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    Re: Circle charts
    From: Andrew Nikitin
    Date: 2009 Jan 31, 12:34 -0800

    Richard,
    
    > In my opinion, the advantage of using your lines directly is being able to
    > pick out the observed angles quite accurately.  The advantage of
    > pre-plotting circles to a larger scale onto an A2 sheet is being able to
    > locate your (reasonably close) position more quickly; and without needing
    > a large plotting table.
    
    That's what I think too. Set of scales is a compromise between immediately
    usable, but somewhat messy and less accurate grid of circles and work
    intensive, but "infinitely accurate" geometric construction method. You still
    have to draw a pair of circles for every single measurement, but it should be
    much easier than to build them from scratch.
    
    I attached another picture that shows grid of circles on the first page and set
    of scales for same stations on the second page.
    
    Also, instead of printing scales on map, how about having several T-shaped
    rulers, short horizontal edge having 2 marks for stations and long vertical
    edge having graduated ticks for centers. Align marks with stations and long
    edge will show location of a center. It is even more work, plus you have to
    have special ruler for each different distance between stations, but the map
    itself remains completely uncluttered.
    
    Same appoach may work with grids of circles: if you print them on transparency
    then it is possible to overlay only those 2 that you currenly use, so that grids
    from unused pairs are not in the way.
    
    > I like that idea for my intended trials down at the river bottoms, with a
    > number of unknown locations smaller than that required for dredging.  I
    
    If you are going to actually try this in real world, I volunteer to make either
    grids or scales for your actual distances between stations in a form of, say, PDF file. Would that help?
    
    > Is your formula for the circle centers on the perpendicular bisectors
    > something like: <(distance between station points)/(2 x tan of the
    > observation angle)>?
    
    I used formula
    x=h/tg(a/2)-h/sin(a)
    where a is observation angle, h is HALF of a distance between stations and x --
    distance from midpoint between stations to a center of a circle.
    It is a difference between h/tg(a/2), which is distance from midpoint to the
    farthest point on a circle, and h/sin(a), which is radius of a circle.
    Whether or not it is equivalent to what you wrote, I do not know.
    
    > Which drawing program did you use?
    
    I use Ghostscript. It is not so much drawing program as a Postscript
    interpreter. It can turn Postscript picture description into variety of vector
    and raster formats, for example, pdf or bmp.
    
    > Looks good at first study; but I think it is preferable to have the middle
    > shore station closest to your expected position to avoid the indeterminate
    > case of the circle of position.  The scales should then be diverging.
    
    I agree. If centers of circles lay close to intersection of scales then circles
    are too close together (intersect at small angle) to find location. "Diverging"
    scales insure that intersection point lays far inland.
    
    
    > We could continue the lines in the other direction for observation angles
    > of greater than 90 degrees, I would think; even though that case may not
    > be as common.
    
    Also negative angles are possible. This is when observer is located on the
    "other" side of the line connecting base station. Or, another way to say it,
    when you view the pair from "behind".
    
    Andrew
    
    
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