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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.
>
> --
> 
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