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    Computing azimuth with the Bygrave in special cases.
    From: Gary LaPook
    Date: 2010 Feb 09, 04:52 -0800

    I have found a way to calculate azimuth when the declination is off the 
    bottom of the cotangent scale of the Bygrave. I have been using a method 
    of approximating the azimuth in this case which produces usable 
    azimuths, agreeing within less than one degree with the correct azimuth, 
    but I was disappointed that I did not have an exact solution. (See 
    excerpts of my prior posts below and the links to prior posts).
    
    My new exact method is to go immediately to the second procedure 
    outlined in the special rules for this situation by exchanging 
    declination and latitude which brings the values within the range of the 
    cotangent scale. You then compute the altitude and the azimuth at the 
    geographic position (G.P.) The altitude is the same calculated at the 
    G.P. and at the  observer's position  (O.P.) but the azimuth calculated 
    at the G.P. is not the same. But we can use this azimuth to derive the 
    azimuth at the O.P. by using the law of sines.
    
    sin a / sin A  = sin b / sin B  = sin c / sin C
    
    so:
    
    sin co-lat / sin azimuth at G.P.  =  sin co-declination  / sin azimuth 
    at observer's position
    
    
    which can be re-written as:
    
    cos lat / sin Az at G.P = cos dec  / sin Az at O.P.
    
    An easy way to solve for azimuth at the O.P. using a normal slide rule 
    or a calculator is to divide sin Az at G.P by cos lat and then multiply 
    by cos dec to produce sin Az at O.P.
    
    sin Az at O.P  =( sin Az at G.P. / cos Lat ) cos dec
    
    But cos dec is  approximately equal to 1 since dec is between 0 and 50' 
    making cos dec  =  1.0 ~.99989 so we can drop that term simplifying it 
    further to
    
    Sin Az at O.P = sin Az at G.P.  /  cos Lat
    
    
    So now I am happy that I have found an exact method but I don't think it 
    is worth the extra effort in practice since the approximate method 
    produces the same azimuth, usually within a fraction of a degree.
    
    
    gl
    
    
    -----------------------------------------------------------------------------------------------------------------------------------------------
    This rule comes from the original Bygrave instructions. Even though the 
    original Bygrave cotangent scale was marked every minute of arc it would 
    be difficult to take out intermediate values and it would be expected 
    that a user would take either the one above or the one below any 
    intermediate reading. Bygrave recognized that the azimuth becomes 
    critical as to the determination of altitude when the azimuth is near 
    90�, and any small error in its determination will have a large effect 
    on the computed altitude. As you noted, the altitudes are computed to a 
    higher level  of precision than is needed for plotting the LOP so for 
    this use such small errors can be ignored but can't be ignored for the 
    determination of altitude when the azimuth is near 90�. So, when azimuth 
    is greater than 85� you start with the normal procedure but when the 
    azimuth comes up greater than 85� you use that azimuth for plotting the 
    resulting LOP but you stop the process at that point and do not go on to 
    calculating altitude. You stop and go back to the beginning on another 
    form and this time you interchange the latitude and the declination. You 
    use the same process and compute an azimuth and then go on to compute 
    altitude. This second azimuth is what would exist at the other corner of 
    the triangle and so is not correct at the observer's position and is 
    disregarded when plotting the LOP. In  most cases the azimuth computed 
    this second time will not exceed 85� so any error in it will not 
    critically affect the altitude. It is possible in some cases that you 
    will still get an azimuth the second time that exceeds 85� but all you 
    can do in this case is recognize that there might be a greater error in 
    the altitude than normal.
    
    I hadn't thought of it before, but is obvious when you do think about 
    it, that the altitude calculated at either the observer's position or at 
    the geographic position of the body must be the same. Since the same 
    calculation will yield the great circle distance between these points, 
    which is the length of this leg, and must be the same length when 
    calculated from either end. But the azimuth will be different which is 
    why you use the first calculated azimuth and ignore the second.
    
    
    If the computed azimuth is greater than 85� the computed altitude will 
    lose accuracy even though the Az is accurate. For azimuths in this range 
    even rounding the azimuth up or down one half minute can change the Hc 
    by ten minutes. So you use the azimuth but you compute altitude by 
    interchanging declination and latitude and then doing the normal 
    computation. You discard the azimuth derived during the computation of 
    altitude and use the original azimuth. When declination is less than 55' 
    on my version (less than 20' on the original) you can't compute "W" 
    because you start the process with declination on the cotangent scale. 
    In this case, Bygrave says to use the same process as when the azimuth 
    exceeds 85�, you simply interchange declination and latitude and compute 
    altitude. But Bygrave didn't tell us how to calculate azimuth in this 
    case. In my testing I have found a method that produces quite accurate 
    azimuths. You simply skip the computation of "W" and simply set "W" 
    equal to declination. The worst case I have found is that the azimuth is 
    within 0.9� of the true azimuth but most are much closer. If the 
    declination is less than one degree and the latitude is also less than 
    one degree, follow this procedure and also assume a latitude equal to 
    one degree. After you have computed the Az you then follow the same 
    procedure discussed above for azimuths exceeding 85� by interchanging 
    the latitude and declination and then computing Hc.
    
    http://www.fer3.com/arc/m2.aspx?i=107947
    
    
    http://www.fer3.com/arc/m2.aspx?i=107414
    
    http://fer3.com/arc/img/106329.bygrave%20manual.pdf
    
    
    
    
    
    

       
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