# NavList:

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

**Still on LOPs**

**From:**Peter Fogg

**Date:**2002 Apr 28, 12:09 +1000

If 3 LOPs meet at a common point then according to the Doctrine of Infinitely Thin Lines the chances of their meeting point being the Actual Position (AP) is zero, if I have understood correctly, just as the chances of the AP falling on any LOP is zero, since an infinitely small point cannot occupy the same space as an infinitely thin line. But knowing this to be true in a theoretical mathematical sense is not of much use to us in a practical navigational sense. To begin with, our AP is not an infinitely small point, it is at least as big as our boat, and could be considered to be as big as our circle of view around the boat, a circle with a radius of a few miles. And as for any LOP, rather than a line of infinite thinness it may be more helpful to think of it as a line of infinite thickness, since the further we go from this line the more remote become the chances of encountering the AP. The converse is presumably also true, so our line looks extremely smudged, gradually merging from white to dark gray then back to white, showing the likeliness of encountering the AP on either side of it. We draw the line with a 2B pencil but it may be helpful to always think of it as this smudged line of infinite thickness, while recognizing that by the time you've traveled 100 miles (or much less) from it, it is almost as white as the paper, reflecting the very small chance the AP is out there. Now we have two of these LOPs crossing. Once again, while theoretically the chances of the AP being at the intersection is zero, if you use my smudged lines then the chance becomes 100%. And this reflects practical navigational practice; based on available data this is our fix position. And this is where this discussion started, when it was contended that 2 LOPs were enough and I proposed that 3 were better, citing as an example the uncertain meeting point of 2 LOPs from similar bearings. When we introduce a third (smudged) LOP we have 3 intersections: each presumably is, 100%, the AP. Knowing we can only be at one point at any one time all 3 cannot be correct, but since based on available data they are equally correct then we find a fix position based on least squares, inside the hat. And if our AP is not contained within the cocked hat, then where is it? If outside, but near one intersection then those 2 LOPs are roughly correct but the third is quite wrong. If outside, and somewhat halfway along one of the LOPs between 2 intersections, then that LOP is roughly correct but the other 2 must be decidedly dodgy. Is this really the case 75% of the time? Normally we don't have any way of knowing how accurate any or all of them are apart from the size of the hat. In a practical sense a small hat is an accurate fix, and a common meeting point is impossibly perfect. But even a small hat still has smudged lines, reminding us that the AP could be outside it, although I never heard an answer to my question of what good knowing this was to us. By the time we've worked out our fix position we're somewhere else, and promptly use the fix to run forward our DR. Thanks to the nav. list I spent some time thinking about this while bowling along dusty unsealed roads beneath a wide deep blue sky, exploring that strange phenomenon; a marsh in near desert country, a mostly dry swamp (although sometimes its full of water - and life) larger than a small European country like Denmark. It was fascinating, and the navigation was easy - the tracks twisted and turned, but apart from the shadows cast by the sun the near full moon was clearly visible for most of each afternoon. We saw lots of kangaroo and emu, and even, somewhere else, a koala in the wild, and that is very rare. I've avoided thinking about how multiple (more than 3) LOPs affect things. Personally I find them hopelessly confusing, as some intersect the hat and others fall outside it. My navigational calculator copes with them very well. If the chance of the AP being within the hat is 25%, then how is this affected by multiple LOPs?