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Re: Loxodromic midpoint
Posted:
Jan 5, 1999 5:32 PM


Erm,
Did you mean...
dv/du = k cos v
??
Robert Hill wrote: > > 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)
 Clive Tooth http://www.pisquaredoversix.force9.co.uk/ End of document



