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    Re: Scale of Intercept
    From: Frank Reed
    Date: 2020 Feb 1, 17:13 -0800

    David C, you wrote:
    "To summarise lat and long are an artificial construct imposed on a sphere."

    Yes, of course. But don't get too carried away with that! All coordinate systems are artificial by nature. But so is every geometric construct ever. There are no circles and squares in Nature. The "artifice" of coordinates is a "feature, not a bug" as a code might say. The arbitrariness of coordinates is one of their great strengths.

    You wrote:
    "If I  completely ignore lat and long and instead think of a sphere with great circles drawn on it it all makes sense."

    Your circle of position, however, is not a great circle. Imagine an observer in the southern hemisphere who has managed to measure the altitude of the faint star sigma Octantis (now officially known, per IAU, as Polaris Australis). And let's just ignore the fact that it's not a perfect pole star and pretend that it's declination is exactly -90.0°. If I measure its altitude as 50.0°, then I must be on a circle centered on the geographic south pole with a radius of 40.0° and that is the same thing as the latitude circle for -50.0° (a.k.a. 50° S). Thus the circle of position corresponds exactly to a circle of latitude. That's a "small circle", like any circle of latitude (except the equator itself). The radius from the geographic pole to the circle of position is an arc along a great circle. 

    Distance off (radius of circle of position): great circle,
    Circle of position itself: small circle.

    But suppose I instead measure the altitude of Canopus, which is over 37° from the souht celestial pole. Suppose its altitude is 50.0°. I can draw a circle on the globe with a radius of 40.000° centered on the subStar point for Canopus (worked out from the Nautical Almanac or other database) and then try to puzzle out how this circle intersects the lines of latitude and longitude. Or, alternatively, I can introduce a new latitude and longitude coordinate system on the globe with its pole at the subStar point for Canopus. In this alternate lat/lon system, circles of "latitude" are drawn around the Canopus subStar point, and lines of "longitude" radiate out from the subStar point. It's just like ordinary latitude and longitude but with the poles in the "wrong" place. Our position in this alternate coordinate system is then very simple, much as if we had measured the altitude of Polaris Australis. The altitude is equal to the latitude. Then to find out where this is in terms of a position on the globe, we pick up the coordinate system and rotate it. It's all the same!

    Well, it's all the same when you ask a computer to do it. :) For us humans, rotating coordinate systems is rather complicated. But it's the concept that counts.

    You added:
    "To change the subject slightly. I have an AP way out in the southern ocean. The intercept is 11°.  Is it valid to draw a 660nm line on google-earth? At that scale I am geting the "sphere" projection of google-earth."

    An intercept?? Why? Just draw the usual circle of position from the subStar point. And yes, 11° distance is exactly 660 nautical miles. In Google Earth, you always get a sphere projection (really, it's a rather specific type of map projection onto the "flat plane" of the display, but it's a naive, visual type of map projection). But  you don't always get a great circle arc. There are little details you have to be wary of. For the most part though, if you ask Google Earth for an arc between two points, you are getting a great circle arc. Note: in Google Maps on the web, things are more complicated since they have brought back theeir old Mercator view, too --this is never a concern in Google Earth.

    Frank Reed

       
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