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Walking Around the World


Date: 02/08/97 at 18:02:14
From: Anonymous
Subject: Calculating new position and heading after a walk around the 
world

I've been trying to solve the following problem, but as I don't know 
any spherical geometry (or whatever it is that's needed). I haven't 
had much luck so far.

Given someone's starting point on the earth (in degrees longitude and
latitude), and the direction he is initially facing, if he travels in 
a straight line a certain distance, where will he end up (in degrees/
radians) and in what direction will he now be facing?

I really want to get this down to a set of formulae which I can put 
the 4 given values into (longitude, latitude, direction, distance) and 
get out the 3 values I'm looking for (new longitude, new latitude, new
direction).


Date: 02/13/97 at 12:59:12
From: Doctor Mitteldorf
Subject: Re: Calculating new position and heading after a walk around 
the world

Dear Kluj,

I've been working periodically on your question about headings, 
latitude and longitude.  I have an answer for you, and I'll describe 
briefly where it comes from.  Let me know if you want more details, or 
if the derivation either too advanced or too elementary.

First, definitions:
  
Start at a point with given longitude, we'll call this long1, and 
given latitude, we'll call this lat1.   

Travel in a compass heading, we'll call this hdng1, and go a certain 
distance, we'll call this dist.  The compass heading is measured from 
due north, which is equal to 0.

Define alpha = dist/[earth radius] to be the angular distance covered 
on the earth's surface.

You're looking for coordinates of the point that you reach, which 
we'll call lat2 and long2.
  
You also want to know your new compass heading, which we'll call 
hdng2.

The answer:

First calculate the new latitude:

1) sin(lat2) = cos(hdng1)*cos(lat1)*sin(alpha) + sin(lat1)*cos(alpha)

Then calculate the new longitude, using your result for lat2:

                            cos(alpha) - sin(lat1)*sin(lat2) 
2)    cos(long2-long1) =   ----------------------------------
                                  cos(lat1)*cos(lat2)

Again, using lat2 you can calculate the new compass heading:

                      sin(lat1) - sin(lat2)*cos(alpha) 
3)    cos(hdng2) =   ----------------------------------
                           cos(lat2)*sin(alpha)

hdng2 is the compass setting from point 2 to point 1; it's just 180 
degrees from the compass heading that one has when traveling from
point 1 to point 2 and continuing on from there.

The derivation:

I'm a physicist, and I find it natural to think in terms of vectors.  
Going back and forth between expressing the vectors in spherical 
coordinates (r, theta, phi) and in cartesian coordinates (x,y,z) gives 
you a wealth of identities, which are sufficient to solve the kind of 
problem you've posed.
   
The first useful result from this is that the scalar product (dot 
product) between two vectors can be expressed in two different ways:  
In cartesian coordinates, it's x1*x2 + y1*y2 + z1*z2.  In polar 
coordinates, it's r1*r2*cos(alpha), where alpha is the angle between 
the two vectors.  Now, on the surface of the earth, the angle alpha is 
just the distance between the points divided by the earth's radius.  
This gives you a very useful expression for the distance between any 
two points on the earth's surface.  The quantities x1/r1, y1/r1 and 
z1/r1 are simply related to latitude and longitude of point 1. By 
definition:

            z1/r1 = sin(lat1) 
            x1/r1 = cos(lat1)*sin(long1)
            y1/r1 = cos(lat1)*cos(long1)

Identifying the cartesian and polar expressions for the dot product 
gives:

cos(alpha)= sin(lat1)*sin(lat2) + cos(lat1)*cos(lat2)*cos(long1-long2)

This is half of your problem right here: it's an expression for the 
distance between any two points on the globe in terms of their 
coordinates.

The other expression you need involves compass headings.  Given that 
you're at point 2 with coordinates as above, and you want to get to 
point 1, what direction should you travel?  The answer is given in 
equation (3) at the top of my answer.  The derivation for this also 
comes from vector manipulations in polar coordinates, but it's a 
little more involved than the one above.

Let r2u be a unit vector from the center of the earth's surface to 
point 2. This unit vector is just a vector that's one unit long, but 
pointing in the same direction as the vector r2.  Let zu be the unit 
vector in the z direction.  Then you can construct a vector that 
points "due north" along the earth's surface from point 2 as: 

       north pointing vector = zu - (zu.r2u)r2u

The period in this expression connotes the scalar product of two 
vectors. Similarly, a vector pointing along the earth's surface from 
point 2 toward point 1 (on a great circle) can be written as:

       heading vector = r1u - (r1u.r2u)r2u

The cosine of the heading angle is just the angle between these two 
vectors. You can get this by taking the scalar product of the two 
vectors and then dividing by the length of each vector separately.  
This is exactly what I did to derive equation (3).

I'm hoping at least the formulas are useful to you, and if you want to 
go through the details of the derivation, please write back.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
College Calculus

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