# NavList:

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

**Maximum likelihood position with triangle on a sphere**

**From:**Bill Lionheart

**Date:**2018 Oct 19, 17:09 +0100

So now we all know the symmedian point (and I expect some of you are a bit fed up with it. Sorry!) For a plane triangle formed by three lines of position with the correct direction but zero mean normally distributed errors in the distances from the true position the maximum likelihood estimate is the symmedian point. I was thinking of related problems on the sphere. Some of which I can solve and some I don't know how to solve yet. For example suppose we have three great circle position lines with some sensible assumption about the errors. For example radio direction finding from distant stations. Or three small circles, as we would get if we were using celestial navigation and had the altitude of three bodies, but the errors were huge so the straight line approximation was not valid. There is a definition of a symmedian point and a Lemoine point for a spherical triangle (but they are different). Of course I could compute a point numerically using an optimization code, but I am looking for geometric descriptions of the maximum likelihood estimate. My question is just if NavList members know of anything in the literature. (..public or secret). Thanks Bill Lionheart