TOPICS
Search

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 a and polar radius c can be specified parametrically by

x=asinvcosu
(1)
y=asinvsinu
(2)
z=ccosv,
(3)

where a>c. Using the second partial derivatives

(partial^2x)/(partialu^2)=-asinvcosu
(4)
(partial^2x)/(partialv^2)=-asinvcosu
(5)
(partial^2y)/(partialu^2)=-asinvsinu
(6)
(partial^2y)/(partialv^2)=-asinvsinu
(7)
(partial^2z)/(partialu^2)=0
(8)
(partial^2z)/(partialv^2)=-ccosv
(9)

gives the geodesics functions as

P=((partialx)/(partialu))^2+((partialy)/(partialu))^2+((partialz)/(partialu))^2
(10)
=a^2sin^2v
(11)
Q=(partialx)/(partialu)(partialx)/(partialv)+(partialy)/(partialu)(partialy)/(partialv)+(partialz)/(partialu)(partialz)/(partialv)
(12)
=0
(13)
R=((partialx)/(partialv))^2+((partialy)/(partialv))^2+((partialz)/(partialv))^2
(14)
=a^2+(c^2-a^2)sin^2v=a^2(1-e^2sin^2v),
(15)

where

 e=sqrt((a^2-c^2)/(a^2))
(16)

is the eccentricity.

Since Q=0 and P and R are explicit functions of v only, we can use the special form of the geodesic equation

u=c_1intsqrt(R/(P^2-c_1^2P))dv
(17)
=c_1intsqrt((a^2(1-e^2sin^2v))/(a^4sin^4v-c_1^2a^2sin^2v))dv
(18)
=intsqrt((1-e^2sin^2v)/((a/(c_1))^2sin^2v-1))(dv)/(sinv),
(19)

where c_1 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

d=a/(c_1)
(21)
cosphi=(dcosv)/(sqrt(d^2-1)),
(22)

F(phi|m) is an elliptic integral of the first kind with parameter m, and Pi(phi|m,k) 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

x+iy=kappa(sigma(a+u))/(sigma(u)sigma(a))e^(u[eta-zeta(omega+a)])
(23)
x-iy=kappa(sigma(a-u))/(sigma(u)sigma(a))e^(-u[eta-zeta(omega+a)])
(24)
z^2=lambda^2(sigma(omega^('')+u)sigma(omega^('')-u))/(sigma^2(u)sigma^2(a))
(25)

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

The equation of the geodesic can be put in the form

 dphi=(sqrt(1-e^2sin^2v)sina)/(sqrt(sin^2v-sin^2a)sinv)dv,
(26)

where a is the smallest value of v on the curve. Furthermore, the difference in longitude between points of highest and next lowest latitude on the curve is

 pi-2(sqrt(1-e^2sin^2a))/(sina)int_0^kappa(dnu-dn^2u)/(1+cot^2asn^2u)du,
(27)

where the elliptic modulus of the elliptic function is

 k=(ecosa)/(sqrt(1-e^2sin^2a))
(28)

(Forsyth 1960, p. 446).


See also

Ellipsoid Geodesic, Great Circle, Oblate Spheroid

Explore with Wolfram|Alpha

References

Forsyth, A. R. Calculus of Variations. New York: Dover, 1960.Gosper, R. W. "Spheroid Geodesic Integral." math-fun@cs.arizona.edu posting, Sept. 9, 1996.Halphen, G. H. Traité des fonctions elliptiques et de leurs applications fonctions elliptiques, Vol. 2. Paris: Gauthier-Villars, pp. 238-243, 1886-1891.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.

Referenced on Wolfram|Alpha

Oblate Spheroid Geodesic

Cite this as:

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

Subject classifications