Ellipsoid Geodesic

An Ellipsoid can be specified parametrically by

$$x = a\cos u\sin v$$ (1)

$$y = b\sin u\sin v$$ (2)

$$z = c\cos v.$$ (3)

The Geodesic parameters are then

$$P = \sin^2 v(b^2\cos^2 u+a^2\sin^2 u)$$ (4)

$$Q = {\textstyle{1\over 4}}(b^2-a^2)\sin(2u)\sin(2v)$$ (5)

$$R = \cos^2 v(a^2\cos^2 u+b^2\sin^2 u)+c^2\sin^2 v.$$ (6)

When the coordinates of a point are on the Quadric

$${x^2\over a}+{y^2\over b}+{z^2\over c}=1$$ (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

$${q\,dq\over\sqrt{q(a+q)(b+q)(c+q)(\theta+q)}}\pm {p\,dp\over\sqrt{p(a+p)(b+p)(c+p)(\theta+p)}}=0,$$ (8)

with $\theta$ an arbitrary constant, and the Arc Length element $ds$ is given by

$$-2{ds\over pq}={dq\over\sqrt{q(a+q)(b+q)(c+q)(\theta+q)}} \pm {dp\over\sqrt{p(a+p)(b+p)(c+p)(\theta+p)}},$$ (9)

where upper and lower signs are taken together.

See also Oblate Spheroid Geodesic, Sphere Geodesic

References

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.