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