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    Slide rule sight reduction accuracy
    From: Paul Hirose
    Date: 2009 Jun 15, 21:07 -0700

    In a recent message I described a method to reduce celestial navigation
    observations with an ordinary slide rule using rectangular coordinates
    instead of spherical trigonometry. At the time I omitted some of the
    mathematics. Here is the full procedure.
    
    Convert LHA to an angle "theta" which is -90° (or 270) at the meridian,
    increasing east. This step isn't really necessary, but omitting it puts
    the rectangular coordinate frame into an unconventional orientation.
    It's no problem mathematically, but I find it awkward to visualize.
    
    theta = -90 - LHA
    
    Convert the body's local hour angle and declination to rectangular
    coordinates in a frame whose +z axis is directed to the north pole and
    +y axis directed to intersect Earth's axis.
    
    x = cos(dec) * cos(theta)
    y = cos(dec) * sin(theta)
    z = sin(dec)
    
    Rotate the coordinate frame about the X axis by the complement of
    latitude. This orients the +z axis to the zenith and +y north. In the
    second equation, y is the old y, not the new y computed in the first
    equation.
    
    y = y *  cos(90-lat) + z * sin(90-lat)
    z = y * -sin(90-lat) + z * cos(90-lat)
    
    Find azimuth. Note that x and y are swapped from their usual positions
    so azimuth will be zero at north, increasing east. This formula yields a
    value in the range -90 to +90. If y < 0, add 180 degrees.
    
    az = arctan(x / y)
    
    Compute the body's distance from the z axis.
    
    r = sqrt(x*x + y*y)
    
    Compute elevation.
    
    el = arctan(z / r)
    
    
    I implemented this in a computer program which simulates slide rule
    accuracy. At each place a slide rule would be used, the result is
    multiplied by a number of the form (1 + x), where x is a random value,
    centered on zero, with Gaussian distribution and .001 standard
    deviation. In other words, the simulated slide rule has .1% accuracy.
    That's the figure commonly quoted for 10 inch slide rules, and in a test
    with one of my own rules I confirmed it.
    
    Sight reduction problems are automatically generated, starting with
    a random azimuth and elevation. In order to evenly distribute the
    targets about the sky, elevation is the arc sine of a random number
    between 0 and 1. (If you simply distribute elevations evenly between 0 
    and 90 degrees, the band of sky from 0 to 10 degrees will have as many 
    targets as the band from 80 to 90, though the latter is much smaller.)
    
    A random latitude is obtained with the same arc sine method. The program
    can restrict elevations and latitudes to specified limits; I restricted
    elevations to 5 - 80 degrees and latitudes to 0 - 70.
    
    Declination and LHA are then computed from azimuth, elevation, and
    latitude. All these values are, for practical purposes, perfectly
    accurate.
    
    Declination, LHA, and latitude are submitted to the sight reduction
    routine, and the returned azimuth and elevation compared to the correct
    values. This occurs in a loop which runs any desired number of problems
    and tabulates the statistics.
    
    With this Monte Carlo simulation program I've found the slide rule sight
    reduction method outlined above is accurate in elevation to 3.1 minutes
    (square root of the mean squared error). About 95% of the results are
    within 6.2 minutes. The worst case results are about 15 minutes off.
    These appear to be due to unfavorable combinations of the random errors;
    I can't see any pattern in the azimuths and elevations where they occur.
    
    Azimuth RMS error is about 3.3 minutes. Worst cases are nearly one
    degree, and always occur when the problem is near the upper elevation
    limit (80 degrees in this test).
    
    My program is designed in a modular fashion so different sight reduction
    algorithms can be plugged in easily. I plan to implement others. If 
    anyone has a burning desire to see a certain method put to the test, 
    speak up. I'll move it to the head of the list.
    
    -- 
    I filter out messages with attachments or HTML.
    
    
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