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Lines of Position It is possible to solve part of the fix problem without knowing your exact location. For example, assume that you are in a strange town and you call a friend to meet you downtown. If you tell this person that you are somewhere on Main Street, your friend can limit any search for you to just Main Street. In this case Main Street is like a line of position (LOP). An LOP is a series of possible positions. It can be a straight line such as a city street or bearing from the ship to a buoy. It can also be a curved line of position such as a radar range from a known return. The one thing that a single LOP gives is a definite clue as to your position.
"A fix gives definite information as to both course and speed over the ground since the last fix." 
An LOP can be obtained through various methods and instruments. You can get an LOP from the depthmeter, radar, a single time differential (TD) from the Loran, a visual compass bearing on any identifiable point, a range using two or more objects, or even from a celestial observation.
Types of LOPs A fix gives definite information as to both course and speed over the ground (COG and SOG) since the last fix. A single LOP can only define either the COG or the SOG, but not both. And in some cases it may not clearly define either. The accuracy obtained from an LOP depends upon the angle at which it intersects the track of the boat, and LOPs are often classified according to this angle.
A LOP that is perpendicular (or nearly so) to the track is called a "speed line." It indicates how far the boat has traveled along the track, and thus is a measure of SOG. A speed line does not indicate whether the vessel is to the right or left of course. If a speed line is in front of the DR, it indicates a SOG greater than the boat's speed through the water and therefore a current that is setting along the course. If the speed line falls behind the DR than the converse is true.
One method of determining a LOP is to establish the direction of the line of sight to a known fixed object. This direction can be determined using the ship's compass, a handheldcompass, lining up two objects, or by using the radar. The direction of the line of sight is the bearing of the object from the vessel. A line plotted on the chart from the object along the reciprocal bearing establishes a line of position. At the time of the observation the boat was somewhere on that line of position. These bearings can be either a true or relative bearing depending on the referenced direction.
A relative bearing is the angle between the foreandaft axis of the boat and the line of sight to the object. A relative bearing is always measured from the bow (000 degrees) clockwise through 360 degrees. If an object is directly abeam the starboard side of the boat, it has a relative bearing of 90 degrees. An object directly astern has a relative bearing of 180 degrees, and one abeam the port side has a relative bearing of 270 degrees. To convert a relative bearing to a true bearing, simply add the relative bearing to the compass course of the boat (RB+CC = TB). Subtract 360 degrees if the total exceeds this amount. For example if the relative bearing is 70 degrees for a vessel on a compass course of 210 degrees, then the true bearing is 280 degrees magnetic.
Relative bearings like compass bearings must be converted to a true bearing before they can be plotted. True and magnetic bearings are interconverted by the use of variation and deviation in the same manner as courses previously discussed. Do you remember the mnemonic "Can Dead Men Vote Twice" for converting from compass to true? This is used to help you remember how to apply variation and deviation to convert from compass to true. In other words you first apply Compass Deviation to the compass reading to get Magnetic and then apply the Variation to get True. See "Understanding and Using the Magnetic Compass".
A distance LOP is determined when the vessel is a known distance from a fixed object of a known height. Distances are determined by radar, range finders, and vertical angles. The LOP is plotted using a compass set to the distance and arced off on the chart from that object. Two or more distance LOPs may be plotted, or a single distance LOP can be used with one or more bearing LOPs to determine a fix. Finally, a range and bearing from the same point may also be used to plot a fix.
Plotting the LOP Locate the object that you used to take the bearing on the chart and place one point of the dividers on it. Then orient the plotter using the calculated true bearing and draw the LOP away from the object toward the boat's position. It is not necessary to calculate the reciprocal as you will find it on the plotter scale under the bearing. Once the LOP is on the chart, label it with the time and the compass bearing. Continue to plot any remaining bearing LOPs in the same way. The intersection of two or more LOPs is your position.
When the fix consists of three LOPs, the fix is usually placed in the center of the small triangle. However if the triangle formed by the LOPs is fairly large, bisect the angles to determine the exact center. Label the fix with the time placed parallel with the top or bottom edge of the chart. Now calculate your DR position for the time of the fix and plot it on the chart. Label the DR with the time placed at a 45degree angle with reference to the bottom edge of the chart. The relationship between the fix and the DR is called set and drift and is the summation of the tidal current, wind, and steering error that affected the boat between fixes. Plot a dashed line between the DR and the fix. The measured direction between the DR to the fix is the set and the distance between them divided by the elapsed time in hours and tenths of an hour from the last fix is the drift. Label the set and drift on the chart near the dashed line. Finally, compare the fix with a GPS or Loran position and the computed set and drift to the predicted set and drift.
It is assumed that all your LOPs are taken almost simultaneously or at the least in less than a minute. On a slowmoving vessel, little error is introduced by making several observations in quick succession. A wise precaution is to take bearings of objects that are nearly ahead or astern first, with beam bearings taken last, and using the latter time as the fix time. Good crew coordination can enable you to do this quickly. One person takes the bearing and says "Mark" while another records the time. The person taking the bearing quickly reads the bearing, gives it to the recorder, and then lines up the second object. Repeat this process until all bearings are completed. In rough seas, you may want to take three bearings on each object and average the readings.
If the time interval between LOPs is several minutes or more, then the LOPs will need to be advanced or retarded to a common time. I will have more to say about this in my next article when I get into the subject of running fixes.
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