# NavList:

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

**Re: Still on LOP's**

**From:**Chuck Taylor

**Date:**2002 Apr 27, 23:32 -0700

Here is some more food for thought: We take 3 bearings on fixed objects. Assume that the error in taking a bearing follows a Gaussian (Normal) distribution with mean 0 and standard deviation 1 degree. These numbers seem to me to be not unreasonable. The probability that any one bearing is within +/- 1 degree of the true bearing is 0.6826. The probability that all 3 bearings are within +/- 1 degree of the true bearing is (0.6826)^3 = 0.32. The probability that any one bearing is within +/- 2 degrees of the true bearing is 0.9544. The probability that all 3 bearings are within +/- 2 degrees of the true bearing is (0.9544)^3 = 0.87. The probability that any one bearing is within +/- 3 degrees of the true bearing is 0.9974. The probability that all 3 bearings are within +/- 3 degrees of the true bearing is (0.9974)^3 = 0.99. Using this information, we can define confidence regions of a sort. Converting 1 degree of angle to distance depends on distance from the landmark. Here is a rough table, computed by multiplying the tangent of 1 degree by the distance to the landmark: Distance Width of to landmark 1 degree 1 nm 0.017 nm 2 0.035 3 0.052 4 0.070 5 0.087 6 0.105 7 0.122 8 0.140 9 0.157 10 0.175 Interpreting these numbers for a particular instance, if all 3 landmarks are at a distance of just under 6 miles, then we can say that each line of position is within +/- 0.1 nm (about 200 yards) of the true line of position with probability 0.32. Similarly, we can say that each line of position is within +/- 0.2 nm (about 400 yards) of true position with probability 0.87, and within +/- 0.3 nm (about 600 yards) with probability 0.99. Note that I have said nothing about whether a particular cocked hat covers the true position. Still, a feel for how far off a particular bearing line might be can be useful information to a navigator. Chuck Taylor Everett, WA, USA