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Re: Q: error in Mathematica Book formulae for surface area of ellipsoid?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg14376] Re: Q: error in Mathematica Book formulae for surface area of ellipsoid?
*From*: mtrott (Michael Trott)
*Date*: Thu, 15 Oct 1998 00:29:14 -0400
*Organization*: Wolfram Research, Inc.
*References*: <6vurhp$8lh@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <6vurhp$8lh at smc.vnet.net> wacb at aplcomm.jhuapl.edu (wacb)
writes:
> I've not been successful calculating the surface area of an ellipsoid,
> based on formulae given in the 3rd edition of the Mathematica Book (pg.
> 938). I have tried to verify that these formulae give value that
> correspond to a sphere and spheroids (both oblate and prolate). The
> values I get do not correspond to those I derive using other texts. For
> example, Mathematica calculates a value for a sphere of only 2 Pi.
>
> I believe the error is due to an incomplete definition for theta: the
> expression given in the book has a trailing "/", which suggests to me
> that something is missing. Can anyone confirm this error or give a
> corrected expression?
>
> Thanks.
>
> Bill Christens-Barry
In comp.soft-sys.math.mathematica article <6vurhp$8lh at smc.vnet.net> you
wrote:
> I've not been successful calculating the surface area of an ellipsoid,
> based on formulae given in the 3rd edition of the Mathematica Book (pg.
> 938). I have tried to verify that these formulae give value that
> correspond to a sphere and spheroids (both oblate and prolate). The
> values I get do not correspond to those I derive using other texts. For
> example, Mathematica calculates a value for a sphere of only 2 Pi.
>
> I believe the error is due to an incomplete definition for theta: the
> expression given in the book has a trailing "/", which suggests to me
> that something is missing. Can anyone confirm this error or give a
> corrected expression?
>
> Thanks.
>
> Bill Christens-Barry
The formula for theta has indeed a typo, a square root is missing.
Thanks for catching this. The corrrect formula is:
\[Theta] = ArcSin[Sqrt[1 - c^2/a^2]]
Whith this definition for theta the formula is correct. Let us check
this numerically.
A parametrization of the ellipsoid.
In[1]:=
x=a Cos[\[Phi]] Sin[\[Theta]];
y=b Sin[\[Phi]] Sin[\[Theta]];
z=c Cos[\[Theta]];
Gauss's surface forms
In[2]:=
e=D[{x,y,z},\[Phi]].D[{x,y,z},\[Phi]]
Out[2]=
\!\(b\^2\ Cos[\[Phi]]\^2\ Sin[\[Theta]]\^2 +
a\^2\ Sin[\[Theta]]\^2\ Sin[\[Phi]]\^2\)
In[3]:=
g=D[{x,y,z},\[Theta]].D[{x,y,z},\[Theta]]
Out[3]=
\!\(a\^2\ Cos[\[Theta]]\^2\ Cos[\[Phi]]\^2 + c\^2\ Sin[\[Theta]]\^2 +
b\^2\ Cos[\[Theta]]\^2\ Sin[\[Phi]]\^2\)
In[4]:=
f=D[{x,y,z},\[Phi]].D[{x,y,z},\[Theta]]
Out[4]=
\!\(\(-a\^2\)\ Cos[\[Theta]]\ Cos[\[Phi]]\ Sin[\[Theta]]\ Sin[\[Phi]] +
b\^2\ Cos[\[Theta]]\ Cos[\[Phi]]\ Sin[\[Theta]]\ Sin[\[Phi]]\)
The resulting numerical integral.
In[5]:=
\!\(ANumerical[{a_, b_, c_}] :=
NIntegrate[
Sin[\[Theta]]
\ at \(a\^2\ b\^2\ Cos[\[Theta]]\^2 +
b\^2\ c\^2\ Cos[\[Phi]]\^2\ Sin[\[Theta]]\^2 +
a\^2\ c\^2\ Sin[\[Theta]]\^2\ Sin[\[Phi]]\^2\), \n
\t\t{\[Phi], 0, 2 \[Pi]}, {\[Theta], 0, \[Pi]}]\)
The corrected symbolic formula.
In[6]:=
ASymbolic[{a_, b_, c_}] :=
With[{m = (a^2 (b^2 - c^2))/(b^2 (a^2 - c^2)),
\[Theta] = ArcSin[Sqrt[1 - c^2/a^2]]},
2 Pi (c^2 + (b c^2 EllipticF[\[Theta], m])/Sqrt[a^2 - c^2] +
b Sqrt[a^2 - c^2] EllipticE[\[Theta], m])]
A numerical check.
In[7]:=
ANumerical[{3.,2.,1.}]
Out[7]=
48.882
In[8]:=
ASymbolic[{3.,2.,1.}]
Out[8]=
48.882
For a nearly sphere we obtain 4 Pi:
In[9]:=
ASymbolic[{1. + 10^-8,1. + 10^-8,1. + 10^-10}]
Out[9]=
12.5664
In[10]:=
N[4 Pi]
Out[10]=
12.5664
Be aware that the formula requires a very careful limit for a sphere
because the module of the incomplete elliptic integrals go to 1 in case
of a sphere. The resulting singularities cancel the square root
singularities in ASymbolic. For a safe numerical evaluation one
should have a > b > c.
--
Michael Trott
Wolfram Research, Inc.
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