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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
>
> 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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