When the coordinates of a point are on the quadric
(7)
and expressed in terms of the parameters and of the confocal quadrics passing through that point (in other
words, having ,
,
,
and ,
,
for the squares of their semimajor axes), then the equation of a geodesic
can be expressed in the form
(8)
with
an arbitrary constant, and the arc length element is given by
Dragovich, V. J. Phys. A29, 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.
and 
of the confocal quadrics passing through that point (in other
words, having 
,
,
,
and 
,
,
for the squares of their semimajor axes), then the equation of a geodesic
can be expressed in the form
an arbitrary constant, and the arc length element 
is given by
