Stewart, Sandy 443-6113

12-24-1996, 12:28 AM

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

************************************************** **********************

* Sandy F.C. Stewart, PhD *

* Hydrodynamics & Acoustics Branch *

* Food & Drug Administration *

* Rockville, MD 20851 *

* *

* 301-443-6113 * fax 301-443-1343 * sxs@fdadr.cdrh.fda.gov *

************************************************** **********************

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

************************************************** **********************

* Sandy F.C. Stewart, PhD *

* Hydrodynamics & Acoustics Branch *

* Food & Drug Administration *

* Rockville, MD 20851 *

* *

* 301-443-6113 * fax 301-443-1343 * sxs@fdadr.cdrh.fda.gov *

************************************************** **********************