# NavList:

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

**Re: "100 Problems in Celestial Navigation"**

**From:**Joel Jacobs

**Date:**2004 Apr 2, 18:10 -0500

Herbert, I knew I would get a scholarly answer from you. I remember using the least squares method in statistical analysis to establish a line of central tendency, used it to separate the fixed and variable elements of costs in financial analysis, and in a nautical sense, to determine fuel burn at various RPM and speeds. I discovered the bother was NOT worth the effort, and did it visually using a see through or plastic rule which seemed to work equally well. You visually average the dispersion of the observations around a line, that on the average, would split them equally above and below what be came the line of best fit. As far as the cocked hat, did you mention that bisecting the apex of all angles would produce a point that has some mathematical, though not necessarily accurate, MPP? I think that it would work for more than three angles. One of the Tamaya navigation calculators had an algorithm that produced a MPP, but I forget which of the later series it was, and certainly don't recall the modeling used. Thanks Herbert, Joel Jacobs ----- Original Message ----- From: "Herbert Prinz"To: Sent: Friday, April 02, 2004 2:55 PM Subject: Re: "100 Problems in Celestial Navigation" > Joel Jacobs wrote: > > > Since you indicate an error may be the result of estimating the MPP within a > > cocked hat, I am interested in what method you recommend to determine the > > MPP? > > Joel, > > Assuming random observation error, the MPP is that point for which the sum of > the squares of the distances from the individual LOPs is a minimum. > > 1) For two LOPs, this is the point of their intersection. > > 2) For three LOPs, there is a way to find this point geometrically. The > construction is based on the fact that if a,b,c are the sides of the cocked hat > triangle and Da, Db, Dc the distances from the MPP to the respective sides, then > a is to Da as b is to Db and c is to Dc. > > Therefore, to construct the MPP, pick two arbitrary distances Da' and Db' that > are in the ratio of a : b, draw parallels at these distances to a and b > respectively (either both inside or both outside the triangle), connect their > intersection point with C and name this line sC. Repeat the same procedure for > b,c and A, obtaining sA. The intersecton of sC and sA is the MPP. > > 3) For more than 3 LOPs, there is no geometrical way. The numeric solution is > the one that is obtained with the algorithm published in the Nautical Almanac, > pp. 277-283. > > Herbert Prinz