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Geodesics on an ellipsoid
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Geodesics on an ellipsoid : ウィキペディア英語版
Geodesics on an ellipsoid

The study of geodesics on an ellipsoid arose in connection with geodesy
specifically with the solution of triangulation networks. The
figure of the Earth is well approximated by an
''oblate ellipsoid'', a slightly flattened sphere. A ''geodesic''
is the shortest path between two points on a curved surface, i.e., the analogue
of a straight line on a plane surface. The solution of a triangulation
network on an ellipsoid is therefore a set of exercises in spheroidal
trigonometry .
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 and, in this case, the
equator and the meridians are the only
closed geodesics. Furthermore, the shortest path between two points on
the equator does not necessarily run along the equator. Finally, if the
ellipsoid is further perturbed to become a triaxial ellipsoid (with
three distinct semi-axes), then only three geodesics are closed and one
of these is unstable.
The problems in geodesy are usually reduced to two main cases: the
''direct problem'', given a starting point and an initial heading, find
the position after traveling a certain distance along the geodesic; and
the ''inverse problem'', given two points on the ellipsoid find the
connecting geodesic and hence the shortest distance between them.
Because the flattening of the Earth is small, the geodesic distance
between two points on the Earth is well approximated by the great-circle
distance using the
mean Earth radius—the relative error is
less than 1%. However, the course of the geodesic can differ
dramatically from that of the great circle. As an extreme example,
consider two points on the equator with a longitude difference of
179°59′; while the connecting great circle follows the
equator, the shortest geodesics pass within
180 km of either pole (the
flattening makes two symmetric paths passing close to the poles shorter
than the route along the equator).
Aside from their use in geodesy and related fields such as navigation,
terrestrial geodesics arise in the study of the propagation of signals
which are confined (approximately) to the surface of the Earth, for
example, sound waves in the ocean and the
radio signals from lightning . Geodesics are
used to define some maritime boundaries, which in turn determine the
allocation of valuable resources as such
oil and mineral rights. Ellipsoidal geodesics also
arise in other applications; for example, the propagation of radio waves
along the fuselage of an aircraft, which can be roughly modeled as a
prolate (elongated) ellipsoid
.
Geodesics are an important intrinsic characteristic of curved surfaces.
The sequence of progressively more complex surfaces, the sphere, an
ellipsoid of revolution, and a triaxial ellipsoid, provide a useful
family of surfaces for investigating the general theory of surfaces.
Indeed, Gauss's work on the
survey of Hanover, which involved
geodesics on an oblate ellipsoid, was a key motivation for his
study of surfaces
. Similarly, the existence of three closed geodesics
on a triaxial ellipsoid turns out to be a general property of
closed, simply connected surfaces; this
theorem of the three geodesics was
conjectured by and proved by
.
== 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 (somewhat less than half the
circumference) that other distinct routes require less distance.
Locally, these geodesics are still identical to the shortest distance
between two points.
By the end of the 18th century, an ellipsoid of revolution (the term
spheroid is also used) was a well-accepted approximation to the
figure of the Earth. The adjustment of triangulation 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
φ1 and longitude λ1 and
''B'' at latitude φ2 and longitude
λ2 (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:
# the ''direct geodesic problem'' or ''first geodesic problem'', given ''A'', α1, and ''s''12, determine ''B'' and α2;
# the ''inverse geodesic problem'' or ''second geodesic problem'', given ''A'' and ''B'', determine ''s''12, α1, and α2.
As can be seen from Fig. 1, these problems involve solving the triangle
''NAB'' given one angle, α1 for the direct
problem and λ12 = λ2 − λ1 for the
inverse problem, and its two adjacent sides.
In the course of the 18th century these problems were elevated
(especially in literature in the German language) to the
''principal geodesic problems''
.
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
and
).
The full solution for the direct problem (complete with computational
tables and a worked out example) is given by .
Much of the early work on these problems was carried out by
mathematicians—for example, Legendre,
Bessel, and Gauss—who
were also heavily involved in the practical aspects of surveying.
Beginning in about 1830, the disciplines diverged: those with an
interest in geodesy concentrated on the practical aspects such as
approximations suitable for field work, while mathematicians pursued the
solution of geodesics on a triaxial ellipsoid, the analysis of the
stability of closed geodesics, etc.
During the 18th century geodesics were typically referred to as "shortest
lines".
The term "geodesic line" was coined by :

Nous désignerons cette ligne sous le nom de ''ligne géodésique'' (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.
When determining distances on the earth, various approximate methods are
frequently used; some of these are described in the article on
geographical distance.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Geodesics on an ellipsoid」の詳細全文を読む



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