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 GPS receivers nowadays use algebraic solutions. Plotting is meant for 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.htmlScroll 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
