Longitude-Latitude Product and DistanceDate: 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/ |
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