# NavList:

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

**Re: Position from crossing two circles : was [NAV-L] Reality check**

**From:**George Huxtable

**Date:**2006 Jun 8, 23:21 +0100

Michael Dorl wrote- | Bear with me... | | I'm not thinking of any kind of graphical solution but rather the spherical | trig behind it. Consider two observations A and B. Call the zeniths ZA and | ZB and the great circle between them AB. Now think of a great circle | passing through ZA offset from AB by some angle X. We can write equations | for coordinates of points on the equal altitude circle for A as a function | of X, ZA, and the observed altitude of A. Those equations represent places | we could have been when we made observation A. Now since the equations give | us a starting position, can't we extend the starting point by the vector | representing the distance we traveled between observing A and B and write | equations for the ending point as a function of X, ZA, and the distance | sailed vector? I understand this requires that we make some assumptions | about the shape of the Earth. Also we end up with whatever errors occur | when translating speed - time - bearing | data to great circle data but these errors will arise in any treatment of | this problem. Ok, so now we have equations for our ending position as a | function of X, ZA, the altitude of A, and the distance sailed vector. Now | we can write equations for the observed altitude | of B from our ending position as functions of X, ZA, the altitude of A, the | distance sailed vector and ZB. Solve these equations for X knowing the | observed altitude of B. There will in general be two solutions on opposite | sides of AB. Knowing X, we can compute our | starting and ending positions. ============================== It's at times like this that we really need that Nav-l blackboard. I've been puzzling over those words, trying to picture what Michael is putting across. We have a great circle AB between the zenith points ZA and ZB. And we have another great circle which also passes through ZA but is offset from AB by some angle X. It's the relation between those great circles that I haven't taken in. In what way is the angle X measured? What does it represent? Michael adds- I understand this requires that we make some assumptions | about the shape of the Earth. Also we end up with whatever errors occur | when translating speed - time - bearing Let's be clear about it. The distortion of a position circle, when every point on it is shifted through the same course and distance, into a non-circle has nothing to do with the details of the shape of the Earth. It occurs for a spherical Earth, which is the normal assumption in all these discussions. Nor does it relate to differences between great-circle and rhumb-line displacements. The distortion occurs in the same way when a position circle is shifted due North, in which case the shift is both great-circle and also rhumb-line. I have little doubt that it would be possible to create some expression that defined the resultant distorted circle, but it's unlikely to be any simple function; more like a numerical construction. It would be centred on the first GP, after that had been displaced through the course and distance of the shift, but instead of being a circle on a polar diagram, defined by radius = 90 - alt, it would be more like radius = (90 - alt ) - some correction term * sin 2 (az - course). That would vary in the right sort of way. And then some form of simultaneous equation would be required to find the intersections with the true circle that surrounds the GP of the body of the second observation, either analytically or numerically. Doesn't seem a simple business, to me. George. contact George Huxtable at george@huxtable.u-net.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.