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Re: Loxodromic midpoint
Posted:
Jan 5, 1999 8:23 PM
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Clive Tooth wrote:
> dv/du = k cos v
Anyway, this is the way I see it...
u is the longitude and v is the latitude.
Let r be the radius of the Earth.
Let a be the angle made by the loxodrome with each parallel of latitude.
Let tan a = k.
Let the initial and final latitudes be Lat0 and Lat1.
Let the initial and final longitudes be Lon0 and Lon1.
Let the latitude and longitude of M be LatM and LonM.
Now,
ds/dv = r cosec a
Integrating from P0 to P1 gives:
s = (Lat1-Lat0) r cosec a
In other words, the distance is _linear_ in the latitude.
So
LatM = (Lat0+Lat1)/2 (1)
Again,
k = sec v dv/du (2)
Integrating from P0 to P1 gives:
k(Lon1-Lon0) = log((sec Lat1 + tan Lat1)/(sec Lat0 + tan Lat0))
So
k = log((sec Lat1 + tan Lat1)/(sec Lat0 + tan Lat0))/(Lon1-Lon0)
(3)
Integrating (2) from P0 to M gives:
k(LonM-Lon0) = log((sec LatM + tan Lat1)/(sec LatM + tan Lat0))
giving
LonM = Lon0+log((sec LatM + tan Lat1)/(sec LatM + tan Lat0))/k
(4)
Where LatM and k are already know from (1) and (3) above.
LatM and LonM are have now been determined, as required.
--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
End of document
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