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Date: 10/21/96 at 9:9:50
From: Jerry Jost
Subject: Re: navigation formula

Hi,

I'm writing a program that deals with aircraft navigation. The only
part of the equations that I'm still having trouble with is finding
the formula that gives the final longitude and latitude if the
starting longitude and latitude, course, and direction of a plane are
known. Let me put it in terms of what the program requires.

I'm taking off in an aircraft from an airport whose latitude and
logitude are known (LatOrigin and LonOrigin). I'm flying a course of
274 degress true (not magnetic). I fly for a distance of 73.9 nautical
(or statute) miles. Given these variables, where would I end up
(LatDest and LonDest)? I guess in mathematical terms, the distance
would be referred to as a great circle distance. Stating the question
in another way; I left Minneapolis airport and flew X nautical miles
at a course of Y degress (true). How do I determine my new current
position (in Latitude and Longitude)?

Thank you very much for your help,
Jerry
```

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Date: 10/24/96 at 16:46:31
From: Doctor Jerry
Subject: Re: navigation formula

Jerry,

Since you refer to a course, I think you must mean a constant compass
heading, that is, all meridians are crossed at the same angle.  If so,
then you are not talking about great circle navigation (except in a
few special cases); rather, you are talking about rhumb line
navigation, which is based on Mercator's projection.

The answer is relatively complicated, depending on your background. I
feel certain that you need some understanding of calculus to do this
problem.

On the chance that you may not have had a calculus course, I won't go
into extreme detail and, moreover, I'll assume a few things to make
the discussion easier for me.  If you indeed want to see the details,
complete with graphs, I guess we'd have to talk about that since it
involves some effort.  I expect that this is written up somewhere, but
I'm not an expert on practical navigation.  What I'll assume is such
things as the radius of the earth is one unit, longitude is measured
from the (x,z)-plane, and I'll use co-latitude instead of latitude.
You can see that I'll be working on the unit sphere, using spherical
coordinates phi and theta, which I'll abbreviate to p and t. These
will be in radians.

I'll give some details, just to give an idea of what's involved.

Under Mercator's projection, a point (p,t) on the sphere is mapped
onto the point (X,Y) = (2t/pi,-(2/pi)ln(tan(p/2))) in an (X,Y)-plane.
This is correct for certain map conventions. Minor changes for other
conventions.  Let Y(p) = -(2/pi)ln(tan(p/2)).

If one flies or sails on a rhumb line, making an angle of b with the
meridians, then p and t are related by the equation
t = (pi/2)[(tan b)Y(p) - k], where k is a constant and can be
determined by knowing a single point (p1,t1) through which the rhumb
line passes.  This could be, for example, your starting point.

Let H(p) = (pi/2)[(tan b)Y(p) - k].

We can now write a parametric vector equation for the rhumb line on
the sphere.  It would be:

r(p) = {cos(H(p))*sin(p),sin(H(p))*sin(p),cos(p)}, p >= p1.

There are formulas for calculating arc length of such a curve.  One
calculates r'(p) and then its length |r'(p)|.  The arc length from
p = p1 to p = p2 is integral from p1 to p2 of |r'(p)|dp.

I know this is too much, too fast.  But you may decide this stuff
isn't for you.

Anyway,  we don't know p2; what we are given is (p1,t1) and the arc
length s.  We must solve the equation for p:

integral from p1 to p of |r'(q)|dq = s

I used q as a "dummy" variable, to avoid confusion with the variable
p.  Newton's method can be used.  Each iteration will require a
numerical integration.

I tested this whole scheme by taking a beginning point (p1,t1), and
ending point (p2,t2), and calculating the arc length s.  I then did
Newton's method starting with (p1,t1) and s, and, to my delight, got
p2.  The number t2 then comes from H(p2).

Are you interested in continuing this discussion?

I wrote up a student project, for a calculus course, on a Mercator's
project last year.  I have been asked to write up a key answer sheet
and your question pushed me into doing this a little earlier than I
planned.  Thanks!  This was good for me.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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