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John,

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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