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    Re: Horizontal distance off measurements
    From: Trevor Kenchington
    Date: 2003 Mar 24, 15:52 -0400

    Doug Royer wrote:
    > On 03-17 @ 212624 hrs. Mr. Kenchington wrote that one cannot use sextant
    > horizontal angles to find distance off measurements or bearings.
    I made no such statement. Doug: If you want to pick an argument, or simply 
    find self-justification, you should at least start off by quoting other 
    people correctly.
    > I must
    > differ with his opinion.Perhaps he did not understand  or misunderstood what
    > I was trying to explain.
    I assumed that what you were trying to explain was the method of horizontal 
    sextant angles, which was the method which Rodney Myrvaagnes had mentioned in 
    a message that Peter Fogg inquired about and you then responded to.
    It now appears that you had a quite different position-fixing method in mind.
    > The following is a simple explaination of the proceedure taken from the
    > books "Elements of Navigation" pgs. 140-143 by W.J. Henderson and "Coastal
    > Pilloting"pgs.89-93 by Commadore R.P. Ferchew.Each book has multiple
    > examples of the proceedures ranging from a simple 90* angle solution to
    > complex solutions for acute and obtuse angles.
    > Using the simple example of one of the objects @ 90* from the observer and
    > table 31,Bowditch and a calculator I did the exercize.
    In my Bowditch (1995 edition), Table 31 deals with correction of barometric 
    readings for altitude, which is clearly not what you intended. Could you 
    explain what Table 31 was in your edition, rather than just using the table 
    number? That way, I might have some idea how your method works.
    > I then used a UTM
    > projection and using 10 digit coordinates measured the distance between
    > electrical towers.I then calculated the distance off 2 of the towers and
    > fixxed a position.To keep the evolution simple I kept the 3 objects as close
    > to parrallel to my LOP as possible.Useing a handheld GPS I then went to the
    > calculated position fixxed from the projection.Using a sextant I then took
    > the Horizontal angles of the towers.I also took the bearings of the
    > towers.After the calculations were complete I used a laser rangfinder to
    > measure each distance.The delta between the chart measurements,sextant
    > measurements and the laser measurements were 5 ft.on the longest leg
    > distance and 2* T on the bearings.To me that proves one can get accurate
    > distance off and bearing measurements from Horizontal sextant angles.
    I don't think that your field experiment was really needed to prove your 
    point. Either you have a way that, geometrically, can produce an estimate of 
    distance off from a horizontal angle or you do not. Whether the necessary 
    measurements can be taken with high precision is a different question 
    > The example:
    >                                                   C                    A
    > B
    >                                                    .                     .
    > .
    >                                                                           .
    >                                                                          O
    > Useing dividers measure the distance AB = 300yds. AC = 307 yds.
    > Useing the sextant find the angles AOB = 11* 32' 13". AOC = 11* 47' 33"
    > dist. off OB = 300yd./(sin 11*32' 13") = 1500 yds.
    > dist. off OC = 307 yd/(sin 11*47' 33") = 1502 yds.
    That can only be true if both the angles OAB and OAC are equal to 90 degrees 
    and hence that A, B and C lie on a straight line. In the special case where 
    that is true and is known to be true, of course a measurement of AOB can be 
    used to calculate the distance OB (and likewise for OC). But unless you 
    already know your position, how can you know that OAB and OAC are right 
    angles? You could by determining the bearing of A from O, along with the 
    bearings of A from B and C. But you would not then be determining distance 
    off by horizontal sextant angles. You would be doing it by the combination of 
    horizontal sextant angles and a bearing.
    > Set dividers to 1500 yd. and swing an arc from B.Set the dividers to 1502
    > yd.and swing an arc from C.Where the 2 arcs intersect will be your
    > approximate position.Using parrallel rules find the bearings in * T of OB
    > and OC.Take the dividers and measure the line OA.This will be the distance
    > off A.
    Sure it will. But you could have plotted your position by any other means and 
    then subsequently measured the distance from A (or any other point) to your 
    plotted fix. That is only indirectly finding your distance off A.
    > One can calculate the dist. off A = 300 yd./(tan 11* 32' 13") = 1469.7
    > yds.
    Under the same assumption that OAB and OAC are right angles and hence with the 
    various caveats I have stated above.
    > A = 307 yd./(tan 11* 47' 33") = 1470.5 yds.
    > There is also a nice technique for finding the ballpark distance off an
    > object over terrain useing a mil type lensatic compass I have used with good
    > results.If anyone is interested I will share it with you.
    There are probably times when a navigator is more interested in distance
    off than in a fix of his position and yet wants to use horizontal
    sextant angles supplemented with other information. A single horizontal
    angle, combined with one bearing and the application of the Cosine Rule
    ought to do it. (No need for the second horizontal angle, except as a
    check.) However, if using horizontal sextant angles alone, you have to
    first fix your position (which you can do with extreme precision) and
    then determine the distance from that point to whatever object interests
    you. I dare say that the mathematicians on this list can figure out how
    to get the position and thence the distance numerically, without
    plotting anything on a chart, but the math would only be doing what the
    rest of us do with a pencil -- finding position first and the distance
    off only indirectly.
    Trevor Kenchington
    Trevor J. Kenchington PhD                         Gadus@iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
                         Science Serving the Fisheries

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