“Oblate Spheroid Geodesic”的概念、定义、习题解题方法及翻译-数学辞典

单词

Oblate Spheroid Geodesic

释义

Oblate Spheroid Geodesic

The Geodesic on an Oblate Spheroid can be computed analytically for a spheroid specified parametricallyby

			(1)
			(2)
			(3)

with

, although it is much more unwieldy than for a simple Sphere. Using the first Partial Derivatives

(4)

(5)

(6)

and second Partial Derivatives

(7)

(8)

(9)

gives the Geodesics functions as



			(10)
			(11)

			(12)

Since

and

are explicit functions of

only, we can use the special form of the Geodesicequation.



			(13)

Integrating gives

(14)

where

			(15)
			(16)

is an Elliptic Integral of the First Kind with Parameter

, and

isan Elliptic Integral of the Third Kind.

Geodesics other than Meridians of an Oblate Spheroid undulate between twoparallels with latitudes equidistant from the equator. Using the Weierstraß Sigma Function and Weierstraß Zeta Function, the Geodesic on the OblateSpheroid can be written as

			(17)
			(18)
			(19)

(Forsyth 1960, pp. 108-109; Halphen 1886-1891).

The equation of the Geodesic can be put in the form

(20)

where

is the smallest value of

on the curve. Furthermore, the difference in longitude between points ofhighest and next lowest latitude on the curve is

(21)

where the Modulus of the Elliptic Function is

(22)

(Forsyth 1960, p. 446).

See also Ellipsoid Geodesic, Oblate Spheroid, Sphere Geodesic

References

Forsyth, A. R. Calculus of Variations. New York: Dover, 1960.

Halphen, G. H. Traité des fonctions elliptiques et de leurs applications fonctions elliptiques, Vol. 2. Paris: Gauthier-Villars, pp. 238-243, 1886-1891.

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