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    Bygrave sight reduction by slide rule
    From: Paul Hirose
    Date: 2017 Dec 13, 14:28 -0800

    In February I offered a version of the Bygrave celestial navigation
    sight reduction formulas. Though valid for all combinations of latitude,
    declination, LHA, and altitude (including negative altitude), I was not
    happy with some deviations from standard practice. For instance, my
    azimuth angle was defined differently from the normal sense of Z.
    
    Here are a new set of rules which conform to the traditional conventions
    of celestial navigation. For convenience on the slide rule, ALL NEGATIVE
    SIGNS ARE IGNORED. For instance, tan -10° = -.176, but you calculate as
    if it were +.176. Similarly, arc tangents are always in the range 0 - 90°.
    
    It's not necessary to normalize LHA to a certain range. For example, 10°
    and 350° work equally well.
    
    This Bygrave variant is intended for the conventional straight slide
    rule. It requires that you know how to read and set the full range of
    possible angles on the trig scales. See my "360 degree slide rule trig"
    series:
    
    http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37154
    http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37223
    http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-feb-2017-g38237
    
    
    FORMULAS AND RULES
    
    W = arctan(tan dec / cos LHA)
    
    If LHA is between 90 and 270, replace W with its supplement: W = 180 - W
    
    If latitude and declination have same name
         X = 90 - lat + W
    If contrary name
         X = 90 - lat - W
    In either case lat is always positive.
    
    If X is not 0 to 180, add or subtract 180 to make it so. If the
    adjustment is necessary, ALTITUDE IS NEGATIVE.
    
    Z = arc tan(tan LHA * cos W / cos X)
    
    If EXACTLY ONE of these is true
         altitude negative
         X less than 90
    replace Z with its supplement: Z = 180 - Z
    
    Azimuth angle Z is reckoned in the conventional sense: zero at the north
    (south) pole if lat is north (south), and increasing east (west) if the
    body is east (west) of the meridian.
    
    Hc = arc tan(cos Z * tan X)
    Apply negative sign if applicable.
    
    There are two places where an arc tangent is (in some cases) replaced
    with its supplement. But if you ignore signs, tangents of supplementary
    angles are equal. E.g., tan 10° = tan 170°. Therefore you can read the
    supplement on the T scale without any calculation.
    
    
    ACCURACY TEST
    
    I worked these 10 examples on a 10 inch slide rule. The problems were
    generated at random by a program which distributes stars uniformly on
    the celestial sphere, and observer positions uniformly on the Earth
    (both hemispheres). It excludes altitudes greater than 80 or within 5°
    of zero, and latitudes greater than 70.
    
      51.7575 lat
      -5.1860 dec
      27.9717 LHA
       5.86   W (not so easy to read on some rules)
      32.382  X
    147.86   Z (error = .113°)
      28.25   Hc (error = -.009°)
    
      -1.1816 lat
      47.0243 dec
    213.7008 LHA
    127.78   W
    141.038  X (alt neg)
    152.25   Z (error = -.032)
    -35.60   Hc (error = -.002)
    
      22.1846 lat
      43.1575 dec
    333.0564 LHA
      46.44   W
    114.255  X
      40.50   Z (error = .033)
      59.37   Hc (error = .005)
    
    -47.9567 lat
    -58.8701 dec
    296.4248 LHA
      74.96   W
    117.003  X
      49.00   Z (error = -.012)
      52.16   Hc (error = -.038)
    
    -11.1965 lat
      30.3719 dec
    304.7456 LHA
      45.80   W
      33.004  X
    129.82   Z (error = -.058)
      22.58   Hc (error = -.035)
    
      40.7939 lat
      -3.3744 dec
    259.7277 LHA
    161.70   W
      67.506  X (alt neg)
      85.82   Z (error = -.005)
    -10.00   Hc (error = .025)
    
    -50.1163 lat
    -51.0710 dec
      97.9351 LHA
      96.37   W
    136.254  X
      47.78   Z (error = .039)
      32.74   Hc (error = -.033)
    
      65.9394 lat
    -17.9946 dec
       9.2966 LHA
      18.20   W
       5.861  X
    171.11   Z (error = -.024)
       5.79   Hc (error = .021)
    That one required cosines of small angles.
    
    -28.7610 lat
    -35.6695 dec
    233.0997 LHA
    129.86   W
      11.099  X (alt neg)
      41.02   Z (error = -.099)
      -8.42   Hc (error = .035)
    
      -9.1664 lat
    -54.4142 dec
      10.8386 LHA
      54.86   W
    135.694  X
       8.76   Z (error = -.001°)
      43.94   Hc (error = -.016°)
    
    Root mean squared altitude error = 2.5 arc minutes.
    

       
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