Errata-Surface Area of Ellipsoids

[SNNAP Board Admin]

There was a subsequent correction presented:

isb.ri.ccf.org/biomch-l/archives/biomch-l-1996-12/00114.html

Also, unless I'm misunderstanding it and this is what is meant,
isn't "s" (geodetic/geographic) LATITUDE, not longitude (with "t"
being the reduced/parametric latitude)?
Additionally, I've found it extremely helpful to define the
ellipticity as an angle: Oz_o = acos{b/a}; Oz_p = acos{a/b};
thus E = sin{Oz} and t = ArcTan{Tan{s} * sec{Oz}}.

~Kaimbridge~

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

Dear Subscribers:

Recently I posted a summary of responses to my questions
concerning the surface area of an ellipsoid. I also summarized
my own derivations, which included equations for sectors of the
M&M- and football-shaped ellipsoids, where the sector is less
than half the ellipsoid. However, I have found an error in the
two equations for the sectors. The correct method follows:

Let s equal the longitude of the circular edge of the sector.
Then calculate the angle t by one of the equations below. For the
M&M ellipsoid (two major axes a and one minor axis b, with a > b),
use:

t = ArcTan(a * Tan(s)/b)

For the football-shaped ellipsoid (two minor axes b and one major
axis a, again with a > b), use:

t = ArcTan(b * Tan(s)/a)

Then use the equations I gave in my original summary.

The reason for this is that the angle t over which the
surface area is integrated is a parametric angle, and is *not* the
same as the limits to integration (angle s), except at t = s = 0
and t = s = pi/2. If the ellipsoid were a sphere, then A = B, and
the angles t and s would be the same. Since A <> B, however, the
two are different.

I apologize for not catching this error earlier, especially
since this is essentially the same error I made that began the
whole enterprise.

Regards,
Sandy Stewart

Kaimbridge