Re: Symmetric rational approximations of the surface area of an ellipsoid

anonymous_at_mathforum.org
Date: 06/30/04 

Date: Wed, 30 Jun 2004 12:13:48 +0000 (UTC)

On 29 Jun 2004, David W. Cantrell wrote:
>David W. Cantrell <DWCantrell@sigmaxi.org> wrote in message news:<20040529195723.225$tP@newsreader.com>...
>> [This post is to supersede my original one, in which the last exponent in
>> the numerator of (1) was erroneously typed as being 3. It should be 2, as
>> now shown below. Sorry for the typo!]
>>
>> A remarkable (new?) approximation for the surface area of an ellipsoid is
>> presented, followed by miscellaneous comments.
>>
>> Let a, b, and c denote the lengths of the semiaxes of an ellipsoid, and let
>> P, Q, and R denote the elementary symmetric polynomials
>>
>> P = a+b+c, Q = ab+bc+ca, and R = abc.
>>
>> Then the surface area of the ellipsoid may be approximated,
>> with |relative error| < 0.0013, by
>>
>> [Please view in a fixed-width font.]
>>
>> 15 Q^3 - 7 PQR - 27 R^2
>> 4pi ----------------------- (1)
>> 15( 2 Q^2 + PR )
>>
>> This symmetric rational approximation is notable for its combination of
>> simplicity and accuracy. I wonder if it is new. [My copy of "The surface
>> area of an ellipsoid" by P. A. P. Moran (pp. 511-518 in _Statistics and
>> Probability: Essays in Honor of C. R. Rao_,
 eds. G. Kallianpur, P. R. Krishnah and J. K. Gosh, 

--
If helpful for a search, a better spelling for the above Indian names
would be :
 G. Kalyanpur, P. R. Krishniah or P. R. Krishna and J. K. Ghosh .
--
 North-Holland, Amsterdam, 1982) has, alas, been
>> "temporarily misplaced". That's the only reference which comes to mind
>> which I think might have already given this approximation.]
>
>I've now had the opportunity to look again at the article by Moran.
>(1) is not given there. I must suppose that it is a new approximation.
>
>> The approximation gives the surface area precisely both in the degenerate case, when an axis is of length zero, and in the spherical
case.
 
Quite interesting. But on the contrary,let us look at exact formula of spherical surface area (FWIW) and read it into axisymmetric oblate/prolate ellipsoid surface SA = (2 Pi a * 2 c) where 2 c is height of spherical segment -- (slidable  anywhere on z axis of sphere,even to a set of infinite ellipsoids with tangential contact at z=c).
We can investigate effect of c on surface area. Even at c = 1, C.R.Rao's formula has about 6 percent error. May be the
coefficients 15,7,27,1 need to be recast so that the SA  on this Mma graph would be tangential to SA = c line at c=1 and would also be an even function of c with minimum 0,5 at c=0, doing away with coefficients of odd c powers. This is easy to do, and I propose a simpler (1+c^2)/2 ! However, :) I do not know how to evaluate back
coefficients of {Q^3, R^2, P*Q*R},{P*R,Q^2)( or the next possibility ) to get (1+c^2)/2 for surface area with any proper criterion. Someone please take it from there, for (a not equal to b ) non-axisymmetric ellipsoid case. 
a = 1 ; b = 1 ; 
P = a+b+c; Q = a b+b c+c a;  R = a b c;
SA = 4 Pi*Simplify[(15 Q^3 - 7 P Q R - 27 R^2)/( 2 Q^2 + P R )/15]
gln=(1+c^2)/2
Plot [{SA/(4*Pi),c,1,gln},{c,.9,1.1}];
Plot [{SA/(4*Pi),c,1},{c,-1.6,1.6}];
SAby2pi=(15+76 c+118 c^2+106 c^3 )/(15(2+10 c+9 c^2));
Plot [{SAby2pi,c,1,gln},{c,-1.6,1.6}];

Relevant Pages