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Reciprocity mismatch using geodetic2aer and aer2geodetic

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Hi All,
I've used geodetic2aer to calculated Azimuth, Elevation & Slant Range between two known locations (lat, lon, h, lat1, lon1, h1).
I found that when you switch between the location coordinates and perform the same calculation, it appears that the results of the azimuth and elevation aren't correlated as expected by 180 degrees in azimuth and (-) in elevation.
I would be interested in understanding how to achieved reciprocant results using geodetic2aer & aer2geodetic functions.
  1 Comment
the cyclist
the cyclist on 14 Jan 2024
Your problem would be easier to understand if you uploaded an example or two of specific coordinates that illustrate the problem.
In particular, it's not clear whether you are talking about a tiny difference (that might a floating-point error), or something larger.

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Answers (1)

David Goodmanson
David Goodmanson on 15 Jan 2024
Edited: David Goodmanson on 16 Jan 2024
Hi Leonardo,
There is no reason for the reciprocity you mentioned to hold true. This has to be the case since the aer coodinate systems at each location are local coordinates for that location, and these differ since the earth is not flat. The reciprocity idea is nearly true for locations that are close to each other and not near the poles, but a couple of illustrative cases show it's not true in general.
Take a great circle that crosses the equator at point p1, and suppose the great circle is heading northwest at p1. Traveling to the west from there by 90 degrees in longitude, the great circle achieves its furthest extent to the north and is headed due west. Put p2 on the great circle there. Then the azimuth from p1 to p2 is northwest, but the azimuth from p2 back to p1 is due east. These two directions are nowhere near 180 degrees from each other.
As you move along a great circle, its local azimuth changes continuously. So if you were a navigator in the navy in the 1800s and wanted to sail by great circle, you had to know your rhumb line for the day.
There are even more extreme examples near the poles, where if p1 and p2 lie almost across the pole from each other, then the azimuth from p1 to p2 is almost directly north, and the the azimuth from p1 to p2 is also almost directly north.
In general the elevations also have to differ from from each other by more than just a factor of -1. For example, take two points on the equator whose longitudes differ by n degrees. The -1 factor would work if the local zenith directions at p1 and p2 were parallel to each other. But the two zenith directions differ by n degrees, so the elevation reciprocity idea does not work.




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