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Re: Loxodromic midpoint
Posted:
Jun 17, 2003 6:12 AM


On 05 Jan 1999, Robert Hill wrote: ><pre> >In article <cSrk2.595$hs6.693@nnrp2.clara.net>, "Bob Street" <bob@belgrave.clara.net> writes: >> >> Axel Harvey wrote in message ... >> >> >Can anyone offer formulas for the latitude and longitude of the midpoint M >> >of a loxodrome, given the coordinates of endpoints P0, P1 and assuming a >> >spherical Earth? (By midpoint I mean that a vessel following the loxodrome >> >will log the same distance from P0 to M as from M to P1.) >> >> >> Well from the shortage of replies, it appears that mine is not the only >> dictionary which doesn't list 'loxodrome' !! >> >> I'd love to know what one is, please.... >> >> (From the question, I guess it's _not_ an arc of a great circle.) > >It's a curve on the sphere which makes equal angles with every meridian >of longitude (or with every parallel of latitude) it intersects. >Equivalently, a path on the sphere which looks straight >on a Mercator projection map. >Equivalently (if we pretend that geographical and magnetic poles coincide), >a compass course. > >Writing u for longitude and v for latitude, it's easy to see >that a loxodrome satisfies the differential equation > > dv/du = k cos u > >where k is the tan of the angle made with each parallel of latitude. >So the equation of the curve is > > v  v0 = k (sin u  sin u0). > >The arc length, s, satisfies > > ds/du = sqrt (1 + k^2 cos(u)^2), > >which I think gives an elliptic integral, so there will not be an >exact elementary expression for the midpoint of an arc (except in >special cases such as when the endpoints are at equal and opposite >latitudes). > > >Robert Hill > >University Computing Service, Leeds University, England > >"Though all my wares be trash, the heart is true." >  John Dowland, Fine Knacks for Ladies (1600) > > ></pre>



