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Todd Frye wrote-

I have found an intriguing example of triangulation from a 16th century
mathematician, Willebrord
Snellius(http://www.sailingissues.com/navcourse4.html). I have searched
other web sources on the web and so far, have had no success in finding a
straight forward explanation of how this method of triangulation works. Have
any of you actually used the "Snellius Construction" technique, and could
you help me understand it as a practical navigation application?

===================

If the name Willebrord Snellius has a familiar ring, that's because it's the
Latinised version of the name of the discoverer of "Snell's Law", governing
the refraction of light across a boundary between two media.

The construction that Todd refers to was being taught as a standard method
in evening-classes in coastal navigation in the UK, all of 50 years ago,
though never attributed to Snell. You will find it treated in Bowditch, as a
"three-point fix"; in my 1977 edition it's in chapter XI of vol 1. He also
refers to a special tool for the job, as a "three-arm protractor", which in
the UK is more usually called a "station pointer". The "poor man's"
alternative is simply to draw, on a piece of tracing paper, three converging
lines, meeting at a point, the angles between those lines corresponding to
the angles between the landmarks. Then slide that tracing paper over the
chart, until the lines pass through all three marks, and you must be at
their crossing-point.

But this thee-point fix is really a tool for coastal surveying, offering
high precision, because  the relative horizontal angles between the
landmarks can be measured with a sextant, much more precisely than with a
compass; however, that level of precision is seldom necessary in coastal
navigation.

The opportunity of finding three identifiable, well-charted, landmarks
seldom offers itself until you are rather close inshore, and if you're in
motion, calls for the angles to be measured simultaneously, or nearly so.
And when close inshore, by the time you've measured the angles and done the
necessary plotting, you've moved on, so the precision is no longer relevant.
That's why it's more useful to hydrographic surveyors, working to high
accuracy from a stationary vessel.

In practical coastal navigation, my preferred approach is the traditional
one by compass, as follows- as soon as one landmark or light has been picked
up and identified, draw a position line on your chart by a compass bearing.
To do that, of course, you have to be confident about compass corrections,
but then most navigators are, unless they are in unfamiliar waters or an
unfamiliar craft. As soon as you identify a second object, draw in both
bearings, and you have a fix. When you pick up a third landmark, draw in all
three, to give a triangle or cocked hat. Only if that is unfeasibly large
should you then worry.

There's  a problem in using the 3-point method, with relative bearings. If
all three landmarks happen to lie on or nearly-on the same circle (which you
don't usually realise until you've made the observations and tried to plot
them) then the two position circles (for which you need the point of
intersection) coalesce.  That means that you could be anywhere on those
circles; there's no way of distinguishing where, and all the precision has
been lost in the resulting ambiguity.

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.


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