Spheroid

A spheroid is an Ellipsoid

$${r^2\cos^2\theta\sin^2\phi\over a^2}+{r^2\sin^2\theta\sin^2\phi\over b^2}+{r^2\cos^2\phi\over c^2} = 1$$ (1)

with two Semimajor Axes equal. Orient the Ellipse so that the $a$ and $b$ axes are equal, then

$${r^2\cos^2\theta\sin^2\phi\over a^2}+{r^2\sin^2\theta\sin^2\phi\over a^2} + {r^2\cos^2\phi\over c^2} = 1$$ (2)

$${r^2\sin^2\phi\over a^2} + {r^2\cos^2\phi\over c^2} = 1,$$ (3)

where $a$ is the equatorial Radius and $c$ is the polar Radius. Here $\phi$ is the colatitude, so take $\delta\equiv \pi/2-\phi$ to express in terms of latitude.

$${r^2\cos^2\delta\over a^2} + {r^2\sin^2\delta\over c^2} = 1.$$ (4)

Rewriting $\cos^2\delta=1-\sin^2\delta$ gives

$${r^2\over a^2} + r^2\sin^2\delta\left({{1\over c^2} - {1\over a^2}}\right)= 1$$ (5)

$$r^2\left({1 + a^2\sin^2\delta {a^2-c^2\over c^2a^2}}\right)= r^2\left({1 + \sin^2\delta {a^2-c^2\over c^2}}\right)= a^2,$$ (6)

so

$$r = a\left({1 + \sin^2\delta {a^2-c^2\over c^2}}\right)^{-1/2}.$$ (7)

If $a>c$, the spheroid is Oblate. If $a<c$, the spheroid is Prolate. If $a=c$, the spheroid degenerates to a Sphere.

See also Darwin-de Sitter Spheroid, Ellipsoid, Oblate Spheroid, Prolate Spheroid