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