Latitude and Longitude of a Point Halfway between Two Points

Date: 10/10/2001 at 12:14:09
From: Chuck Baker
Subject: Determining lat and long of a point between two given points

Hello all,

I've read some interesting questions about how to determine the 
distances between two points, but how can one determine a position 
halfway between two points? I would like to know how to determine the 
latitude and longitude of a point halfway between New York and Los 
Angeles.

My guess is that I have to first determine the length of arc from the 
North Pole to the desired point, and then use the half angle formula 
to find the other unknowns. I'm a little rusty, so how does one do 
this?

Date: 10/11/2001 at 11:41:08
From: Doctor Rick
Subject: Re: Determining lat and long of a point between two given 
points

Hi, Chuck.

This will probably be a little more complicated than you think. The 
easiest way for me to do it is to think in terms of vectors. Some of 
the items in our Archives on the topic of latitude and longitude use 
the vector approach, so you can see them for background. For example:

   Distance Between Two Points on the Earth
   http://mathforum.org/library/drmath/view/51722.html   

Let's call the two points A and B, and choose a rectangular coordinate 
system in which the equator is in the x-y plane and the longitude of 
point A is in the x-z plane. Let lat1 be the latitude of A, let lat2 
be the latitude of B, and let dlat be the longitude of B minus the 
longitude of A. Finally, use distance units such that the radius of 
the earth is 1. Then the vectors from the center of the earth to A and 
B are

  A = (cos(lat1), 0, sin(lat1))
  B = (cos(lat2)*cos(dlon), cos(lat2)*sin(dlon), sin(lat2))

Point C, the midpoint of the shortest line between A and B, lies along 
the sum of vectors A and B. (This works because A and B have the same 
length, so the sum of the vectors is the diagonal of a rhombus, and 
this diagonal bisects the angle.)

  A+B = (cos(lat1)+cos(lat2)*cos(dlon),
         cos(lat2)*sin(dlon),
         sin(lat1)+sin(lat2))

To get the actual vector C, we need to scale this vector to length R 
so it ends at the surface of the earth. Thus we have to divide it by 
|A+B|, that is, the length of vector A+B. That would get pretty messy. 
But we can find the latitude lat3 and longitude difference (lon3-lon1) 
by looking at ratios of the coordinates of A+B, and these ratios are 
the same whether we scale the vector or not. To see this, look back at 
the formula for vector B. Knowing that vector, we can recover lat2 and 
dlon:

  dlon = tan^-1(B_y/B_x)
  lat2 = tan^-1(B_z/sqrt(B_x^2+B_y^2))

Here, B_x, B_y, and B_z are the x, y, and z coordinates of vector B.

We can do the same thing with vector A+B to find the latitude and 
longitude of point C:

  dlon3 = tan^-1(cos(lat2)*sin(dlon)/
                 (cos(lat1)+cos(lat2)*cos(dlon)))
  lat3 = tan^-1((sin(lat1)+sin(lat2))/
                sqrt((cos(lat1)+cos(lat2)*cos(dlon))^2+
                     (cos(lat2)*sin(dlon))^2))

That's the formula you seek. Since both formulas involve division, we 
must consider the special cases. The longitude calculation fails when 
C_x = 0, that is, when C is 90 degrees away from A in longitude, so 
dlon3 will be +90 or -90; the sign depends on the sign of dlon. The 
latitude calculation fails when C_x and C_y are both zero, thus we 
know that in this case, lat3 = 0. A complete algorithm will check for 
these cases, but they won't occur if you're interested only in the 
continental US.

When I plug in the latitudes and longitudes for LA and NYC, I get:

LA 34.122222 118.4111111
NYC 40.66972222 73.94388889
midpt 39.54707861 97.201534

I hope this is helpful to you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/