Maximum Difference, Longitude and Latitude

Date: 04/10/2001 at 00:32:40
From: Mini Thomas
Subject: Maximum longitude and latitude difference between two points 
separated by a given distance

I want to find the maximum longitude difference and maximum latitude 
difference between two points A and B on Earth that are separated by 
a distance of 1000km.

I tried the formula theta = cos (LatA)* cos(LatB)* cos(LonA-LonB)+ 
Sin(LatA)* Sin(LatB), where theta in radians =1000KM/EarthDistance.
But I am stuck finding max(LatA - LatB) and max (LonA-LonB) which 
can be applied to within this distance.
Can you please help me?

Date: 04/10/2001 at 14:31:14
From: Doctor Rick
Subject: Re: Maximum longitude and latitude difference between two 
points separated by a given distance

Hi, Mini.

Do you want the maximum latitude and longitude differences for *any* 
pair of points separated by 1000 km? If A is within 1000 km of the 
north or south pole, then B could have any longitude whatsoever; the 
maximum longitude difference is thus 180 degrees.

The maximum latitude difference is the same regardless of where point 
A is, and it isn't hard to calculate. If you draw a circle of radius 
1000 km (that is, a radius following the curvature of the earth) 
around point A, this circle will be tangent to latitude lines at 
points due north and south of A. These points are therefore the 
extremes of latitude difference, and the latitude difference there (in 
radians) is just 1000 km divided by the radius of the earth (R = 6367 
km).

If you're interested in the maximum longitude difference for a *given* 
point A, then we have more work to do. As with latitude, we find the 
points of maximum longitude difference by looking for the points where 
the circle drawn around point A is tangent to a line of longitude. At 
these points, the line joining A and B is perpendicular to the line of 
longitude at B. In other words, A can be reached from B by starting 
out due east or west and following a great circle route. But since 
lines of latitude are not great circles, A and B are NOT at the same 
latitude (unless they are on the equator, which is a great circle).

To find the longitudes in question, pass a line through the center of 
the earth O, perpendicular to the line (circle) of longitude we are 
looking for. Let this line intersect the surface of the earth at C. 
Every line (great circle) from C to the line of longitude meets it at 
right angles, just as lines of longitude meet the equator at right 
angles. The line through C and A therefore meets the line of longitude 
at right angles; thus the point where this line meets the line of 
longitude is the point B that we seek. Since angle AOB is (1000 km/R) 
radians (as in the latitude calculation), and angle COB = pi/2 
radians, and the four points are coplanar, the angle COA must be 
(pi/2 - 1000/R) radians.

Here is where we are so far: given the latitude and longitude of point 
A, we need to find the point C on the equator that is (pi/2 - 1000/R) 
radians of arc away from A. (Actually, there will be two such points, 
one east and one west.) Then we add (or subtract, depending on which 
point we found) pi/2 radians to the longitude of C, and this is the 
longitude of point B. Subtract the longitude of A to find the maximum 
longitude difference at A's latitude.

To find point C, we can use vector algebra, discussed in this 
connection in several places in our archives, including this answer:

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

Letting the radius of the earth be 1 unit, we can identify point C 
(which is on the equator) as (1, 0, 0), on the x-axis of my coordinate 
system. Then point A will be 

  (cos(lat1)*cos(lon1), cos(lat1)*sin(lon1), sin(lat1))

where lon1 is the *difference* in longitude between C and A. The 
scalar product of these two vectors is equal to the product of their 
magnitudes times the cosine of the angle between them. Thus,

  cos(COA) = A.C = cos(lat1)*cos(lon1)

We know lat1, and we know that angle COA is pi/2 - 1000/R. Thus we can 
solve for lon1, the longitude difference between A and C:

  lon1 = arccos(cos(pi/2-1000/R)/cos(lat1)
       = arccos(sin(1000/R)/cos(lat1)

The longitude of point B in the coordinate system that I have chosen 
is pi/2. Therefore the maximum longitude difference is

  max_dlon = pi/2 - lon1
           = arcsin(sin(1000/R)/cos(lat1))

Let's check out this formula. When lat1 = 0 (A is on the equator), we 
expect the maximum longitude difference to be 1000/R. In fact, since 
cos(0) = 1, we get the arcsine of the sine of 1000/R, which is 1000/R.

When lat1 = pi/2 - 1000/R (so that B could be at the north pole), we 
expect the maximum longitude difference to be pi/2 (since the 1000-km 
circle is tangent at the pole to the longitude line 90 degrees from 
A's longitude). In fact, since cos(pi/2 - 1000/R) = sin(100/R), we 
have max_dlon = arcsin(1) = pi/2. At higher latitudes, 
cos(lat1) < sin(1000/R), and the arcsin of a number greater than 1 
does not exist. As we saw above, at these latitudes it is actually 
possible for point B to have any longitude.

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