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

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Robert Hill

Posts: 529
Registered: 12/8/04
Re: Loxodromic midpoint
Posted: Jan 5, 1999 1:43 PM
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In article <cSrk2.595$>, "Bob Street" <> 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

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)

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