A spheroid is an Ellipsoid
![\begin{displaymath}
{r^2\cos^2\theta\sin^2\phi\over a^2}+{r^2\sin^2\theta\sin^2\phi\over b^2}+{r^2\cos^2\phi\over c^2} = 1
\end{displaymath}](s2_1332.gif) |
(1) |
with two Semimajor Axes equal. Orient the Ellipse so that the
and
axes are
equal, then
![\begin{displaymath}
{r^2\cos^2\theta\sin^2\phi\over a^2}+{r^2\sin^2\theta\sin^2\phi\over a^2} + {r^2\cos^2\phi\over c^2} = 1
\end{displaymath}](s2_1333.gif) |
(2) |
![\begin{displaymath}
{r^2\sin^2\phi\over a^2} + {r^2\cos^2\phi\over c^2} = 1,
\end{displaymath}](s2_1334.gif) |
(3) |
where
is the equatorial Radius and
is the polar Radius. Here
is the colatitude, so take
to express in terms of latitude.
![\begin{displaymath}
{r^2\cos^2\delta\over a^2} + {r^2\sin^2\delta\over c^2} = 1.
\end{displaymath}](s2_1336.gif) |
(4) |
Rewriting
gives
![\begin{displaymath}
{r^2\over a^2} + r^2\sin^2\delta\left({{1\over c^2} - {1\over a^2}}\right)= 1
\end{displaymath}](s2_1338.gif) |
(5) |
![\begin{displaymath}
r^2\left({1 + a^2\sin^2\delta {a^2-c^2\over c^2a^2}}\right)= r^2\left({1 + \sin^2\delta {a^2-c^2\over c^2}}\right)= a^2,
\end{displaymath}](s2_1339.gif) |
(6) |
so
![\begin{displaymath}
r = a\left({1 + \sin^2\delta {a^2-c^2\over c^2}}\right)^{-1/2}.
\end{displaymath}](s2_1340.gif) |
(7) |
If
, the spheroid is Oblate. If
, the spheroid is Prolate. If
, the spheroid degenerates to a Sphere.
See also Darwin-de Sitter Spheroid, Ellipsoid, Oblate Spheroid, Prolate Spheroid
© 1996-9 Eric W. Weisstein
1999-05-26