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    Re: Position from altitude and azimuth.
    From: Bill Lionheart
    Date: 2020 May 21, 17:25 +0100

    Dear Robin
    
    Yes we have worked out it is very much not a small circle. Indeed it
    is (a subset of) the intersection of a quartic "cylinder" with the
    sphere. The quartic can have one or two components and can get quite
    triangular in shape.
    
    You are correct that the stereographic projection is indeed 3rd order
    (my colleague Gabor Megyesi, an algebraic geometer, just told me
    this).
    
    I am working on some animations in Mathematica. ... but I am a bit ill
    at the moment so please be patient. Brain only firing on two
    cylinders.  (My Covid 19 test kit has just arrived)
    
    Best wishes
    
    Bill
    
    On Thu, 21 May 2020 at 17:17, Robin Stuart  wrote:
    >
    > It may be a bit late now but in order to determine whether a isoazimuth is a 
    small circle I decided to construct a counter example. The plan was to obtain 
    4 points lying on an isoazimuth. Choosing 3 of those points calculate the 
    centre of the small circle on which they fall. Then choose a different 
    combination of 3 out of the 4 points and calculate the centre on the circle 
    that those fall on. If the centres are different then an isoazimuth is not a 
    small circle.
    >
    > Circles great and small are conveniently handled analytically by 
    stereographic projection onto the complex plane as described here. Circles on 
    the sphere map to circles on the plane. Moreover since the Littrow projection 
    is a simple analytic function of the stereographic projection this offers a 
    consistent approach for the treatment of equal altitude circles and 
    isoazimuths. In an earlier post by Bill Lionheart provided a link to a useful 
    article which stated "If the complex z-plane is the Littrow map and the 
    w-plane is the polar stereographic map, then z = w +1/w". After some time 
    wrestling with this and trying to make it behave I realized there is a typo. 
    It should read "z = w -1/w" and that makes all the difference.
    >
    > In any case here's the counter example generated by appealing to the 
    properties of the Littrow projection
    >
    > Assume the GP of the body being observed is at 16°46.8'N 45°51.1'E. An observer at any of the points
    > P1 = 10.306896°N 55.375875°E
    > P2 = 0.248451°S 70.71412°E
    > P3 = 25.433371°S 95.262638°E
    > P4 = 48.38593°S 102.746085°E
    >
    > The points P1, P2 and P4 all lie on a small circle with centre or pole at
    > c124 = 37.001705°S 28.211937°E
    > this can be checked calculating their distances from the centre and seeing that they are all the same.
    >
    > The points P1, P2 and P3 all lie on a small circle with centre or pole at
    > c123 = 43.700701°S 18.268654°E
    >
    > These centres are not the same hence isoazimuths are not in general small circles.
    >
    > Under stereographic projection onto the complex plane isoazimuths appear to 
    satisfy 3rd order equations. It may still be possible to calculate the 
    intersection of 2 isoazimuths or an isoazimuth with a small circle. This 
    would be analogous to the algebraic solution of the double altitude problem 
    discussed here.
    >
    >
    > Robin Stuart
    >
    > View and reply to this message
    

       
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