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Ellipsoid Geodesic


An ellipsoid can be specified parametrically by

x=acosusinv
(1)
y=bsinusinv
(2)
z=ccosv.
(3)

The geodesic parameters are then

P=sin^2v(b^2cos^2u+a^2sin^2u)
(4)
Q=1/4(b^2-a^2)sin(2u)sin(2v)
(5)
R=cos^2v(a^2cos^2u+b^2sin^2u)+c^2sin^2v.
(6)

When the coordinates of a point are on the quadric

 (x^2)/(a^2)+(y^2)/(b^2)+(z^2)/(c^2)=1
(7)

and expressed in terms of the parameters p and q of the confocal quadrics passing through that point (in other words, having a+p, b+p, c+p, and a+q, b+q, c+q for the squares of their semimajor axes), then the equation of a geodesic can be expressed in the form

 (qdq)/(sqrt(q(a+q)(b+q)(c+q)(theta+q)))+/-(pdp)/(sqrt(p(a+p)(b+p)(c+p)(theta+p)))=0,
(8)

with theta an arbitrary constant, and the arc length element ds is given by

 -2(ds)/(pq)=(dq)/(sqrt(q(a+q)(b+q)(c+q)(theta+q)))+/-(dp)/(sqrt(p(a+p)(b+p)(c+p)(theta+p))),
(9)

where upper and lower signs are taken together.


See also

Great Circle, Oblate Spheroid Geodesic

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References

Dragovich, V. J. Phys. A 29, L317, 1996.Dragović, V. math-ph/0008009 (2000).Eisenhart, L. P. A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover, pp. 236-241, 1960.Forsyth, A. R. Calculus of Variations. New York: Dover, p. 447, 1960.Fricke, R. Kurzgefasste Vorlesungen über verschiedene Gebiete der höheren Mathematik mit Berücksichtigung der Anwendungen. Leipzig, Germany: Teubner, 1900.Joachimsthal, F. Anwendung der Differential- und Integralrechnung auf die allgemeine Theorie der Flächen un der Linien doppelter Krümmung. Leipzig, Germany: Teubner, 1890.Knörrer, H. Invent. Math. 59, 119, 1980.Kravchenko, N. N. Vestnik Mosk. Univ Ser. 1, No. 4, 69, 1996.Prasolov, V. and Solovyev, Y. Elliptic Functions and Elliptic Integrals. Providence, RI: Amer. Math. Soc., 1997.Tabanov, M. B. Russian Acad. Sci. Dokl. Math. 48, 438, 1994.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.Toth, J. A. Ann. Phys. 130, 1, 1995.Tricomi, F. Elliptische Funktionen. Leipzig, Germany: Geest und Portig, 1948.Viesel, H. Archiv Math. 22, 106, 1971.Wiersig, J. and Richter, P. H. Z. Naturf. 51a, 219, 1996.

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Ellipsoid Geodesic

Cite this as:

Weisstein, Eric W. "Ellipsoid Geodesic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipsoidGeodesic.html

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