Geometry of an oblate spheroid

Posted on by John

We all live on an oblate spheroid [1], so it could be handy to know a little about oblate spheroids.

Eccentricity

Conventional notation uses a for the equatorial radius and c for the polar radius. Oblate means a > c. The eccentricity e is defined by

e = \sqrt{1 - \frac{c^2}{a^2}}

For a perfect sphere, a = c and so e = 0. The eccentricity for earth is small, e = 0.08. The eccentricity increases as the polar radius becomes smaller relative to the equatorial radius. For Saturn, the polar radius is 10% less than the equatorial radius, and e = 0.43.

Volume

The volume of an oblate spheroid is simple:

V = \frac{4}{3}\pi a^2 c

Clearly we recover the volume of a sphere when a = c.

Surface area

The surface area is more interesting. The surface of a spheroid is a surface of rotation, and so can easily be derived. It works out to be

A = 2\pi a^2 + \pi \frac{c^2}{e} \log\left(\frac{1 +e}{1 - e} \right)

It’s not immediately clear that we get the area of a sphere as c approaches a, but it becomes clear when we expand the log term in a Taylor series.

A = 2\pia^2 + 2\pi c^2\left( 1 + \frac{e^2}{3} + \frac{e^4}{5} + \frac{e^6}{7} + \cdots \right)

This suggests that an oblate spheroid has approximately the same area as a sphere with radius √((a² + c²)/2), with error on the order of e².

If we set a = 1 and let c vary from 0 to 1, we can plot how the surface area varies.

plot c vs area

Here’s the corresponding plot where we use the eccentricity e as our independent variable rather than the polar radius c.

plot of area as a function of eccentricity

More differential geometry posts

[1] Not in a yellow submarine as has been suggested.

2 thoughts on “Geometry of an oblate spheroid”

  1. Phil

    Pardon me if I’m wrong. I’m here to be intellectually stretched.

    For a sphere then e is zero and the second term in the first area equation would get log(1). That’s also zero and cancels the second term, leaving the usual results for spheres because a is r. True?

  2. John

    Phil,

    In the first equation for area, you can’t let e -> 0 directly because the log term goes to zero but the 1/e term goes to infinity.

    In the second equation for area, you can let e-> 0 and you get 4 pi r^2, since e -> 0 implies a and c converge to a common r.

    You could use L’Hopital’s rule to take the limit in the first equation and you’d end up at the same place.

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