A ``squashed'' Spheroid for which the equatorial radius
is greater than the polar radius
, so
. To
first approximation, the shape assumed by a rotating fluid (including the Earth,
which is ``fluid'' over
astronomical time scales) is an oblate spheroid. The oblate spheroid can be specified parametrically by the usual
Spheroid equations (for a Spheroid with z-Axis as the symmetry axis),
with
,
, and
. Its Cartesian equation is
![\begin{displaymath}
{x^2+y^2\over a^2}+{z^2\over c^2}=1.
\end{displaymath}](o_15.gif) |
(4) |
The Ellipticity of an oblate spheroid is defined by
![\begin{displaymath}
e\equiv\sqrt{a^2-c^2\over a^2},
\end{displaymath}](o_16.gif) |
(5) |
so that
![\begin{displaymath}
1-e^2 = {c^2\over a^2}.
\end{displaymath}](o_17.gif) |
(6) |
Then the radial distance from the rotation axis is given by
![\begin{displaymath}
r(\delta) = a\left({1+{e^2\over 1-e^2}\sin^2\delta}\right)^{-1/2}
\end{displaymath}](o_18.gif) |
(7) |
as a function of the Latitude
.
The Surface Area and Volume of an oblate spheroid are
An oblate spheroid with its origin at a Focus has equation
![\begin{displaymath}
r={a(1-e^2)\over 1+e\cos\phi}.
\end{displaymath}](o_24.gif) |
(10) |
Define
and expand up to Powers of
,
Expanding
in Powers of Ellipticity to
therefore yields
![\begin{displaymath}
{r\over a} = 1-{\textstyle{1\over 2}}(e^2+e^4-2e^4+6e^6)\sin...
...6)\sin^4\delta-{\textstyle{15\over 8}} e^6\sin^6\delta+\ldots.
\end{displaymath}](o_36.gif) |
(14) |
In terms of Legendre Polynomials,
The Ellipticity may also be expressed in terms of the Oblateness (also called Flattening), denoted
or
.
![\begin{displaymath}
\epsilon \equiv {a-c\over a}
\end{displaymath}](o_44.gif) |
(16) |
![\begin{displaymath}
c = a(1-\epsilon)
\end{displaymath}](o_45.gif) |
(17) |
![\begin{displaymath}
c^2 = a^2(1-\epsilon)^2
\end{displaymath}](o_46.gif) |
(18) |
![\begin{displaymath}
(1-\epsilon)^2 = 1-e^2,
\end{displaymath}](o_47.gif) |
(19) |
so
![\begin{displaymath}
\epsilon=1-\sqrt{1-e^2}
\end{displaymath}](o_48.gif) |
(20) |
and
![\begin{displaymath}
e^2 = 1-(1-\epsilon )^2 = 1-(1-2\epsilon +\epsilon^2) = 2\epsilon -\epsilon^2
\end{displaymath}](o_49.gif) |
(21) |
![\begin{displaymath}
r = a\left[{1 + {2\epsilon -\epsilon^2\over (1-\epsilon)^2} \sin^2\delta}\right]^{-1/2}.
\end{displaymath}](o_50.gif) |
(22) |
Define
and expand up to Powers of
Expanding
in Powers of the Oblateness to
yields
![\begin{displaymath}
{r\over a} = 1-{\textstyle{1\over 2}}(2\epsilon+3\epsilon^2-...
...lon^2+6\epsilon^3)\sin^4\delta+8\epsilon^3\sin^6\delta+\ldots.
\end{displaymath}](o_58.gif) |
(26) |
In terms of Legendre Polynomials,
To find the projection of an oblate spheroid onto a Plane, set up a coordinate system such that the z-Axis is
towards the observer, and the
-axis is in the Plane of the page. The equation for an oblate spheroid is
![\begin{displaymath}
r(\theta)=a\left[{1+{2\epsilon-\epsilon^2 \over (1-\epsilon)^2}\cos^2\theta}\right]^{-1/2}.
\end{displaymath}](o_62.gif) |
(28) |
Define
![\begin{displaymath}
k\equiv {2\epsilon-\epsilon^2 \over (1-\epsilon)^2},
\end{displaymath}](o_63.gif) |
(29) |
and
. Then
![\begin{displaymath}
r(\theta) =a[1+k(1-x^2)]^{-1/2} = a(1+k-kx^2)^{-1/2}.
\end{displaymath}](o_65.gif) |
(30) |
Now rotate that spheroid about the
-axis by an Angle
so that the new symmetry axes for the spheroid are
,
, and
. The projected height of a point in the
Plane on the
-axis is
To find the highest projected point,
![\begin{displaymath}
{dy\over d\theta} = {{a\sin(B-\theta)}\over (1+k\cos^2\theta...
...theta)\cos\theta\sin\theta \over (1+k\cos^2\theta)^{3/2}} = 0.
\end{displaymath}](o_74.gif) |
(32) |
Simplifying,
![\begin{displaymath}
\tan(B-\theta)(1+k\cos^2\theta)+k\cos\theta\sin\theta = 0.
\end{displaymath}](o_75.gif) |
(33) |
But
Plugging (34) into (33),
![\begin{displaymath}
{\sqrt{1-x^2}\tan B-x \over\sqrt{1-x^2}+x\tan B} [1+k(1-x^2)]+kx\sqrt{1-x^2} = 0
\end{displaymath}](o_79.gif) |
(35) |
and performing a number of algebraic simplifications
![\begin{displaymath}
(\sqrt{1-x^2}\tan B-x) (1+k-kx^2)+kx\sqrt{1-x^2}(\sqrt{1-x^2}+x\tan B) = 0
\end{displaymath}](o_80.gif) |
(36) |
|
|
|
(37) |
![\begin{displaymath}
(1+k)\tan B\sqrt{1-x^2}-kx(1-x^2)-x+kx(1-x^2) = 0
\end{displaymath}](o_83.gif) |
(38) |
![\begin{displaymath}
(1+k)\tan B\sqrt{1-x^2}=x
\end{displaymath}](o_84.gif) |
(39) |
![\begin{displaymath}
(1+k)^2\tan^2 B(1-x^2)=x^2
\end{displaymath}](o_85.gif) |
(40) |
![\begin{displaymath}
x^2[1+(1+k)^2\tan^2 B]=(1+k)^2\tan^2 B
\end{displaymath}](o_86.gif) |
(41) |
finally gives the expression for
in terms of
and
,
![\begin{displaymath}
x^2={\tan^2 B(1+k)^2 \over 1+(1+k)^2\tan^2 B}.
\end{displaymath}](o_87.gif) |
(42) |
Combine (30) and (31) and plug in for
,
Now re-express
in terms of
and
, using
,
so
![\begin{displaymath}
1+k = \left({a\over c}\right)^2.
\end{displaymath}](o_94.gif) |
(45) |
Plug (44) and (45) into (43) to obtain the
Semiminor Axis of the projected oblate spheroid,
We wish to find the equation for a spheroid which has been rotated about the
-axis by Angle
, then the
-axis by Angle
Now, in the original coordinates
, the spheroid is given by the equation
![\begin{displaymath}
{{x'}^2\over a^2}+{{y'}^2\over c^2}+{{z'}^2\over a^2} = 1,
\end{displaymath}](o_107.gif) |
(48) |
which becomes in the new coordinates,
|
|
|
(49) |
Collecting Coefficients,
![\begin{displaymath}
Ax^2+By^2+Cz^2+Dxy+Exz+Fyz=1,
\end{displaymath}](o_110.gif) |
(50) |
where
![$\displaystyle A$](o_111.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {\cos^2 P+\sin^2 B\sin^2 P\over a^2}+{\cos^2 B\sin^2P\over c^2}$](o_112.gif) |
(51) |
![$\displaystyle B$](o_113.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {\sin^2 P+\sin^2 B\cos^2 P\over a^2}+{\cos^2 B\cos^2 P\over c^2}$](o_114.gif) |
(52) |
![$\displaystyle C$](o_115.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {\cos^2 B\over a^2}+{\sin^2 B\over c^2}$](o_116.gif) |
(53) |
![$\displaystyle D$](o_117.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 2\cos P\sin P\left({{1-\sin^2 B\over a^2}-{\cos^2 B\over c^2}}\right)$](o_118.gif) |
|
|
![$\textstyle =$](o_7.gif) |
![$\displaystyle 2\cos P\sin P\cos^2 B\left({{1\over a^2}-{1\over c^2}}\right)$](o_119.gif) |
(54) |
![$\displaystyle E$](o_120.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 2\sin B\cos B\sin P\left({{1\over b^2}-{1\over a^2}}\right)$](o_121.gif) |
(55) |
![$\displaystyle F$](o_122.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 2\sin B\cos B\cos P\left({{1\over a^2}-{1\over b^2}}\right).$](o_123.gif) |
(56) |
If we are interested in computing
, the radial distance from the symmetry axis of the spheroid (
) corresponding to
a point
![\begin{displaymath}
Cz^2+(Ex+Fy)z+(Ax^2+By^2+Dxy-1) = Cz^2+G(x,y)z+H(x,y) = 0,
\end{displaymath}](o_124.gif) |
(57) |
where
can now be computed using the quadratic equation when
is given,
![\begin{displaymath}
z={-G(x,y)\pm\sqrt{G^2(x,y)-4CG(x,y)}\over 2C}.
\end{displaymath}](o_130.gif) |
(60) |
If
, then we have
and
, so (51) to (56) and
(58) to (59) become
![$\displaystyle A$](o_111.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {1\over a^2}$](o_134.gif) |
(61) |
![$\displaystyle B$](o_113.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {\sin^2 B\over a^2}+{\cos^2 B\over b^2}$](o_135.gif) |
(62) |
![$\displaystyle C$](o_115.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle {\cos^2 B\over a^2}+{\sin^2 B\over b^2}$](o_136.gif) |
(63) |
![$\displaystyle D$](o_117.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 0$](o_137.gif) |
(64) |
![$\displaystyle E$](o_120.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 0$](o_137.gif) |
(65) |
![$\displaystyle F$](o_122.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle 2\sin B\cos B\left({{1\over a^2}-{1\over b^2}}\right)$](o_138.gif) |
(66) |
![$\displaystyle G(x,y)$](o_125.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle Fy = 2y\sin B\cos B\left({{1\over a^2}-{1\over b^2}}\right)$](o_139.gif) |
(67) |
![$\displaystyle H(x,y)$](o_127.gif) |
![$\textstyle \equiv$](o_28.gif) |
![$\displaystyle Ax^2+By^2-1$](o_140.gif) |
|
|
![$\textstyle =$](o_7.gif) |
![$\displaystyle {x^2\over a^2}+y^2\left({{\sin^2 B\over a^2}+{\cos^2 B\over b^2}}\right)-1.$](o_141.gif) |
(68) |
See also Darwin-de Sitter Spheroid, Ellipsoid, Oblate Spheroidal Coordinates, Prolate Spheroid,
Sphere, Spheroid
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 131, 1987.
© 1996-9 Eric W. Weisstein
1999-05-26