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

(1) | |||

(2) | |||

(3) |

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

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) | |||

(11) | |||

(12) |

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

(13) |

Integrating gives

(14) |

(15) | |||

(16) |

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

(17) | |||

(18) | |||

(19) |

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

The equation of the Geodesic can be put in the form

(20) |

(21) |

(22) |

**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.

© 1996-9

1999-05-26