# NavList:

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

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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
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
```
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