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    Fwd: Lehmann's Rules and surveying
    From: Peter Fogg
    Date: 2007 Mar 17, 18:39 +1100
    This message came directly to me, rather than to the group, which I assume was not intended, so have forwarded to the NavList.

    Hope this is OK with you, Richard. It does not seem to be in the nature of a personal message.

    ---------- Forwarded message ----------
    From: Richard <rmpisko1{at}telus.net>
    Date: Mar 17, 2007 4:36 PM
    Subject: Lehmann's Rules and surveying
    To: peter.fogg---.com

    On Thu, 15 Mar 2007 23:30:01 -0700, P F <peter.fogg---.com > wrote:

    > Lehmann's Rules were used by surveyors, mainly for orienting the plane
    > table
    > (essentially a drawing board mounted on a tripod).
    >
    I'm glad that you mentioned that analogy, as I had been thinking that way
    for some time.  However, no analogy is perfect.

    Note that even in the standard three point resection problems there are
    four cases:
    1.  The true location is within the great triangle formed by the three
    reference objects
    2.  The true location is outside of the great triangle, but within the
    great circle running through the three reference objects
    3.  The true location is on the Great Circle (the indeterminate case)
    4.  The true location is outside the great circle.

    Only in case "1." will the true location be found inside of the little
    triangle on paper (cocked hat) formed by sighting and drawing with a
    misaligned table (systematic error).

    Case "3." will always show as one of many perfect intersections on the
    Great Circle no matter what the table orientation, so find some other
    reference objects or change the plane table setup location.  You could try
    the Italian method instead, which will show the c-B and a-B intersection
    (x) to lie right on top of the b point on the paper.

    Italian method, three steps:  From left to right on the ground A, B, C.
    On the paper, a scale plot of the objects called a, b, c.  For the
    indefinite case, assume you set up on P very close to the fourth corner of
    a square A-B-C-P.   Place the alidade on "c" and "a", pivot the table so
    the alidade and table rotate together, until the alidade now points
    c-a-A.   Lock the table.  Rotate the alidade only about the point "c", and
    draw a line from "c" toward B.

    Set the alidade in the opposite direction compared to step one, so as to
    line up with "a" to "c".  In step two, pivot the table so the alidade will
    line up a-c-C and lock the table.  Pivot the alidade about the point a on
    the paper so it points to B on the ground and draw a line from c toward
    B.  The two lines, c-->B and a--> B will intersect at a point "x" on the
    paper.

    Step three:  Draw a dotted line through x and b on the paper, set the
    alidade along this line, release the table clamp and point the alidade
    aligning x-b-->B (or perhaps b-x-->B) on the ground.  The table is now
    properly oriented, so resecting from A-a->p and C-c->p will give your true
    position "p" on the paper and "P" on the ground.  But, if you are on the
    "Great Circle" along with A, B, and C; point "x" will land nearly on top
    of "b", and it will be obvious that it is impossible to orient the table
    with any accuracy in step three, and thus "p" cannot be found.

    Generally, fairly good intersection angles can be obtained if you chose
    reference objects and set up such that you are closest to the middle
    object . . . but all this is for mapping, not navigation.  You (correctly,
    I think) mentioned the stars could be thought of as being all at the same
    distance.

    --
    Richard . . .
    Using Opera's e-mail client since Dialog, "the Dog", died.

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