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Michael Dorl wrote:

> At 11:29 AM 6/8/2006, you wrote:
>
>> Michael Dorl wrote:
>>
>>> Since one has a good idea of the course and speed between any two
>>> observations,
>>> it seems to me the problem is to determine points on the equal altitude
>>> circles separated by that vector.
>>
>
> and Herbert Prinz replied
>
>> Right. And this has to be done point for point along the circle, not by
>> just moving its center by that vector. The "circle" isn't a circle on a
>> mercator projection to start with and changes its shape when being moved
>> north or south. Said vector translates to different delta longitude at
>> different latitudes, so by how much do you shift the GP of the star? You
>> know this only after you have the latitude of your fix.
>
>
> Bear with me...
>
> I'm not thinking of any kind of graphical solution but rather the
> spherical
> trig behind it.


Michael,

Yes, so do I. I was just pointing out that the mercator projection is at
the heart of the transformation that you are looking for. I am not sure
why George Huxtable insists so adamantly that mercator or the loxodrome
has "NOTHING to do with it".

What does it mean to say "Since my last observation, I sailed for 4
hours at 9 knots on COG 075, then I took another one."? It means that
for each point on the position line of my first observation I have to
find its mercator-representation in x,y coordinates, advance the
position by 36*(sin 75, cos 75), convert back into spherical and
intersect the resulting egg with the second LOP. Since there is no
closed form representation of the egg, I will do this point by point as
I go along. The deformation of the circular LOP comes entirely from the
properties of the mercator projection.

Of course, the change in shape over short distances is not dramatical.
So there are solutions that consist of advancing the GP, keeping the
circular LOP and correcting the small error in the resulting position.
With the last two sentences in the quote above I addressed the problem
that the correction is dependent on the observer's position and so it
must either be roughly known beforehand or the algorithm becomes
iterative. In that it resembles the intercept method and it looses all
its potential advantages over the latter.

Herbert Prinz

   

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