A NEW METHOD FOR PLOTTING THE POSITION LINE: THE GOLEM SOLUTION

E. Gradsztajn

Peer Reviewed

Abstract:	Nowadays the navigator, in order to obtain the position line has the choice between the St. Hilaire method (1875) which can be called geometrical, and any analytical method yielding two or several points of the position line. We shall compare these two kinds of methods in section III. Captain T. Sumner, after he discovered the existence of the position line (1837) proposed the so-called longitude method where a point of the position line is obtained by cutting the terrestrial position circle by a parallel of latitude. Unfortunately this simple method cannot be used when the sight is taken near the meridian (noon time). One can overcome this difficulty, as St. Hilaire mentioned in 1873, if the position circle is cut by a meridian, but then the same difficulty appears near the prime vertical. In order to decide how to cut the position circle it can be suggested to test the value of the derivative of the latitude as a function of the hour angle dL/dHA = cos L tan Z where Z can be the measured azimuth and L the estimated latitude. If dL/dHA is smaller than unity, cut the position circle by a meridian, otherwise, by a parallel. But this does not avoid the difficulty arising when the chosen parallel or meridian does not cut the position circle at all. These complications led St. Hilaire to cut the position circle by the vertical circle of the observed body and to introduce the concept of the estimated altitude (1875). In the method presented here, the coordinates of any point M of the position circle (Fig. 1) are calculated as functions of the position angle E. This center angle is defined by two great circles: the hour circle PO and the vertical circle MO. The entire circle can be built in this way, as soon as the coordinates of the geographical position 0 and the radius of the circle are known. The way the navigator chooses the interesting fraction of the position circle will be described in section II. Two points will be sufficient in usual cases to plot accurately and very easily the position line. It is obvious that this method is valid for any altitude or latitude, even very high, and for any kind of chart. We shall also see in the following section that the distance of the estimated position from the position circle does not introduce any approximation in the calculations, no matter how long it can be. This new method is of particular interest if used with a hand-held calculator having a built-in program. The calculation may also be performed with a programmable or an ordinary calculator or even with trigonometric tables or special tables.
Published in:	NAVIGATION: Journal of the Institute of Navigation, Volume 26, Number 1
Pages:	70 - 77
Cite this article:	Gradsztajn, E., "A NEW METHOD FOR PLOTTING THE POSITION LINE: THE GOLEM SOLUTION", NAVIGATION: Journal of The Institute of Navigation, Vol. 26, No. 1, Spring 1979, pp. 70-77.
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