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### Longitude-Latitude Product and Distance

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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.

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
```

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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

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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