A sign somewhere in the United States says that London, England, is 4386 miles away and that Tokyo, Japan, is 6182 miles away.
Where in the U.S. is the sign?
(Consider the earth to be a perfect sphere of radius 3959 miles, and that London is at 51°30’ North, 0°10’ West and Tokyo is at 35°40’ North, 139°45’ East.)
This ones sure been waiting a long time...
While what follows is no solution in itself, it is a method that could be used to find the solution.
Latitude and Longtitude closely relate to the spherical coordinate system of locating points. While the cartesian (most common) system labels points with an x,y and z value which gives a unique 3 dimensional location, and cylindrical coordinates (an extension of polar coordinates) locate points with a planar angle from a reference line (generally the x-axis), a distance from the origin within that plane and the relative height to the point from the plane, spherical cooridinates locate points by the quantity of a first rotation (about a reference plane) between 0 and 360 (similar to longitude), a second rotation (perpendicularly away from the reference plane) between -90 and +90 (similar to latitude), and a distance from the origin (which would be the radius of the Earth). There are easily calculated formulas to transfer a known point from one coordinate system to another.
Using this infomation, and assuming a coordinate system that is convenient to the problem (i.e. the reference plane contains the equator, and the reference line goes through longitude 0 degrees), we can use the lat, lon and radius to create cartesian coordinates for any location on earth.
Next we use the definition of vector dot product to calculate the absolute angle formed between these two surface points, and from there the distance of the great circle arc between them.
This is all math (and somewhat challenging but not prohibitively), and so far there is no need to make restrictive or error inducing guesses/assumptions.
I'm not certain how to proceed from here in a rigorous fashion, however, with a good map of the States (which I don't have), you could probably find the solution within a degree (still a lot of error) with 10 tries or so, and further narrow your result by performing more iterations until you've the accuracy you require. Of course, there's most likely a better way to do it.
concept
