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Rafael,

Aha. So the methods you were investigating were intended for extrapolating great circles! These are critically important for long distances, but for the problems we've been discussing, this skyline problem and the problem of crossing a pair of two-point celestial lines of position, they are overkill, and in the celestial case, actually inappropriate. Our celestial lines of position are short sections of small circles on the globe. Extrapolating them as great circles would be fine over short distances (apart from being overkill), but technically that's incorrect.

So how do we do it without plotting? I have two pairs of lat/lon values representing buildings in Manhattan and structures on the shore of Coney Island. I can calculate the "slope" of each line in the usual fashion. In this case, since the lines are oriented nearly north-south, I want dLon/dLat. Call that LonLatSlope. I set this up in a little spreadsheet. The goal is to calculate a longitude of some new point, Lon1, on each line of position for a given offset in latitude, DLat, given the initial position, Lat0, Lon0 and given the value of that LonLatSlope. The math is trivial: Lon1 = Lon0 + LonLatSlope · DLat. I set that up for each line of position, and then I experiment by trial and error with different DLat values until the Lon1 values for each LOP are identical, within some small limit. Then I get the latitude from Lat1 = Lat0 + DLat. And I'm done. This is equivalent to drawing and crossing lines, but given the narrow crossing angle and the relatively precise locations of the buildings and the structures on Coney Island, calculating beats drawing. 

Frank Reed

PS: Here's a skyline photo I took Friday while driving. Now that we're all NYC skyline experts, this one should be easy! Where am I?

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