Longitude-Latitude Product and Distance

Date: 02/14/2001 at 01:01:55
From: Tariq Abdullahi
Subject: Longitude and Latitude

I am currently working on a school project that involves determining 
the proximity of locations on the earth's surface in relation to each 
other, using longitude and latitude in decimal degrees. For example, 
I would take the product of the longitudinal and latitudinal 
coordinates (in decimal degrees) of one location (call it A), and 
compare it to the product of another location (call it B), to decide 
which is the closest point to another location C. Or I would take the 
long-lat product of point A and compare it to a list of long-lat 
coordinate products to determine which is the closest location to 
point A.

I have tried to implement this with a number of examples, and found 
that it works in those cases. I have also tried to do a mathematical 
proof to determine if it is always the case that the closer the 
long-lat product of point A is to the long-lat product of any point 
C, the closer the locations actually are. I haven't had much luck as 
I am at a loss as to how to approach it. I haven't taken much 
geometry or trig so studying textbooks at the library didn't help. It 
is a most unorthodox approach, but I wonder if it can be proven 
mathematically.

I came across your service while I was looking for information 
related to my project, and decided to give it a try. Please, if you 
have any information in regards to this, I would like to hear it.

Thank you.
Tariq in Cairo

Date: 02/14/2001 at 08:53:28
From: Doctor Rick
Subject: Re: Longitude and Latitude

Hi, Tariq.

Are you saying that you multiply together the latitude and longitude 
of one location, repeat for a second location, and then take the 
difference?

     Location 1: latitude a1, longitude b1
     Location 2: latitude a2, longitude b2

     f(1,2) = |a1*b1 - a2*b2|

Then you propose that this function is a monotonically increasing 
function of the distance between locations 1 and 2?

The locations (long 30, lat 40) and (long 40, lat 30) have the same 
long-lat product, so the difference is zero. There are certainly 
locations that are closer together but have a greater long-lat 
product difference than zero. Your proposal is disproved by 
counterexample.

In general, if we look at the neighborhood of one location A and map 
out the values of the long-lat product for nearby locations, we'll 
find that the locus of points with a particular long-lat product is a 
line running roughly northwest-southeast (in east longitude and north 
latitude, where you are). Thus a point that is northwest or southeast 
of A will have a long-lat product that is closer to that of A than a 
point that is northeast or southwest of A and the same distance away.

You can find plenty of information on how to calculate the actual 
distance between two points by going to our Search Dr. Math page and 
searching the archives for the words   latitude longitude  . I don't 
know of anyone who has asked what you are asking (to find a function 
that is monotonically related to distance but is easier to compute), 
but I don't think you'll find any formula that is nearly as simple as 
your proposal.

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