NavList:

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From: George Huxtable <george@hux.me.uk>
To: NavList@fer3.com
Sent: Sun, June 13, 2010 8:52:24 AM
Subject: [NavList] Re: Graphical Solution to Double Altitude Sight using Stereographic Projection

On 8 June, Robin Stuart referred to a possible use of a stereographic
projection (in this case, from the South) in obtaining a graphical solution
to the intersection of position circles.

He asked "Has anyone seen this approach referred to or used?".

However, I've seen no responses. All I can do is to add my own, and say-

No, I don't recall ever having come across such a method. It seems
completely novel, to me, and rather interesting, if applicable to only a
restricted set of practical cases.

I wonder if there's an analogy, here, with the use of the astrolabe
(astronomers' not mariners'), which maps the stars of the Northern sky with
exactly such a projection

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
----- Original Message -----
From: "Robin Stuart" <robinstuart@earthlink.net>
To: <NavList@fer3.com>
Sent: Tuesday, June 08, 2010 12:18 AM
Subject: [NavList] Graphical Solution to Double Altitude Sight using
Stereographic Projection


In my talk at the 5 June Navigation Weekend at Mystic I showed, as an
aside, how position could be determined from a double altitude sight by
purely graphical (ruler and compass) methods with a stereographic
projection chart. The problem considered was taken from Lecky, S. T. S.,
"Wrinkles" in Practical Navigation, George Philip & Son, Liverpool, 1886;
http://books.google.com/books?id=dmbOAAAAMAAJ
and involves two observations of the Sun roughly 4 hours apart. The
relevant values for Greenwich Hour Angle (GHA), declination (dec) and
zenithal distance (ZD) of the Sun are
GHA1 = 0h59m59.10s  dec1 = -4 59'19.9"  ZD1 = 40 00m00.0s
GHA2 = 4h59m58.97s  dec2 = -4 55'26.0"  ZD2 = 56 43m15.0s
The solution is shown under the usual Mercator projection in Lecky's
chartlet (attached). The circles of positions (CoP's) on the chartlet are,
of course, not circles or any convenient shape for that matter. The same
problem is shown using stereographic projection with the Geographic
Position (GP) of the Sun at the 2 observations being labelled zp1 and zp2.
Note that the CoP's remain circles on this chart but their centres are not
at zp1, zp2. The actual locations of the centres, labeled zc1 and zc2, and
the radii of the circles can be found graphically and the observer's
location immediately follows.

The centre and radius of each circle on the chart can be found as described
below:
1) Locate the vertices (farthest North and South points) of the circle.
These have the same longitude as the GP of the Sun and latitude (dec + ZD)
and (dec - ZD). For the first circle the vertices are at
35 00'40.1"N 14 59'46.5W (0h59m59.10s)
44 59'19.9"S 14 59'46.5W (0h59m59.10s)
Plot these points on the chart and draw a line between them.
2) Use ruler and compass to locate the midpoint of the line (a standard
trick that every school pupil learns) and hence find the centre of the
circle.
3) Set the compass using the distance between the center and the vertex.
4) Draw the circle

Accuracy is limited by the scale of the chart but for this particular
problem if we assume a chart of 100cm in height (radius), a 1mm plotting
error corresponds to about 5 n.m. error in position. In certain situations,
e.g. for sights too far off the zenith, the CoP's may become too large in
stereographic projection to be drawn and render the method impractical.

Has anyone seen this approach referred to or used?

Robin Stuart
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