The Geodesic on an Oblate Spheroid can be computed analytically for a spheroid specified parametrically
with , although it is much more unwieldy than for a simple Sphere.
Using the first Partial Derivatives
and second Partial Derivatives
gives the Geodesics functions as
Since and and are explicit functions of only, we can use the special form of the Geodesic
is an Elliptic Integral of the First Kind with Parameter , and
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
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
where the Modulus of the
Elliptic Function is
(Forsyth 1960, p. 446).
See also Ellipsoid Geodesic, Oblate Spheroid, Sphere Geodesic
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.
© 1996-9 Eric W. Weisstein