Calculating the track of a plane or other object through concentric circles is not complicated. In fact, it is a derivative of the same math used every day to make precision eyeglasses, telescopes, and other optical instruments that require precise “depth of arc” measurements. As a rule, the math revolves around measurements of the “Sagitta” or “Sag”: a term of Latin derivation that refers to the center of an arch or Arc.
In this instance, we need to work with both the Sagitta and the “Annulus”. The latter is the empty space between successive congruent rings. Exactly half of the width of the annulus (between successive rings) gives us the Sagitta in these exercises. The following illustration shows an annulus between congruent circles. But in calculating the annulus in this instance, we only want a small part of it. Not the whole Chimichanga.
The Annulus is not shaded in the illustration below, but it is easily identified as the area between the White 4th ping ring and the Black 5th ping ring. It’s important because, when we add the mirror copy of the 5th ping ring (both shown in black), we want the mirror ring to cross the baseline exactly where the White 4th ping ring crosses it.
Once mirror ping rings are in place, Annulus Chords can be extended as shown with the Magenta line that terminates at Pin #5.
These steps are repeated for each ping ring of interest. With MH370, the final ping was of greatest interest because families still want to confirm the plane’s terminal location. Many probably also hope the Black Boxes will be recovered and will still be useful in completing our understanding of the tragedy that took their loved ones so cruelly.
While this “Sagitta of the Annulus” approach is ambiguous in that it always gives us two possible solutions, it remains an accurate and powerful tool. Perpendicular bisector solutions are always a little ambiguous, but it is a small annoyance. Some may liken this approach to quantum physics when it comes to outcome uncertainty. But knowledge of Quarks is not required to find MH370.
This math does not return a GPS as presented here, but if used in conjunction with GPS-mapped software like Google Earth, precise GPS values are easily obtained.