" routes on a great circle between two points. Inside the
astroid four geodesics intersect at each point. Four such geodesics are
shown in Fig. 16 where the geodesics are numbered in order of
increasing length.
The two shorter geodesics are stable, i.e.,,
so that there is no nearby path connecting the two points which is
shorter; the other two are unstable. Only the shortest line has. All the geodesics are tangent
to the envelope which is shown in green in the figure.
The astroid is the evolute of the geodesic circles
centered at. Likewise, the geodesic circles are
involutes of the astroid.

Area of a geodesic polygon

A geodesic polygon is a polygon whose sides are geodesics.
It is analogous to a spherical polygon, whose sides are great circles.
The area of such a polygon may be found by first computing the area between a
geodesic segment and the equator, i.e., the area of the quadrilateral
in Fig. 1. Once this
area is known, the area of a polygon may be computed by summing the
contributions from all the edges of the polygon.
Here an expression for the area of
is developed following. The area of any closed
region of the ellipsoid is
where is an element of surface area and
is the Gaussian curvature. Now the
Gauss–Bonnet theorem applied to a geodesic polygon states
where
is the geodesic excess and is the exterior angle at
vertex. Multiplying the equation for
by, where is the
authalic radius, and subtracting this
from the equation for gives
where the value of for an ellipsoid
has been substituted.
Applying this formula to the quadrilateral, noting
that, and performing
the integral over gives
where the integral is over the geodesic line.
The integral can be expressed as a series valid for small
.
The area of a geodesic polygon is given by summing
over its edges. This result holds provided that the polygon does not
include a pole; if it does, must be added to the
sum. If the edges are specified by their vertices, then a
convenient expression
for the geodesic excess is

Solution of the direct and inverse problems

Solving the geodesic problems entails mapping the geodesic onto the
auxiliary sphere and solving the corresponding problem in
great-circle navigation.
When solving the
"elementary" spherical triangle for in Fig.
5,
Napier's rules for quadrantal triangles can be employed,
The mapping of the geodesic involves evaluating the
integrals for the distance,, and the longitude,
, Eqs. and and these depend on
the parameter.
Handling the direct problem is straightforward, because
can be determined directly from the given
quantities and.
In the case of the inverse problem, is given;
this cannot be easily related to the equivalent spherical angle
because is unknown.
Thus, the solution of the problem requires that be
found iteratively.
In geodetic applications, where is small, the integrals
are typically evaluated as a series

. For arbitrary
, the integrals and can be found by
numerical quadrature or by expressing them in terms of
elliptic integrals .
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
for the WGS84 ellipsoid; however the inverse method
fails to converge for nearly antipodal points.
continues the expansions to sixth order which suffices to provide full
double precision accuracy for
and improves the solution of
the inverse problem so that it converges in all cases.
extends the method to use elliptic
integrals which can be applied to ellipsoids with arbitrary flattening.

Geodesics on a triaxial ellipsoid

Solving the geodesic problem for an ellipsoid of revolution is, from the
mathematical point of view, relatively simple: because of symmetry,
geodesics have a constant of motion, given by Clairaut's relation
allowing the problem to be reduced to
quadrature. By the early 19th century
,
there was a complete understanding of the properties of geodesics on an
ellipsoid of revolution.
On the other hand, geodesics on a triaxial ellipsoid 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, discovered a
constant of the motion allowing this problem to be reduced to quadrature
also.

The triaxial coordinate system

Consider the ellipsoid defined by
where are Cartesian coordinates centered on the
ellipsoid and, without loss of generality,
employed the ellipsoidal latitude and longitude
defined by
.
In the limit,
becomes the parametric latitude for an oblate ellipsoid, so the use of
the symbol is consistent with the previous sections.
However, is different from the spherical
longitude defined above.
Grid lines of constant and
are given in Fig. 17. These
constitute an orthogonal coordinate system: the
grid lines intersect at right angles. The principal sections of the
ellipsoid, defined by and are shown in
red. The third principal section,, is covered by the
lines and or
. These lines meet at four
umbilical points where the
principal radii of curvature are equal. Here
and in the other figures in this section the parameters of the ellipsoid
are, and it is viewed in an orthographic
projection from a point above,
The grid lines of the ellipsoidal coordinates may be interpreted in three
different ways:
  1. They are "lines of curvature" on the ellipsoid: they are parallel to the directions of principal curvature.
  2. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets.
  3. Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points. For example, the lines of constant in Fig. 17 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.

    Jacobi's solution

Jacobi showed that the geodesic equations, expressed in ellipsoidal
coordinates, are separable. Here is how he recounted his discovery to
his friend and neighbor Bessel,
The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal.
Königsberg, 28th Dec. '38.

The solution given by Jacobi
is
As Jacobi notes "a function of the angle equals
a function of the angle. These two functions are
just Abelian integrals..." Two constants and
appear in the solution. Typically
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. However, for geodesics
that start at an umbilical points, we have and
determines the direction at the umbilical point.
The constant may be expressed as
where is the angle the geodesic makes with lines of
constant. In the limit,
this reduces to, the
familiar Clairaut relation. A derivation of Jacobi's result is
given by ; he
gives the solution found by for general quadratic
surfaces.

Survey of triaxial geodesics

On a triaxial ellipsoid, there are only three simple closed geodesics, the
three principal sections of the ellipsoid given by,
, and.
To survey the other geodesics, it is convenient to consider geodesics
that intersect the middle principal section,, at right
angles. Such geodesics are shown in Figs. 18–22,
which use the same ellipsoid parameters and the same viewing direction
as Fig. 17. In addition, the three principal ellipses are shown
in red in each of these figures.
If the starting point is,
, and, then
and the
geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic
oscillates north and south of the equator; on each oscillation it
completes slightly less than a full circuit around the ellipsoid
resulting, in the typical case, in the geodesic filling the area bounded
by the two latitude lines. Two examples
are given in Figs. 18 and 19. Figure 18 shows
practically the same behavior as for an oblate ellipsoid of revolution
; compare to Fig. 9.
However, if the starting point is at a higher latitude
the distortions resulting from are evident. All
tangents to a circumpolar geodesic touch the confocal single-sheeted
hyperboloid which intersects the ellipsoid at
If the starting point is,
, and
, then
and the geodesic encircles the ellipsoid
in a "transpolar" sense. The geodesic oscillates east and west of the
ellipse ; 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
and.
If, all meridians are geodesics; the effect of
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
; in this case, the ellipsoid becomes a
prolate ellipsoid and Fig. 20 would resemble Fig. 10
. All tangents to a transpolar geodesic touch the
confocal double-sheeted hyperboloid which intersects the ellipsoid at
If the starting point is,
, and
, then
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
becomes closer to or
so that asymptotically the geodesic lies on the
ellipse ,
as shown in Fig. 22. A single geodesic does not
fill an area on the ellipsoid. All tangents to umbilical geodesics
touch the confocal hyperbola that intersects the ellipsoid at the
umbilic points.
Umbilical geodesic enjoy several interesting properties.
If the starting point of a geodesic is not an umbilical
point, its envelope is an astroid with two cusps lying on
and the other two on
. The cut locus
for is the portion
of the line between the cusps.

Applications

The direct and inverse geodesic problems no longer play the central role
in geodesy that they once did. Instead of solving
adjustment of geodetic networks as a
two-dimensional problem in spheroidal trigonometry, these problems are
now solved by three-dimensional methods.
Nevertheless, terrestrial geodesics still play an important role in
several areas:
By the principle of least action, many problems in physics can be
formulated as a variational problem similar to that for geodesics. Indeed,
the geodesic problem is equivalent to the motion of a particle
constrained to move on the surface, but otherwise subject to no forces
.
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: