# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 Hirosewrote: > 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. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---