# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Fix from 2 Bearings**

**From:**UNK

**Date:**2003 Jul 11, 14:28 -0400

On Friday, July 11, 2003 1:06 PM Dan Allenasked: > > On Friday, July 11, 2003, at 08:30 AM, Smith_Peter{at}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!