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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 5:32 PM
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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/
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