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.