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    Re: Geodesic versus great circle
    From: Frank Reed
    Date: 2018 Nov 16, 10:30 -0800

    Bill,

    • It's too small to worry about.
    • Algorithms have already gone way beyond the pure geometry issue. Weather routing is normal routing.
    • Great circle paths have a "cultural" element to them. By non-explicit agreement, they serve as a standard of comparison despite the fact that they are not reigorously the shortest paths on the surface of the ellipsoidal globe.
    • Navigators of the hyper-nerd school already obsess over great circle courses. We don't need to give them anything more to obsess over!!

    To satisfy my math curiosity, about 25 years ago I went through the geodesic math, even messing around with curvature tensor components and all that. If you're sailing from, let's say, Florida to the mouth of the English Channel, you get a slightly shorter ground distance by deviating just a bit to the north of the standard great circle. At most it's one part in 298, which is the oblateness of the Earth. And that deviation in track to higher latitude makes sense: the earth is "smaller" at higher latitude. The simpest case is travelling from a point on the equator to its antipodal point on the equator 180° away in longitude. It's clear that the shortest route by a small amount goes over the pole instead of along the equator or on any arc at intermediate peak latitude. 

    Even the most trivial examination of weather and wind and currents would overwhelm that small geometric effect. On any vessels where efficiency matters, some weather and current routing are employed. Current routing has been a thing since the 18th century. There's even a connection to the whaling industry and the Folger family (another Folger famously discovered the descendants of the Bounty mutineers at Pitcairn).

    Despite being slightly incorrect mathematically and questionable on various grounds, great circle paths and distances have become a standardized global distance scheme. It's part of the culture of mapping.

    One could argue that the "through the earth" distance more "accurately" represents the distance from one point to another. If I could tunnel through the earth, the distance from Wellington, New Zealand to the antipodal point close to Salamanca, Spain would be about 7920 statute miles or about 360·60/pi nautical miles. But if you "ask the internet", you'll get roughly 12,400 statute miles or about 180·60 nautical miles. That's the great circle distance. But why is that preferred? I'm suggesting that it's just a cultural standard. It is at least convenient that the projection of the "through the earth" tunnel onto the earth's surface is the same as the normal great circle track (and that makes good sense: the straight line is the shortest distance in three-dim space while the great circle is the set of points on the globe closest to that straight line).

    Frank Reed

       
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