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    Re: Applications of Complex Analysis to Celestial Navigation
    From: Robin Stuart
    Date: 2009 Oct 26, 16:53 -0700

    You may have seen the paper by now but if not let me explain that the method 
    does not make use trigonometric functions with complex arguments but just 
    simple arithmetic operations on complex numbers. In a certain sense it is 
    analogous to the use of complex numbers in AC circuit analysis. A point on 
    the surface of a sphere is identified with a complex number by stereographic 
    projection. A single complex number encapsulates both of the angular 
    coordinates (e.g. latitude/longitude, declination/GHA or altitude/azimuth). 
    If z is a complex number representing the declination & GHA of a celestial 
    body and w is a complex number representing its altitude & azimuth then the 
    two are related by
    w = ( a * z + b ) / ( -Conjugate[b] * z + Conjugate[a] )
    where a and b are complex numbers that depend only on the latitude and 
    longitude of the observer. Altitude and azimuth are computed simultaneously 
    by a single equation.
    As far as Andres' referring to quaternions as a special case of tensor 
    calculus, I took that to mean that the algebra of quaternions is isomorphic 
    to that of the Pauli matrices. I would hesitate to place quaternions on the 
    same level as complex numbers however. The latter, of course, give rise to 
    the whole field of complex function analysis which is not really replicated 
    for quaternions,
    Robin Stuart
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