Some years ago we presented a procedure for finding the most likely position based on a plot of of 3 lines of position (LOP). We focused on finding that position when the random uncertainties in each of the LOPs was different, and we also added the crucial factor that there could be a fixed systematic error that applied to all triangles. We will return to that subject in the near future, with more details on the mathematics behind that solution.
In the meantime, we step back to the simpler case where the random uncertainties in all three of the lines are the same, and there is no systematic error to the set of sights—which could be cel nav sextant sights, or compass bearings, or any means of piloting that puts 3 LOPs on the chart. In this case, the most likely position is located at a very special point in the triangle called the symmedian point.
The concept of the symmedian was first reported in 1803, but the name "symmedian" did not come about until 1883. It comes from an abbreviation of the original French name "symétrique de la médiane," which was meant to convey "symmetrical counterpart of the median" — in that the symmedian line of a triangle is obtained by reflecting the median over the bisector.
The discovery that this point reflects the minimum of the sum of the squares of the distances to the sides of a triangle, and thus could represent the most likely position from three LOPs, was presented in 1877 by another French mathematician, but the point had not been named at that time.
Looking at the picture below, for any point P, there is a distance to each side of the triangle (dashed lines), and the argument is that the most likely position is where it is the closest it can be to all three sides, but since a distance can be negative outside of the triangle, the most likely position is the location where the sum of the squares of all the distances is minimum.
One can solve for that location several ways, and always end up with the answer being the symmedian point K.
Here is an interactive online app that you can use to show that K is the minimum of the sums.
Least squares demo (to do it yourself)
Or watch this video on the process.
Navigating underway, we want to know how to find the point in any triangle of LOPs we run across. There are several ways to do this, maybe dozens! We go over here the basic method of reflecting the median over the bisector, which is all plotting, and then we combine a computation with plotting for a faster solution.
The traditional way reflecting the medians over the bisectors, but with some short cuts.
A hybrid approach of doing a couple computations first and then plot the results. Download the free online app that computes the distances needed.
To get it on your phone, open that link in your phone and then share it to your home screen.
I will be back with more details on this topic as we proceed with our 3-LOPs rejuvenation program.
The app computations are based on this discovery from a German article in 1827:
