Area of an Ellipsoid

Date: 09/28/2001 at 11:32:39
From: Jason Barnabas
Subject: Area of an ellipsoid

How do you calculate (or even estimate) the area of an ellipsoid that 
is neither oblate nor prolate (a<>b<>c<>a)?

Date: 09/28/2001 at 16:34:09
From: Doctor Mitteldorf
Subject: Re: Area of an ellipsoid

The solution when the two axes are equal can be found in the Dr. Math 
FAQ:

   Ellipsoid, Torus, Spherical Polygon Formulas
   http://mathforum.org/dr.math/faq/formulas/faq.ellipsoid.html   

I'm more comfortable with numerical methods, and that's how I can tell 
you to solve it. I'll just outline the procedure - if you need 
more details, please write back.

Let your ellipsoid be (x/a)^2 + (y/b)^2 + (z/c)^2 = 1. Set up a double 
integral over the portion of the x-y plane enclosed by the z = 0 
projection of the ellipsoid:  

  X integral from -a to + a
     Y integral from -b(1-(x/a)^2) to b(1-(x/a)^2)

The integrand is the ratio of the size of a little bit of area in the 
ellipsoid to dxdy in the xy plane; this ratio is 1/cos(phi), where phi 
is the angle between the local normal to the ellipsoid and the z axis.  

To find the normal vector, use the fact that the function

   F = (x/a)^2 + (y/b)^2 + (z/c)^2

is constant all along the surface. Therefore, the direction in which F 
changes most steeply (del F) is the local normal to the surface. 

The components of del F are, respectively, the partial derivatives of 
F with respect to x, y and z. Once you have this vector, you need to 
divide by its magnitude to make a unit vector, and then the z 
component is the very cosine that you are seeking.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 09/28/2001 at 16:38:47
From: Doctor Rob
Subject: Re: Area of an ellipsoid

Thanks for writing to Ask Dr. Math, Jason.

The exact area cannot be found in closed form in terms of familiar
functions of a, b, and c, and familiar constants. It can be computed
in terms of the value of the integral of an elliptic integral (!).

I know of no good way to approximate the surface area in symbolic
form. Once fixed values are given to a, b, and c, one may use a
convergent power series. One may also do numerical integration to
evaluate the elliptic integral, and then its integral.

In my view, it is a minor miracle that the prolate and oblate
spheroids have closed-form formulas, given the above facts.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/