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It's satisfying that so much serious interest is being taken by this
list into the question of position-finding by intersecting circles,
and the effect of a vessel's "run" on such a position circle.
Particularly for me, as I have a sent a refutation, to go into the
next issue of the RIN's "Journal of Navigation", of the recent
Zevering paper in the January issue. That may already be set in stone,
but even so, confirmation of my view at this stage by Nav-l pundits
(or even the converse, of course; perhaps especially the converse) is
of great interest.

First, I would like to resolve an apparent disagreement with Herbert
Prinz over the question of the distortion of a position circle, when
every point on that circle is shifted through the same course and
distance, to become a non-circle drawn on the Earth's surface. It may
be only a matter of semantics, but I would like to be sure it lies no
deeper than that.

Let's call such a shift of all points on a position circle as
"advancing" it through a course and distance, whatever its resulting
shape.

Herbert wrote-

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.

What I am trying to say is that introduction of a Mercator projection
is a red-herring, which confuses matters by adding extra, and
unnecessary, complication. The trouble is that representing ANY true
non-tiny circle on a Mercator projection turns it into a non-circle on
the paper; then advancing that circle adds additional distortion. It's
simpler to think about the distortion of a true circle drawn directly
on the Earth's sphere, or on a globe model of it, without bothering
about how it might be represented on plane paper.

Perhaps a simple numerical example may help, similar to one I have
previously offered in the pages of Navigator's Newsletter. It assumes
a spherical Earth.

At some moment, the Sun is directly overhead some point in the Gulf of
Guinea, at lat, long (0N, 0W). That's its Geographical position (GP)
at that moment.

At that moment, it's true altitude is measured from three vessels, to
be exactly 30 degrees. No further obsevations will be made, so there's
no need to bother about the Sun's subsequent motion.

Those vessels happen to be at the following positions-

A is at 60N, 0W,
B is at 0N, 60W,
C is at 45N, 45W.
(ignore the complication that there may be land in the way)

All three vessels lie exactly on a position circle of radius 60 deg (=
90 - alt) or 3600 miles, centred on the Sun's GP, just as you would
expect. That is the "locus" of all possible positions of such vessels,
which measure the Sun's altitude to be 30deg at that moment. For A and
B, that radius is obvious without calculation. The great-circle
distance around the Earth from C to GP can be checked using a
spherical trig formula or using altitude-azimuth tables, and will be
found to be 3600 miles also.

Next, all vessels sail 60 miles due North. Choosing due North avoids
any argument about rhumb-line versus great-circle; in the due North
case, it's both.

So their new positions are-

A moves to 61N, 0W,
B moves to 1N, 60W,
C moves to 46N, 45W.

Note that we can mark these new positions on a globe and there has
been no need for any Mercator chart.

The three vessels now lie on a new locus,  the original position
circle advanced North through 60 miles. What can we say about that
locus? Is it still a circle, and is its radius unaltered, at 3600
miles? If so, where is its centre?

Just by simple considerations of symmetry, we can state immediately
that the new locus must be symmetrical about the 0W meridian.

If it's a circle with radius still at 3600 miles (60 degrees), then to
keep the distance of the centre to the new position of A correct, it
must be centred at 1N, 0W.

If so, the distance to B, now at 1N, 60W, will be 3599.4 miles, by
spherical trig or altitude tables. So as near as dammit the assumption
seems a good one, of the new locus being a centre, with unchanged
radius, and its position simply shifted through the course and
distance of the travel. So far, that is. But wait-

The problem arises when we work out the great circle distance between
the new position of C, at 46N, 45W, and the presumed new centre, at
1N, 0W, by spherical trig or by altitude table, which we find to be
3585.4 miles. That is 14 miles short of the  "radius" of the new locus
when measured in the directions of the cardinal compass directions. So
advancing the original position circle (and by only 60 miles) has
caused the "radius" of the new locus to shrink by over 14 miles, at
that azimuth, from what it would be if it had remained a circle. The
distortion is worst in the inter-cardinal directions, NW, NE, SW, SE.
It turns what was a circle into a sort-of egg-shape.

Note that nowhere in the above example has recourse been made to a
Mercator or other projection. The distortion arises from, and is
demonstrated by, spherical geometry of the Earth's surface.

Because the distortion varies with azimuth then, as Herbert states,
it's amount is unknown until we know the final position of the vessel,
which was the object of the exercise. So some form of iteration will
be called for, destroying the simplicity of the intersecting-circles
concept.

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.

   

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