Oblate Spheroid Geodesic

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

			(1)
			(2)
			(3)

where . Using the second partial derivatives

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

gives the geodesics functions as

			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

where

(16)

is the ellipticity.

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

			(17)
			(18)
			(19)

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

u=-(e^2F(phi|((d^2-1)e^2)/(d^2-e^2))-d^2Pi(d^2-1,phi|((d^2-1)e^2)/(d^2-e^2)))/(sqrt(d^2-e^2)),

(20)

where

			(21)
			(22)

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

			(23)
			(24)
			(25)

(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