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Topic: Loxodromic midpoint
Replies: 8   Last Post: Jun 17, 2003 6:12 AM

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Clive Tooth

Posts: 1,824
Registered: 12/6/04
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/
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