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Tony,

        I went back and rederived the altitude and azimuth formulas in using 3D methods (see the summary at the end of this post). The results are necessarily the same as what Lars gave but inn case it’s not clear I want to emphasize how very useful the interpretation to r is in LoP sight reductions.

The procedure becomes:

Calculate

Pol{ [ cos(Lat) · sin(Dec) - sin(Lat) · cos(Dec) · cos(LHA) ] , [-cos(Dec) · sin(LHA) ] }

and then

H_c = cos^-1 r and θ = Z_n

or possibly more efficiently in terms of keystrokes

Pol{ [ cos(Lat) · tan(Dec) - sin(Lat) · cos(LHA) ] , [-sin(LHA) ] }

and then

H_c = cos^-1( r cos(Dec) ) and θ = Z_n

There is no need for a separate calculation of the altitude!!! One proviso, as Lars indicated, is that it only gives the correct answer for objects above the horizon.

Regards,

Robin Stuart

Summary of key results

Define topopcentric coordinates x,y,z in which the x-axis is horizontal pointing East, the y-axis is is horizontal pointing North and the z-axis is along the vertical. Relating the global Greenwich hour and declination GHA, δ coordinates to local azimuth and altitude Z_n, H_c (details on request) gives

x = sin Z_n cos H_c = -cos δ sin LHA

y = cos Z_n cos H_c= cos δ cos L - cos δ sin L cos LHA

z = sin H_c= sin δ sin L + cos δ cos L cos LHA

which satisfy x²+y²+z² = 1 and hence x²+y² = 1- z² = cos² H_c