PLOTTING LINES OF POSITION

You may recall, from an earlier discussion, that with a large enough chart and a very large pair of dividers you could plot the G.P. of the body on the chart, and draw a circle with a radius equal to zenith distance, or 90˚ minus ho, with the G.P. as its center. Your position would lie somewhere on that circle, so it would be a line of position (lop).

We rarely plot the line of position this way in practice. To have the G.P. and the vessel close enough to be on the same chart or plotting sheet, ho would have to be close to 90˚, and it is very difficult to get an accurate sextant observation of a body nearly overhead.

You do draw a small part of the arc of that circle. The radius of the circle is so large as, much as 4000 miles, that the curvature of the arc isn't even noticeable. The short arc of the circle you use for a line of position LOP can be represented as a straight line with no serious error.

That straight line is a tangent of the circle, or a line that touches the circle at one point. A radius drawn from the center of the circle to the point where the tangent touches the circle will be at a right or 90˚ angle to the tangent.

Even though you can't plot the G.P. on the chart and draw the complete radius, you can draw part of it. When you used 229, you found the computed altitude (Hc) of the body, and its azimuth, (Zn) or true bearing from the assumed position to the G.P. a straight line drawn from the assumed position A.P. in the direction of the Zn is part of the radius. The line of position LOP must lie at right angles to it.

If you had actually been at the A.P. when you made the observation, the lop would pass through the A.P. at right angles to the radius of the circle. It will be more convenient to call the part of the radius you draw from the A.P. the azimuth line.

You'll probably never see that happen. Hc would have to equal ho, which would give you an altitude intercept (a) of zero. The odds are greatly against that happening because you chose the a.p. to avoid interpolation for latitude and LHA. The intercept (a), the difference between Hc and Ho, is the distance in miles from the a.p along the azimuth line to the point where the lop crosses the azimuth line. That distance will be measured from the a.p. in the direction of the Zn if (a) is named toward (Ho greater than Hc), or in the opposite direction if (a) is named away (Hc greater than Ho)

Notice that in the preceding examples we drew the azimuth lines as dashed lines and the LOP's solid. The reason is that when you plot three or more stars from A.P.'s pretty close together, it would be easy to confuse azimuth lines with LOP's. The fix is where the LOP's cross. The azimuth lines are merely used to construct the LOP's and have nothing to do with the fix. Using different colored pencils to distinguish azimuth lines from LOP's would be great if you could find colored pencils that will stay sharp. A sharp pencil and careful plotting is very important. Concerning accuracy of plotting, some tolerance has to be allowed. On the usual plotting sheet, a line drawn with a freshly sharpened #2 pencil might be two tenths of a mile wide. The most experienced navigator who plots the same fix twice will not get exactly the same latitude and longitude twice. Most people get unduly concerned when they're answer to "what is the latitude and longitude of the 1500 fix?" doesn't come out exactly like the book answer. In problems that don't involve drawing, you should get the book answer. In the practice plotting problems, you will most likely differ from the book. When you're still a beginner, be happy if your result is within two or three miles of the book answer. That will prove that you haven't made any gross mistakes and that your procedure is correct. Your plotting accuracy will improve with practice, but nobody can ever get rid of all plotting errors. Relax and enjoy the plotting exercises. Allow yourself a little "slop" and concentrate on procedure--drawing everything in the right direction--at first. For more practice, rework -the same exercises on a fresh sheet. When your fixes start coming within a mile of the book answers, your plotting work will be plenty good enough to pass a coast guard test.

Another thing that worries students unnecessarily is finding a three line fix that doesn't cross to a pinpoint but forms a small triangle.

It is customary to assume the fix is in the center of the triangle, which you can judge quite accurately by eye. No doubt the person who made up the problem started with a pinpoint and worked backwards to create a problem, intending for you to get the same pinpoint. But a normal and perfectly acceptable amount of plotting error can cause your three lines to form a triangle as large as a mile on a side. Eyeball the center of your little triangle and go with it. It won't differ from the author's pinpoint by more than a half mile. That will be close enough to pick out the correct multiple choice answers.

If, however, you wind up with a large triangle several miles across, something is wrong with your work. One or more of the LOP'S is out of place. There is no way to tell which is wrong. Probably the quickest way to correct the mistake is to work the whole thing over again on a fresh sheet. Going over your existing work trying to find a mistake is usually a waste of time. We all tend to believe what we see on paper, especially if we put it there. In no time a man can convince himself that his own work is correct and the mistake must be in the coast guard (or book) problem. That's possible, but not very likely.

Here are five exercises for plotting one LOP at a time. Set up one of the universal plotting sheets for latitudes 27˚-30˚ to work the problems. The answers aren't the sort you'll see on the test. That's just a convenient way of checking to see that the LOP is in the right place.

	Assumed Lat	Assumed Long	Zn	a	Where does your LOP cross?
1	29˚N	113˚41′E	040˚T	10.5T	114˚E
2	28˚N	114˚10′E	101˚T	17.0A	28˚N
3	28˚N	114˚52′E	355.5˚T	22.5T	115˚E
4	27˚N	116˚12′E	175˚T	12.2A	116˚E
5	29˚N	116˚23′E	268˚T	3.6T	29˚N

Answers
1	29˚00.0′N
2	113˚51.0′E
3	28˚22.0′N
4	27˚11.5′N
5	116˚19.5′E

If you had pretty good results plotting one LOP at a time, try this one with three lines which will form a fix. When working with more than one at a time, it's easy to get mixed up and plot the LOP for Deneb from the A. P. that goes with Fomalhout, for example. It's a good idea to label the A.P.'s to prevent that mistake. You can number them to keep your plot from getting too cluttered.

PROBLEM: At 1740 the navigator and two assistants observe simultaneously three stars, with the following results:

	Fomalhaut	Deneb	Aldebaran
Hc	28˚10.3′	34˚59.6′	39˚52.8′
Ho	28˚05.3′	35˚05.6′	39˚46.8′
Zn	210˚.0	308˚.7	089˚.3
Assumed Lat.	28˚00.0′N	28˚00.0′N	28˚00.0′N
Assumed Long.	42˚31.7′W	42˚29.0′W	42˚23.2′W

There was something a little unrealistic about that last problem. The navigator had two assistants capable of making observations at the same time he did! More likely you'll be working by yourself and several minutes will pass between observations. If your boat is going anywhere during that time, your fix won't plot properly unless you adjust all the LOP's to a common time.

You could plot all the LOP's, then move two of them forward or back parallel with the track to meet the third. It's much easier, and makes for less cluttered plot, to advance or retire the A.P.'s before you plot the lines of position. You get the same result with fewer lines drawn on the chart to add to the confusion.

The 1800 DR position of a ship is Lat. 27˚02.2′N, Long. 170^˚17.0′W. The ship is on course 045˚, speed 14 knots. During evening twilight the navigator observes three stars, with the following results:

	Dubhe	Altair	Spica
Time	1815	1821	1830
Hc	34˚45.2′	22˚11.8′	47˚24.8′
Ho	34˚51.3′	22˚15.7′	47˚20.4′
Zn	331.5˚	090.3˚	219.9˚
Assumed Lat.	27˚00.0′N	27˚00.0′N	27˚00.0′N
Assumed Long.	170˚10.2′W	170˚05.0′W	169˚54.8′W

But before you work it, here's what we mean by advancing A.P.'s. The boss wants an 1830 fix, so you can plot the Spica line from its A.P. as it stands. The Dubhe A.P. will have to move in the same direction and the same distance the ship traveled during the fifteen minutes from the time Dubhe was shot until 1830. Altair's A.P. must be moved the same direction at boat speed for nine minutes. Just DR the A.P.'s ahead the way you would the boat.

Then plot the LOP's from the advanced A.P.'s (Your plot won't look exactly like the example--it's drawn to a different scale from your plotting sheets.) On this next one, the man wants a 1942 fix. You'll leave the Regulus A.P. alone, move Saturn forward in the same direction as the ship for eleven minutes at 16 knots, and bring the Rigel Kent A.P. back in the opposite direction for nine minutes at 16 knots.

The 1930 DR position of a ship is lat. 29°10.5'5, long. 12˚35.4′W. The ship is on course 320˚, speed 16 knots. During evening twilight the navigator observes a planet and two stars, with the following results:

Did the appearance of a planet in that last problem bother you? It shouldn't have, because you were given all the information needed to do the problem without having to reduce the sight.

1 Introduction	9 Altitude Intercept	17 Latitude by Meridian Altitude
2 Corrections to Sextant	10 Using Position Plotting Sheets	18 Latitude and Azimuth by Polaris
3 Altitude Corrections	11 Plotting Lines of Position	19 Running Fixes
4 Time	12 Summary of 1 thru 11	20 Time of Sunrise Sunset
5 Finding GHA and Declination	13 Finding Deviation or Gyro Error	21 Star Identification/Selecting for Fix
6 Assumed Position and Local Hour Angle	14 Finding Azimuth by 229	22 Time Tick Problems
7 Computed Altitude and Azimuth	15 Amplitudes	23 Deviation Table Construction
8 Interpolation	16 Time of Local Apparent Noon	24 Sample Final Exam