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    Re: slide rule sight reduction accuracy
    From: Greg Rudzinski
    Date: 2009 Jun 16, 09:07 -0700

    Paul,
    
    Would it be possible to simulate for both 10" and 20" slide rules
    using the altitude sight reduction formula
    
    ALT = Inverse SIN ( COS meridian angle x COS declination x COS
    latitude
                        +/- SIN declination x SIN latitude)
    
    and to analyze separately 0 to 30�, 30� to 45�, 45� to 75� altitude
    zones for
    latitudes from 0 to 70�?
    
    Greg
    
    
    
    
    On Jun 15, 9:07�pm, Paul Hirose  wrote:
    > 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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