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George Huxtable wrote




>It's at times like this that we really need that Nav-l blackboard

Amen brother!

>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?

It's simply a variable introduced to sweep out all possible positions on
A's equal altitude circle. I thought of it as a clockwise rotation but
that's not important.  The point being that one could write equations for
the coordinates of points on A's equal altitude circle as a function of ZA,
X, and the observed altitude of A.

Another perhaps better way to think about X would be as the bearing of the
starting position on A's equal altitude circle as measured from ZA relative
to ZB.


>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'm not suggesting moving the equal altitude circles.  Leave them fixed in
place as defined by the observations. The problem is to find the two points
on these circles separated by the distance sailed vector.  My rather
involved word picture was meant to address Herbert Prinz's objection to my
original post.  His point (as I understand it) being that one cannot
translate the distance sailed vector to a change in latitude longitude
without assuming some position.  Since we have equations giving the
starting coordinate (points on A's equal altitude circle) as a function of
X, ZA, and the altitude of A, we can compute the ending point of the
distance sailed vector. I.e.
We can write an equation giving the ending coordinates as a function of ZA,
X, the altitude of A, and the distance sailed vector {I just realized this
is exactly the curve George refers to below.  We now need to find the
intersection of that curve with B's equal altitude circle.}

We can now write an equation to compute the altitude of B as seen from the
ending point as a function of X, ZA, altitude of A, and the distance sailed
vector. Since we know the observed altitude of B, we can solve that
equation for X.  That allows us to compute our starting point on A's equal
altitude circle.

I have not attempted to follow this recipe and suspect as George does that
a closed form general solution for X might be difficult or impossible  thus
requiring some iterative solution for actual observations.

>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.

I don't think the solution would involve simultaneous equations but it
might be very messy and it might be difficult or impossible to solve
explicitly for X thus requiring some sort of iterative solution for a given
pair of observations.


>George.

My thanks to George for trying to understand my babbling, Mike

   

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