Planning the Fix
In selecting bodies for observation, one should generally consider azimuth primarily and such factors as brightness, altitude, etc., secondarily. If all observations were precisely correct in every detail, the resulting LOPs would meet at a point. However, this is rarely the case. Three observations generally result in LOPs forming a triangle. If this triangle is not more than 2 or 3 miles on a side under good conditions and 5 to 10 miles under unfavorable conditions, there is normally no reason to suppose that a mistake has been made. Even a point fix, however, is not necessarily accurate. An uncorrected error in time, for instance, would require the entire fix to be moved eastward if observations were early and westward if observations were late, at the rate of 1 minute of longitude for each 4 seconds of time.
In a two-LOP fix, the ideal cut of the LOPs is 90°. In Figure 11-12, a 90° cut with a 5 NM error in one LOP causes a 5 NM error in the fix. If the acute angle between the LOPs is 30°, a 5 NM error in one LOP causes a 10 NM error in the fix. Thus, with a two-LOP fix, an error in one LOP causes at least an equal error in the fix; the smaller the acute angle between the LOPs, the greater the fix error caused by a given error in one LOP. Of course, if both LOPs are in error, the fix may be thrown off even more. In a three-LOP fix, the ideal cut of the LOPs is 60° (star azimuths 120° apart). With this cut, a 3 NM error in any one LOP causes a 2 NM error in the fix. With any other cut, a 3 NM error in any one LOP causes more than a 2 NM error in the fix. In a three-star fix, the cut will be 60° if the azimuths of the stars differ by 60° or if they differ by 120°. If there is any unknown constant error in the observations, all the Hos will be either too great or too small.
Notice in Figure 11-13 that, if stars are selected whose azimuths differ by 120°, this constant error of the Hos causes a displacement of the three LOPs, either all toward the center or all away from the center of the triangle. In either case, the position of the center of the triangle is not affected. If you use any three stars with azimuths outside a 180° range, any constant error in observations tends to cancel out.
The three-star fix has two distinct advantages over the two-star fix. First, it is the average of three observations. Second, selecting the stars carefully can counteract the effect of constant errors of observation. There is also a third advantage. Each pair of two LOPs furnishes a rough check on the third. In resolving an observation into a LOP, you might possibly make a gross error; for example, obtaining an LHA that is in error by a whole degree. Such an error might not be immediately apparent. Neither would such a discrepancy come to immediate attention in a two-LOP fix. However, this third advantage does not apply when a single LHA is used in solving all LOPs, such as is done when precomputing and using motion corrections to resolve all LOPs to a common time. Because of these three advantages, it is evident that a three-star fix should be used, rather than a two-star fix, when possible.
Whatever the number of observations, common practice, backed by logic, is to take the center of the figure formed unless there is reason for deviating from this procedure. Center is meant as the point representing the least total error of all lines considered reliable. With three LOPs, the center is considered that point within the triangle equidistant from the three sides. It may be found by bisecting the angles, but is usually located by eye.