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    Re: Position lines, crossing.
    From: Dave Walden
    Date: 2006 Dec 19, 15:00 -0800

    The previous post answers were based on a Monte-Carlo result.  There's
    no doubt, that Monte-Carlo methods are a powerful tool.  But I feel
    sometimes they're "too powerful."  They make is so easy to get to a
    numerical result, one can "stop thinking" about the problem too soon.
    
    After producing the result above, I felt sure the results were "in the
    equations" somewhere, if I just looked hard enough.  Indeed, as
    expected, thats true. The below compact FORTRAN program reproduces the
    average distance results for various numbers of equally spaced lines of
    position based on analytics.  (Well, some numerics, since the
    exponetial integrals involved have no closed form solutions.)  No
    comments/clues are provided so you can enjoy the chase yourself.
    
    Details if there's interest.
    
        pi=4.*atan(1.)
        print*,"number of                   average distance error"
        print*,"equally spaced LOP's   (SD obs error 0.5)"
        do 10 ilop=3,8
        dx=0.
        do 9 jj=1,ilop
        dx=dx+((ilop/2.)/(ilop/2.)**2*sin((-75.+jj*360./ilop)*pi/180.))**2
    9   continue
        sd=sqrt(dx*.5*.5)
        sum=0.
        do 1 i=-300,300,5
        do 1 j=-300,300,5
        x=i/100.
        y=j/100.
        sum=sum+1./sqrt(2.)*sqrt(x**2+y**2)*1./(sqrt(2.*PI)*SD)*EXP(-X**2/(2*SD**2))
         .     *1./(SQRT(2.*PI)*SD)*EXP(-Y**2/(2.*SD**2))
    1   continue
        print*,'      ',ilop,'                 ',sum*.05**2*sqrt(2.)
    10  continue
        end
    
    
    knoppix@ttyp0[knoppix]$ g77 big.f -obig.exe
    knoppix@ttyp0[knoppix]$ ./big.exe
     nunmber of                   average distance error
     equally spaced LOP's   (SD obs error 0.5)
            3                   0.511632323
            4                   0.443074971
            5                   0.396285892
            6                   0.361745924
            7                   0.334897667
            8                   0.313253939
    knoppix@ttyp0[knoppix]$
    knoppix@ttyp0[knoppix]$
    
    
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