Welcome to the NavList Message Boards.


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

Compose Your Message

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

    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
                        +/- 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�?
    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.
    Navigation List archive: www.fer3.com/arc
    To post, email NavList@fer3.com
    To unsubscribe, email NavList-unsubscribe@fer3.com

    Browse Files

    Drop Files


    What is NavList?

    Join NavList

    (please, no nicknames or handles)
    Do you want to receive all group messages by email?
    Yes No

    You can also join by posting. Your first on-topic post automatically makes you a member.

    Posting Code

    Enter the email address associated with your NavList messages. Your posting code will be emailed to you immediately.

    Email Settings

    Posting Code:

    Custom Index

    Start date: (yyyymm dd)
    End date: (yyyymm dd)

    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site
    Visit this site