An interesting and fruitful way of dealing with errors in positions obtained from Astronomical observations involves a consideration of the bisectors of pairs of position lines. The minimum amount of information necessary for fixing a vessel’s position consists of two -intersecting position lines. Any pair of intersecting straight lines produces two bisectors mutually perpendicular to each other, but here we are concerned only with that bisector which not only bisects the position lines but also bisects the angle contained between the bearings of the observed bodies which produce the position lines.

Fig. 36-1 illustrates two Astronomical position lines obtained from simultaneous observations of celestial bodies whose azimuths at the time of the observations are indicated by *X and *Y. It will be seen that BB, the bisector which bisects the angle XOY, also bisects the position lines.

Now consider the two position lines illustrated in fig. 36-2. These position lines intersect at F which, if there is an error in one or both of the position lines, is a false fix. Let us suppose that the position lines in fig. 36-2 have been plotted with the same systematic error – an error perhaps due to the application of the wrong I.E., or a constant personal error – affecting the two position lines. Let this systematic error result in an error in each of the position lines amounting to e, as illustrated. The true position of the vessel is, therefore., at T.

It can easily be seen that regardless of the magnitude or the sense of e, the true position of the vessel lies .on the bisector of the two position lines, provided that the same systematic error, and no random error, affects both position lines. It must be emphasised that the true position lies on the bisector of two position lines only if systematic error alone influences the observations.

The principal and most useful feature of a bisector is that by using it as a position line, a systematic error can be entirely eliminated. A second important feature, which can be demonstrated mathematically, is that random errors are averaged. It follows that a more reliable position is possible by crossing two bisectors than by crossing three or four position lines.

The position lines obtained from Astronomical observations of more than two bodies seldom intersect at a common point. Because of errors in observation, computing or plotting., three Astronomical position lines usually intersect to form a cocked hat. By taking pairs of position lines obtained from simultaneous sights of three stars, three bisectors may be drawn, and these three bisectors will always intersect at a common point.

The most likely reason why three Astronomical position lines do not intersect at a common point is that the altitudes obtained from observation are incorrect. This leads to the displacement of one or more of the three position lines from the vessel’s true position. This, in turn, results in the formation of a cocked hat. In: general, the bigger is the cocked hat, the bigger is the error in the position lines.

In fig. 36-3, P represents a position used to compute intercepts I, I and I from observations of three stars *A, *B and *C, Those azimuths differ by about 120º respectively.

The resulting position lines AA BB and CC intersect to form a cocked hat. If the same systematic error has affected each of the three position lines, the vessel’s true position will lie on each of the three bisectors X, Y and Z. Those bisectors intersect at T which is the centre of the in-circle of the cocked hat.

A common method of dealing with a cocked hat is to apply a trial-and-error method by moving each of the position lines (in the imagination) through the same distance either all towards or all away from the directions of the observed bodies until they intersect at a common point which is taken to be the vessel’s probable position. Such a fix is sometimes called a cartwheel fix. The principal employed in this method is the same as that used in the bisector method.

In the example illustrated in fig. 36-3, the three intercepts are named towards. Had they been of different names, or had they been all three away, the vessel’s true position would still have been inside the cocked hat, and this will always apply when the systematic error affects each of the position lines and the observed bodies are space equally, or nearly so, around the observer. However, fig. 36-4 serves to demonstrate that if all the sights are taken on one side of the observer within a sector of 180º or less, the vessel’s true position will lie outside the cocked hat when the same systematic error affects all the sights.

In fig. 36-4 P is a position used to compute the three intercepts I, I and I. The three resulting position lines form a cocked hat, but the three bisectors intersect at T, outside the cocked hat, this being the vessel’s true position if the same systematic error has affected each of the three position lines. The point T could have been found by the cartwheel principle, in which case each position line would have been moved through a distance d as illustrated in fig. 36-4 It follows that T is the centre of the circle which touches the three position lines at a. b. and c. respectively.

A cocked hat provides information that an error has affected the sights, but it gives no indication of whether the error is systematic or random. It is true that if the navigator is sure that systematic error exists, he may fix his vessel reliably using bisectors or the cartwheel method but, in practice, the navigator is never sure that random error is not present. There is, however, a way by which random error may be detected and, in this event, a navigator may be able to determine the degree of reliability of his fix.


From a consideration of the properties of bisectors, it would appear that the ideal requirements for fixing by Astronomical observations consist of four position lines obtained from simultaneous sights of four stars equally spaced in azimuth around the observer.

Referring to fig. 36-5 suppose that four stars *W, *X, *Y, and *Z. produce the four position lines AA, BB, CC, DD respectively. It will be noticed that the azimuths of the four stars are each directed away from the centre of the square formed by the intersection of the position lines.

By using the cartwheel principle, or the bisectors of parallel pairs of position lines, the vessel’s probable position is at F. the centre of the square.

The pairs of position lines AA and CC, and BB and DD, are separated by the same distance. This coupled with the fact that, relative to F. each of the position lines lies towards the direction of the observed body which produced it, suggest that a common systematic error has affected all four observations. In other words, if four position lines cross as they do in fig. 36-5, there is every possibility that a systematic error only has affected the sights and the vessel’s probable position at the inter-section of the bisectors, as illustrated, is a reliable fix.

However, had the four position lines obtained from the observations of four stars, *W, *X, *Y and *Z intersected as shown in fig. 36-6. the fix at the intersection of the bisectors would not be such a reliable position as that shown in fig. 34-5. It will be noticed in fig. 36-6 that by using the cartwheel principle the effect of moving each position line a given amount towards, or away from the direction of the appropriate observed body, the shape of the area cut off by the intersecting position lines becomes rectangular, but not square. This suggests that random error (in addition perhaps to systematic error) has affected the sights.

It follows that by using four position lines which intersect at right angles or nearly so, the navigator may be provided with evidence of the existence of random error in his sights. By using bisectors, systematic error will automatically be eliminated, and this sort of error, therefore, need not worry the navigator unduly. A navigator who has a positive personal error will, when using four position lines in the way described, normally find that the azimuths point away from the centre of the area of intersection of the position lines. The reverse will be the case if personal error is negative.

Knowledge of one’s personal error can therefore be put to good use in assessing the reliability of a four-star fix. Conversely, using four-star observations of the type described, an effective way of ascertaining one’s personal error is provided. When taking star sights in order to remove or reduce the possibility of faulty sights, a series of three or five shots of each of the observed stars should be taken in quick succession.

After first checking the differences between successive observed altitudes and times and comparing them with the rate of change in altitude so that a faulty sight may be detected, the results should be averaged (see Ex. No. 4 in § 33), and the average values for each of the observed bodies used in the sight reduction. The resulting position lines should then be analysed for systematic and random error as described in this section.

A fix obtained from a series of single shots of each observed body should not be regarded as favouring the production of a reliable observed position and, in general, analysis of position lines obtained in this way is not regarded as being a fruitful procedure.

For the discussion of the treatment of navigational errors in these Chapters we are indebted to two sources of reference. The use of bisectors for analysing star sights was brought to the notice of navigators by Admiral L. Tonta of the Royal Italian Navy in 1931 in an article in Hydrographic Review Vol. 8, and more recently by Captain Mario Bini, of the Italian Navy, in a valuable paper which appears in Volume 8 (1955) of the Journal of the Institute of Navigation.

These chapters concludes that part of the Ocean Navigation which deals exclusively with the determination of position at sea by practical observation of celestial bodies. The Self Assessment which follows takes the form of a revision exercise covering the various aspects of this subject which have been discussed in the preceding chapters.

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