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A bit more on the meridional distance (pardon my continual misspelling).<BR>
<BR>
The kernel of the integral in the manual is simply the radius of the earth's surface in<BR>
the plane of the meridian:<BR>
<BR>
rho = a(1-e^2)/(1-e^2sin^2(phi))^(3/2)<BR>
<BR>
where a:earth's semi-major axis, e: eccentricity and phi: geodetic latitude.<BR>
Note that if eccentricity becomes 0 then then rho = a and integrating for<BR>
meridional distance simply becomes a*phi.<BR>
<BR>
The traditional solution has been to do a binomial expansion of the denominator<BR>
and follow with a term-by-term integration of the series. Very messy and keeps<BR>
one out of the bars for a while. The difference between the multiple angle<BR>
series (sin(n*phi)) versus the power series ((sin(phi))^n) depends upon which set<BR>
of integral tables one wishes to use. The power series is computationally<BR>
desirable in minimizing computer time by drastically cutting time spent calling<BR>
the sine function (another power series).<BR>
<BR>
The elliptic integral method (which is also a power series) came about when<BR>
I happened to locate the integral in Gradsbteyn & Rysbik along with a method<BR>
to evaluate the elliptic integral that gave solution which converged to machine<BR>
precision fairly rapidly. The method is slightly slower than the last power<BR>
series method in PROJ4 but much more accurate.<BR>
<BR>
Testing was made by numerical integration with GP-PARI multiple precision package<BR>
although any multiple precision package ought to work.<BR>
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--
_____________________________________________________________
Jerry and the Low Riders: Daisy Mae and Joshua
"The most certain test by which we judge whether a country is
really free is the amount of security enjoyed by minorities"
---Lord Acton, 1907
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