# NavList:

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

**Re: 3 Lop's**

**From:**George Huxtable

**Date:**2004 Nov 17, 19:54 +0000

Fred Hebard wrote- >With two LOPs, there are no degrees of freedom left to estimate the >accuracy of the fix. With three, if the "cocked hat" is reasonably >small, you have some confidence; if large, you know it's not so good. >I expect a least squares solution also could not estimate errors from >two LOPs. I don't disagree with what Fred wrote, but wish to point out that care is needed in interpreting a single "cocked hat". What is certainly true is that one single LARGE cocked hat immediately shows that there are errors somewhere in the observations or their processing: no question about it. However, a single SMALL cocked hat does not, of itself, prove freedom from such errors. A small cocked hat can happen by accident, even if there is considerable random scatter in the observations, so that the result may appear to be precise, even if it isn't. Position lines from two bodies of a fix must always cross at some point (well, two points actually, but let's presume there's no ambiguity about which is the "right" one). Now draw in the position line from a third body. This can be better represented as a band rather that a narrow line. If errors are entirely random ones, that band will be centred on the true position of the body, and its width will be related to the random scatter in that third observation. The third position line will be somewhere within that band. So, by chance, that third line might happen to pass through, or close to, the intersection of the first two lines, in which case a tiny cocked hat would result, and an observer might deduce that he has made a particularly accurate observation. Even the least-squares fit procedure, described at the back of the Nautical Almanac, would deduce a small or zero error, and the AstroNavPC package, based on that procedure, would display on its screen a tiny error-ellipse. There's nothing else it can do, with those three observations. But it could all be quite illusory, if the statistical scatter is in fact high, and if that small cocked-hat came about only by chance. Of course, if a SERIES of three-point fixes gives rise to a series of small cocked hats, then the observer is entitled to draw extra confidence, from that evidence, that he is navigating precisely. Also if, in a single observation-set, he takes more than three altitudes (a situation that the least-squares fit method can handle without difficulty), perhaps many more, then the resulting error-ellipse (no longer based on a simple cocked-hat triangle) will show more robustly the true scatter in his observations. ================= Fred ended- "BTW, where is George? It's unusual for him not to have posted lately." It's nice to know one's been missed, Fred. George. ================================================================ contact George Huxtable by email at george@huxtable.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================