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    Re: Fix from 2 Bearings
    From: UNK
    Date: 2003 Jul 11, 14:28 -0400

    On Friday, July 11, 2003 1:06 PM
    Dan Allen  asked:
    > On Friday, July 11, 2003, at 08:30 AM, Smith_Peter@EMC.COM wrote:
    >> I used a spreadsheet I wrote that solves a two-LOP fix using rhumb
    >> lines determined by mid-latitude.
    > Is this available?  Could you share the details of how you solve this?
    Since attachments don't work well on the list, I'll mail you a copy of
    the Excel spreadsheet backchannel -- and to anyone else who'd like a copy.
    To solve for the FIX, I place the two targets on a Cartesian grid,
    derive their normalized equations (y = slope * x + intercept), then
    solve the two equations for their common root, which will be the grid
    coordinates of the FIX. Convert that back to lat/lon and we're found.
    First, I arbitrarily put the first target (Mt Rainier) at (0,0). Using
    mid-latitude conversion (yes, meridonal parts would be slightly more
    accurate) I compute the second target (Mt Baker) as 115.48 miles north
    and 2.15 miles west, thus at (-2.15,115.48) on our grid.
    From these two points and the bearings we can derive the equations of
    the two lines.
    The true bearing to Rainier is 158.8d; the slope (dy/xy) of the line is
    therefore tan(90-158.8) = -2.5759
    The true bearing to Baker is 068.8d; the slope (dy/dx) of the line is
    therefore tan(90-68.8) = 0.38821 = dy/dx
    Since Rainier is at (0,0), the y-intercept of its line is 0, so the
    normalized equation is: y = -2.5759 * x .
    Knowing the slope of Baker's line, and knowing a point it passes
    through, we can solve for the y-intercept by multiplying the slope times
    the known x-coordinate less the known y-coordinate
    (0.38821 * -2.15 -  115.48 = 116.32), making the normalized equation:
    y = 0.38821 * x +  116.32 .
    At the FIX, the two equations are equal, so:
        -2.5759 * x = 0.38821 * x + 116.32
    which becomes
        x = 116.32 / (-2.5759 - 0.38821) = -39.242
    which is the x-offset from Rainier (0,0) to the FIX. Plugging this x
    into  either equation will yield the y-offset of the FIX; using
    Rainier's equation: -2.57593 * -39.242 = 101.08, so the FIX is at grid
    (-39.241,101.08). Converting back into lat/lon puts the FIX at
    48d 32.3'N  122d 44.0'W.
    Wah La!

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