The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much
more unwieldy than for a simple sphere . A spheroid with
equatorial radius and polar radius can be specified parametrically by

where .
Using the second partial derivatives

gives the geodesics functions as

where

(16)

is the eccentricity .

Since
and
and
are explicit functions of only, we can use the special form of the geodesic
equation

where
is a constant depending on the starting and ending points. Integrating gives

(20)

where

is an elliptic integral of the first
kind with parameter , and is an elliptic
integral of the third kind .

Geodesics other than meridians of an oblate spheroid undulate between two parallels
with latitudes equidistant from the equator. Using the Weierstrass
sigma function and Weierstrass zeta function ,
the geodesic on the oblate
spheroid can be written as

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

The equation of the geodesic can be put in the form

(26)

where
is the smallest value of on the curve. Furthermore, the difference in longitude between
points of highest and next lowest latitude on the curve is

(27)

where the elliptic modulus of the elliptic
function is

(28)

(Forsyth 1960, p. 446).

See also Ellipsoid Geodesic ,

Great
Circle ,

Oblate Spheroid
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References Forsyth, A. R. Calculus of Variations. New York: Dover, 1960. Gosper, R. W. "Spheroid
Geodesic Integral." math-fun@cs.arizona.edu posting, Sept. 9, 1996. Halphen,
G. H. Traité
des fonctions elliptiques et de leurs applications fonctions elliptiques, Vol. 2.
Paris: Gauthier-Villars, pp. 238-243, 1886-1891. Tietze, H. Famous
Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity
to Modern Times. New York: Graylock Press, pp. 28-29 and 40-41, 1965. Referenced
on Wolfram|Alpha Oblate Spheroid Geodesic
Cite this as:
Weisstein, Eric W. "Oblate Spheroid Geodesic."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/OblateSpheroidGeodesic.html

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