“Great Circle”的概念、定义、习题解题方法及翻译-数学辞典

单词

Great Circle

释义

Great Circle

The shortest path between two points on a Sphere, also known as an Orthodrome. To find the greatcircle (Geodesic) distance between two points located at Latitude and Longitude of and on a Sphere of Radius , convert SphericalCoordinates to Cartesian Coordinates using

(1)

(Note that the Latitude

is related to the Colatitude

of Spherical Coordinates by

, so the conversion to Cartesian Coordinates replaces

and

, respectively.) Now find the Angle

between

and

usingthe Dot Product,



			(2)

The great circle distance is then

(3)

For the Earth,

the equatorial Radius is

km, or 3963 (statute)miles.

Unfortunately, the Flattening of the Earth cannot be taken into account in this simplederivation, since the problem is considerably more complicated for a Spheroid or Ellipsoid (each of which has aRadius which is a function of Latitude).

The equation of the great circle can be explicitly computed using the Geodesic formalism. Writing

			(4)
			(5)

gives the

, and

parameters of the Geodesic (which are just combinations of the PartialDerivatives) as

			(6)
			(7)
			(8)

The Geodesic differential equation then becomes

(9)

However, because this is a special case of

with

and

explicit functions of

only, the Geodesic solution takes on the special form



			(10)

(Gradshteyn and Ryzhik 1979, p. 174, eqn. 2.599.6), which can be rewritten as

(11)

It therefore follows that

(12)

This equation can be written in terms of the Cartesian Coordinates as

(13)

which is simply a Plane passing through the center of the Sphere and the two points on the surface ofthe Sphere.

See also Geodesic, Great Sphere, Loxodrome, Mikusinski's Problem,Orthodrome, Point-Point Distance--2-D, Sphere

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, pp. 26-28 and 62-63, 1974.

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