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    Re: Direct methods
    From: Gary LaPook
    Date: 2007 Nov 09, 01:30 -0800

    Gary LaPook writes:
    Your post reminded me that back in 1979 I wrote a similar program to
    run on my Texas Instruments TI-59 which was programed off of small
    magnetic cards and ran a machine language code. I found the coding for
    the program but I apparently threw away the books that came with the
    TI-59 so I can't decode the formula I used to do the computation and I
    can't find a copy of the formula. I then adapted and expanded the
    program to run on a Casio PB-1000  and I printed out that coding but
    its been so long I am having a hard time figuring out just what the
    program is doing. I am trying to figure out the formula I used but it
    looks different than yours since it uses sines and cosines of the Zn
    and these quantities squared plus the intercept but no usage of tans.
    The program worked very well. I would enter in the departure point and
    destination, the heading since the last fix and airspeed, the fix time
    and the names, times and altitudes of two or three star shots. The
    program would compute the coordinates of the bodies, correct the
    sextant altitudes (including coriolis), would adjust the sights for
    movement of the plane and of the body to the common fix time and
    display the fix location. Then it would display the track and ground
    speed since the last fix and the actual wind encountered since the
    last fix using the inputted airspeed and heading combined with the
    track and ground speed it had calculated. It then displayed the
    heading to hold to reach the destination to correct for the wind that
    had been encountered, the predicted ground speed (also based on the
    computed wind) and the estimated time of arrival. It then remembered
    all of these values so at the next fix the only new information that
    needed to be entered was fix time, names, times and altitudes of the
    sights, airspeed and heading and it would do the computation all over
     Since I am not a great mathematician I had to use an approximation to
    calculate a three body fix. The program figured out the latitude and
    longitude of the two body fix and then used that as the AP for
    calculating the intercept and Zn for the third body. It then used the
    bisector of the LOPs at the first fix in the direction of the third
    LOP and moved the fix along that bisector two thirds the length of the
    third intercept as an approximation of the center of the "cocked hat"
    and then also displayed the length of the third intercept to provide
    some sense of the accuracy of the fix. I was happy with the results
    even though the programing was inelegant.
    On Nov 1, 3:53 pm, "George Huxtable" 
    > d walden's first posting with its attachment came over successfully to me.
    > It may present problems to anyone that doesn't have Excel aboard as his
    > spreadsheet.
    > He wrote-
    > |    It is of course possible, as has been pointed out, to calculate the
    > latitude and longitude of the points of intersection of two circles of
    > position directly with neither an estimated nor an assumed position.  Nor in
    > fact are altitudes and intercepts or any plotting needed.  The intersection
    > of cones method, described by Frank in the discussion of latitude by lunars,
    > and shown in a previously posted FORTRAN and Maxima example can be used.  A
    > few minor changes to the programs posted are needed.
    > |
    > |  The problem is, in fact, slightly easier.  It can be done with
    > intersecting planes.  The attached little spread sheet is an example of how.
    > One enters with the altitude, declination, and GHA of two bodies.  Out come
    > the latitudes and longitudes of the two intersection points.  If you have
    > three bodies, pick a different pair, and two of the points should wind up
    > quite close.  There you are.
    > |
    > |  Seems to be working for me, but no guarantees.
    > ===================
    > For completeness, if nothing else, I resend an attachment which originally
    > went out to the old Nav-L list on 12 June 06, under the threadname
    > "positions from crossing two circles". That provided a routine, written in
    > Bastard-Basic for a Casio pocket-calculator, to determine the two possible
    > positions that result from crossing two observations. It also explains how
    > the thing works.
    > I remember that Andres followed it up with a posting in which he had
    > converted it to a program written in C.
    > George.
    > contact George Huxtable at geo...@huxtable.u-net.com
    > or at +44 1865 820222 (from UK, 01865 820222)
    > or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    >  intersecting circles.rtf
    > 41KDownload
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