V. P. Stokes

02-01-1997, 05:34 AM

This is a follow-up to a previously posted message by Dr. Sandy

Stewart on the surface area of an ellipsoid. This was a very

interesting posting since what would intutively seem to be a

rather simple problem (since an ellipsoid is such a simple

object) was not simple at all.

The attached file (a 2-page PDF document) gives a short derivation

of the definite integral for the surface area of an ellipsoid.

You will need Reader 3.0 to display/print it which is free and can be

downloaded from http://www.adobe.com/. This definite integral can

be used to obtain the surface area of a general (triaxial) ellipsoid:

2 2 2

x + y + z

--- --- --- = 1

2 2 2

a b c

for any a,b,c > 0. The actual numerical integration is easily

accomplished in Mathematica with the precision of the

estimated area chosen by the user.

Also, it may be reassuring to know that the last symbolic

formula given (surface area of an asymmetric ellipsoid) in

Dr. Stewart's summary is quite likely correct -- at least I

was able to find the same equation. However, two warnings in

using this equation: 1) it can be numerically unstable when

the semi-axes (a,b,c) or close (in value) even if a > b > c,

2) the elliptic integrals given in some references can have

opposite ordering to those shown in the symbolic equation. Also,

it is interesting that after extensive testing of the numercial

evaluation of the definite integral and direct evaluation of

the symbolic equation it seems that the numerical evaluation

is more accurate!

--VPS

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

V. P. Stokes *

BMC, Neuroscience *

Karolinska Institute Quality Has No Fear *

Box 5626 of Time *

S-114 86 Stockholm *

SWEDEN *

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

Stewart on the surface area of an ellipsoid. This was a very

interesting posting since what would intutively seem to be a

rather simple problem (since an ellipsoid is such a simple

object) was not simple at all.

The attached file (a 2-page PDF document) gives a short derivation

of the definite integral for the surface area of an ellipsoid.

You will need Reader 3.0 to display/print it which is free and can be

downloaded from http://www.adobe.com/. This definite integral can

be used to obtain the surface area of a general (triaxial) ellipsoid:

2 2 2

x + y + z

--- --- --- = 1

2 2 2

a b c

for any a,b,c > 0. The actual numerical integration is easily

accomplished in Mathematica with the precision of the

estimated area chosen by the user.

Also, it may be reassuring to know that the last symbolic

formula given (surface area of an asymmetric ellipsoid) in

Dr. Stewart's summary is quite likely correct -- at least I

was able to find the same equation. However, two warnings in

using this equation: 1) it can be numerically unstable when

the semi-axes (a,b,c) or close (in value) even if a > b > c,

2) the elliptic integrals given in some references can have

opposite ordering to those shown in the symbolic equation. Also,

it is interesting that after extensive testing of the numercial

evaluation of the definite integral and direct evaluation of

the symbolic equation it seems that the numerical evaluation

is more accurate!

--VPS

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

V. P. Stokes *

BMC, Neuroscience *

Karolinska Institute Quality Has No Fear *

Box 5626 of Time *

S-114 86 Stockholm *

SWEDEN *

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