Polygon Methods

What are Polygons

A polygon is a shape composed of straight lines joined into a closed chain. Closed simply means that the last line of the chain is joined back to the first. Triangles, rectangles, squares (which are just special rectangles) are all polygons. A polygon may have any number of sides greater than or equal to three. Polygons may be simple or complex, convex or concave, regular or irregular, We will be concentrating on polygons which are simple, convex and cyclic. But what do these three terms mean?

Simple polygons have sides which do not cross each other, while complex polygons have sides which do cross each other. In this diagram the triangle and rectangle are simple polygons. The other two are complex. In a convex polygon, all the interior angles are less than 180°. Again the triangle and rectangle are convex, the other two are not. With a cyclic polygon all the corners lie on a single circle. And again the triangle and the rectangle are cyclic, the other two are not.

So simple, convex, cyclic polygons to the rescue! If you recall from the section on geometry we collected a number of points that we hoped were all on a single circle. If we can arrange our points so that they are the corners of a simple, convex, cyclic polygon then we just need to find the centre of the single circle to locate the emergency transmitter. Some polygons are better than others however.

But before we continue we need some more terminology. The single circle upon which all the corners of a cyclic polygon lie is know as the circumcircle for the polygon. The centre of the circumcircle is know as the circumcentre.

Triangles

At first look one might think that triangles are a good choice. Those who are already familiar with Aural Null procedures will know (and the rest of us will soon learn) that Procedure B is based on the triangle. All triangles are simple, convex and cyclic. That means that for any three aural null points we collect we will get exactly one triangle, one circumcircle and one circumcentre. Why is this a bad thing? Well, it means that if any of the three points aren't really aural null points but are mistakes, we will still get a triangle with a circumcircle and a circumcentre. It would be much better if mistakes resulted in no circumcentre, or some other indication that there have been errors. Sadly with triangles this is not the case.

The circumcircle and circumcenter of a triangle are found using a method very similar to how we found the centre of a circle using perpendicular bisectors from a starting chord. We use the three points to draw the triangle, then construct perpendicular bisectors on each side. Where the bisectors cross is the circumcentre of the triangle:

If you still don't believe that the triangle is such a poor choice, Math Open Reference has a very good page dedicated to the circumcircles of triangles. It even has an interactive triangle. You can move the corners around as much as you like and see that each new triangle has a circumcircle. If you use this procedure to find the circumcentre of a triangle and the three bisectors don't all cross at one point, that would be due to sloppy geometry, not due to errors of marking the points.

More is Better

So four or more sides are better, but not fool proof. Regular polygons, polygons where all the sides are the same length, and rectangles are, like triangles, always cyclic. Luckily it would be very difficult to fly an aircraft so that the aural null points form a regular polygon. It is much more likely that the sides will be different lengths giving us an irregular polygon. Some irregular polygons have circumcircles and are therefore cyclic polygons. Certainly we are hoping that our aural null points will form a cyclic polygon. If they form an irregular, cyclic polygon the odds are in our favour that is because the points are all on the boundary of a circular signal area, not because of random chance.

The procedure from the geometry section gave us four points. So we can use the same tools, perpendicular bisectors, to find the circumcircle of the four sided simple, convex, cyclic polygon formed by those four points. In this case if the bisectors don't all cross at one point it could be due to sloppy geometry. But it could also be due to errors in the point location due to irregularities of the signal area or navigation errors. Of course it would most likely be a combination of all of those. It is difficult to perform good geometry in the cockpit of an aircraft in flight. That is where software comes in very handy by eliminating the sloppy geometry and the effort of performing manual geometry as well. Since the software uses GPS the likelihood of navigation errors is greatly reduced (but not eliminated).

This is the genesis of what we are calling Procedure C. In stead of relying on only three points and locating the circumcentre of a triangle, which does not give any indication of the quality of the fix, we collect at least four points to construct a simple, irregular, convex polygon. This provides multiple intersecting bisectors. The intersections of the bisectors will diverge more and more as the quality of the points is reduced.

Now that we are familiar with the geometry involved it is time to see how the aircraft radio can help us with the search.
Comments