Geometry

Terminology

Terminology of a Circle
A good place to start is with some terminology used in discussing the geometry of a circle. Of course a circle may be defined by the location of its centre and its radius or diameter. Less commonly know term that we will also be using are the secant and the chord. We will not be using tangent in this guide.

A chord may be obtained by terminating a secant at the points where it crosses the circumference of the circle, or by drawing a straight line between any two points on the circumference. Any chord that passes through the centre of the circle defines the diameter.

Recall from our introduction that we have postulated an imaginary device that will indicate when the search aircraft is within some unspecified distance from the emergency transmitter. If the crew is able to mark where the aircraft moves from an area where there is no such indication (outside the circle) to an area with that indication (inside the circle), those marks would be points on the circumference of a circle centred on the transmitter. The task then is to determine the location of the centre given those points. There is a direct method that starts with two points on the circumference. Of course the crew can not see the circumference, they can only find points on the circumference by flying from the inside of the circle to the outside and back. 

Finding the Centre


How to Find the Center of a Circle

The video at the left shows the procedure given the circle. Adapting it to our case where we have points on the circumference, but not the circle the procedure is described below.

Draw circles to define a perpendicular bisector.
The first two points (black) are collected as the aircraft flies a secant through the transmitter signal during a search. The first point when the aircraft flies into the signal, the second when if flies out. Draw the chord by joining the two points with a straight line (blue). Draw a perpendicular bisector of the cord as shown in the video, draw two circles each with the same radius, centred on the chord so that the circumference of these two circles each pass through one end point of the chord (green).

Fly the perpendicular bisector to collect points for a chord.
Draw circles to define a new perpendicular bisector.
The perpendicular bisector (purple) passes through the two points where the circles intersect. This new line is a chord that passes through the centre of the circle (the diameter). The crew then flies along the perpendicular bisector in each direction from the first chord to locate a point where the bisector crosses the circumference of the original circle, still unseen.

With these two new points the bisector may in turn be bisected.
Draw the final bisector to find the centre.
The tree chords and the original circle.
Where this new bisector (red) crosses the first is the centre of the circle. We can now draw the circumference of the circle we've been searching for (black far right). Those already familiar with aural searches should recognize this an Aural Null Procedure A. In practice the lines are not bisected using circles, but using a ruler and a right triangle. All forms of the Aural Null are based similarly on points collected by the search crew which are joined to form chords. The chords are then divided with perpendicular bisectors.

While this procedure is quite straight forward with pen and paper, implementing it in a search aircraft presents some difficulties. The crew must plot all the points accurately. They must also fly the purple bisector accurately otherwise the points they collect will define a chord that is not the diameter and thereby introduce errors. Also, although the four points collected should be mathematically related, this procedure does not take advantage of that relationship to provide any information about the amount of error. Luckily there are other geometric methods of finding the centre of a circle which we will discover in the next section Polygon Methods.

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