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I propose that a computer simulation might help settle this argument as to the
probablity that one's actually position lies within the "cocked hat".  I am
hoping that someone has the time and inclination to carry out the simulation.
The simulation would proceed as follows:

1. On a Cartesian grid (like ordinary graph paper with the x- and y-units equal
in length), pick three points to be the "landmarks", and a point between them to
be our "true position".  (If this grid bothers you, pretend that you are near
the equator. A square grid makes the computations easier without loss of
generality.)

2. Select the scale as desired, but 2 to 4 miles between the true position and
the landmarks might be realistic.

3. Construct the equations of the lines from each landmark to the true position.
Compute the true bearing for each in degrees.

4. Repeat the following 1000 times:

a. For each bearing line, generate a random number from a Normal distribution
with mean 0 and standard deviation 1. (This is the distribution associated with
the classic bell-shaped curve; about half the time this number will be negative
and about half the time positive.) Treat these numbers as the bearing errors in
degrees. Add these numbers to the bearings.  These will give the simulated
bearings for one iteration. With this distribution most of the simulated
bearings will be within about +/- 3 degrees of the actual bearing, which seems
fairly reasonable for a good hand-bearing compass.

b.  Using the simulated bearings, compute the equations of simulated bearing
lines from each landmark.

c.  Compute the pairwise intesections of the three simulated bearing lines.
Thise will be the vertices of the triangle (the "cocked hat").

d.  Determine analytically whether or not the true position lies within the
triangle.  If yes, tally one for the "yes" column. If no, tally one for the "no"
column.

5. When 1000 iterations have been completed, the number in the "yes" column
divided by 1000 will be our estimate of the percentage of the time we can expect
our actual position to lie within the cocked hat.


The key idea here is that the randomness is in the bearing, and we need a way to
translate that into the desired probablilty.  I suppose one could do a triple
integral to solve the problem analytically, but a simulation sounds more
appealing to me.  Any takers?

Chuck Taylor
Everett, WA, USA

   

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