From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Calculus-Latitude Calculation
Date: 27 Nov 1995 20:38:43 GMT

In article <48lgd5$j3o@usenetp1.news.prodigy.com>,
Rodney Small <DQHR18A@prodigy.com> wrote:
>The formula for determining the distance in one degree of latitude, 
>according to the International Astronomical Union Ellipsoid, is (in 
>meters) 111,133.35 - 559.84 cosine 2x + 1.17 cosine 4x, where x is the 
>mid-latitude of the arc.  At the equator, therefore, where x = 0, one 
>degree of latitude = 110,574.68 meters.  My question is, what is the 
>value of half a minute (1/120 of a degree) of latitude at the equator?  
>If one simply divides 110,574.68 by 120, the quotient is 921.45567 meters;
>however, this figure is slightly too high because it is the average half 
>minute in one degree of latitude at the equator, not the half minute 
>exactly at the equator.  Because my calculus is rusty, I solicit help on 
>calculating this figure.   Thanks.

It's not calculus, really, it's trigonometry.

If  g(x) is the length of the line from the equator to latitude  x, then
they are saying  g(x+.5)-g(x-5)= a + b cos(2x) + c cos(4x). There is
actually insufficient information to determine  g  uniquely from this
information, but presumably  g  [which is only an approximation to 
reality anyway] is assumed to be a "small" trigonometric polynomial. This
forces  g  to have the form  g(x) = Ax + B sin(2x) + C sin(4x). In order
to determine A, B, and C, compute  g(x+.5)-g(x-.5) = A + 2B sin(1) cos(2x) +
+ 2C sin(2) cos(4x). Thus,  A = a, B = b/(2 sin1), C = c/(2 sin2).

Then you can get the information you want. For example, a curve
centered at latitude  x  and extending an amount  d  in either direction
has length  g(x+d)-g(x-d) = A(2d) + 2B sin(2d) cos(2x) + 2C sin(4d) cos(4x),
which is  a*(2d) + b*(sin(2d)/sin(1)) cos(2x) + c*(sin(4d)/sin(2)) cos(4x). 
As you can see, this recovers the original length when  d=0.5  and the
dominant term is proportional to  d, but as you guessed, the other
terms are not (although the difference between  sin(2d)/sin(1)  and  2d
is pretty slight when  d  is small).

In your case (d=1/240, x=0), I estimate the length to be 
	a*(1/120) + b*(.00833376) + c*(.00833503)=921.45543 meters.  
You can see this is very close to (1/120)*(a+b+c), i.e., to (1/120)
of the value obtained by simply plugging in zero. You have to be
pretty concerned about accuracy to see the difference; indeed, the
difference between the simple calculation and the correct calculation
is sure to be exceeded by the difference between the correct
calculation and the true length of an arc on earth. (Just how accurate
is that 111133.35, anyway?)

dave