# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Finding The Symmedian
From: UNK
Date: 2010 Dec 18, 20:06 +0000

```On 2010-12-17 18:22, Greg Rudzinski wrote:
>
> The linked diagram shows an alternate route to finding the symmedian
> point by using two sides with corresponding vertex. A short spread on
> a divider compass expedites bisecting of angles without the aid of a
> protractor. Dividing a side in half is done by a few trials of the
> divider. Transposing the bisected vertex and bisected side is
> facilitated with a circle.
>
> It is not practical to be using a symmedian point when laying down a
> round of stars but I do see a place for it where a round of GPS
> satellites are automatically analyzed for ultra precise positioning.
>
>

I find it hard to believe that there be little Gremlins in our GPS
receivers plotting the solution to the minimum square sum problem. Most
people, especially for those of us who would rather approximate a
solution within the accuracy limits of a rough plot than tackle an
involved algebraic solution.

In some of the recent posts I have come across a strange notion that
"eyeballing" is a method in its own right to determine the position of
the MPP,  quasi an alternative to various constructive methods. How can
this be? How can one find a point by "eyeballing", when one does not
know what to look for in the first place? Once you know that the point
in question is the symmedian point , you can eyeball it easily by
exploiting some of its unique properties. Or by doing the construction
in your head. Experience helps. "Eyeballing" is a way of implementing a
given method, not an alternative to it.

Several more or less obscure alternatives to the symmedian point have
been proposed without any mathematical foundation or reasoning
whatsoever. It is totally backwards to come up with a substitute for the
symmedian point that is arguably more or less different from the MPP
(dependent on the geometry of the situation) and then plot this wrong
point with perfect accuracy. It makes by far more sense to find a quick
and dirty method to plot the correct point.

Anybody who takes the time to look at Villarceau's diagram that Frank
has kindly posted will be hard pressed to come up with an easier or
faster construction. You don't even need dividers! As I mentioned before
here on this forum, his construction is based on the property of the
symmedian point that x:y:z = a:b:c, where x,y,z are its distances from
the sides a,b,c.

The only tool you need for Villarceau's method is a  90-45-45
plexiglass  plotting triangle. It has with a scale on its hypotenuse
running outwards in both directions from the center, and it has a
perpendicular grid for drawing the parallels and perpendicular lines.
Leave the dividers to Euclides and the purists! To cut down on the
required plotting area, I recommend:  1) Instead of plotting the
parallels at distances a,b,c, use the half distances. That's where the
symmetric scale of the plotting triangle comes in handy. 2) Instead of
plotting the parallels outside of the triangle, plot them on the other side.

Not practical? You must be kidding.

Coming back on "eyeballing". When I spoke in Mystic about finding the
MPP geometrically, I started out with drawing a rectangle on the
blackboard. Then I passed a laser pointer around the audience and
everyone had to "eyeball" the MPP. I marked every guessed point with
chalk. You can see the results at

http://www.fer3.com/mystic2008/afterphotos.html

Scroll down until you see me scribbling on a blackboard with some
drawings and then zoom in on the triangle in the top left corner. The
correct point is at the cross. Most people tend to be too far near the
acute corner. One person guessed it right. It should not come as a
surprise that he is an experienced navigator. You will also see that I
cheated: How could I be sure that I would find the MPP instantly with
perfect accuracy? My triangle has a right angle. At the end of the
presentation, I drew the altitude to the hypotenuse, cut it in half, and
voila!

Every triangle that I have ever seen is close to one of the following
special cases: Equal sides, isosceles, rectangular, degenerate (i.e.
very very acute, very very obtuse). Know where the symmedian point is in
each of these trivial cases and you will never fail to make an educated
guess about the MPP. Art or science? In the olden days there was no
difference.

Herbert Prinz

```
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