Chapman Piloting & Seamanship

Lines of Position • Visual Observations • Positioning Procedures • Radio & Radar Navigation • Using Depth Information • Specialized Techniques

It is your duty as skipper to make a careful judgment as to the precision and frequency with which your vessel’s position should be fixed, and then to see that these fixes are carefully obtained and recorded. This is true whether you are doing the piloting or someone else is—you can delegate the function, but not the responsibility.

The procedures described in this chapter may seem old-fashioned or out-of-date in these days of instant, highly accurate electronic navigation, primarily GPS. Navigation with a GPS receiver or chartplotter is introduced in Chapter 16, and the use of GPS to simplify calculations of the effects of current set and drift on a boat’s position, course over ground, and speed of advance is covered in Chapter 17. But remember, (1) your GPS equipment may fail; (2) your source of electrical power may be lost; or (3) the worldwide GPS system itself may fail. You must be able to navigate manually should the need arise.

This chapter will also explore how to augment visual piloting skills with radar and a depth sounder. Either one can give you good lines of position, and radar can give you highly accurate ranges.

The time-honored procedures for position determination may vary widely in practice. When you are proceeding down a narrow channel, positioning is informal and a chart plot may not be maintained. In this case, position determination is not being omitted; rather it is being done continuously by visual reference to aids to navigation. During an open ocean passage, a dead reckoning plot will be determined only three or four times each day, or even only once a day by noon sights.

Normal cruising situations in pilot waters lie between these two extremes. Cruising just offshore or in larger inland bodies of water, a skipper will usually maintain a DR plot of his track with periodic fixes, perhaps every 15 or 30 minutes, perhaps at hourly intervals.

Your ability to determine your position with appropriate accuracy under any conditions of visibility or sea is critical. Your limitations in these skills must restrict the extent of your boating activities, setting the boundaries of the waters and weather conditions that you can accept without endangering your boat or its crew.

Remember that knowledge of where you are, extended from a recent position determination, is critical if you must call for help. If another boat is in trouble, you can set a direct course to render assistance only if you are both certain of each boat’s location.

A LINE OF POSITION (LOP) is a line along which an observer can be presumed to be located—actually two lines: one “real,” one drawn on a chart; see Figure 18-01. In the absence of other information, the observer may be anywhere along the LOP. An LOP may result from an observation or measurement—from visual, electronic, or celestial sources.

It may be straight or curved. A circular LOP is sometimes called a circle of position; see Figure 18-02. An LOP, in a running fix (defined below), may be “advanced” to a later time according to the movement of the vessel; or, in rare cases, it may be “retarded” to a previous time as needed.

Figure 18-01 A line of position (LOP) is a line along which the observer is located. Although a single LOP does not determine position, it does tell you where the boat is not located, and such information can be useful in itself in many situations.

Figure 18-02 A line of position can be circular as well as straight. A circular LOP results from a measure of distance from an identified object. Usually it is plotted as an arc through the most likely area of position.

A BEARING is the direction toward an object from the observer. Bearings are expressed in degrees as three-digit numbers: 005°, etc. A true bearing is one measured with reference to true North. A magnetic bearing is one measured with reference to local magnetic North, and a compass bearing is one taken over a compass and affected by the compass deviation at the time it was taken. (Refer to Chapter 13 for a discussion of true, magnetic, and compass directions.)

A RELATIVE BEARING is one measured with reference to the vessel’s heading. It is measured clockwise from the fore-and-aft line, with 000° as dead ahead, 090° as broad on the starboard beam, 180° as dead astern, etc.

A RANGE consists of two objects that can be observed in line with each other and the observer—all three are on the same straight line; see Figure 18-03.

Figure 18-03 When two objects can be observed in line, this is a special LOP called a range. It is labeled only with the time of observation above the plotted line. Some range objects will have been set up specifically for that purpose, but any identifiable pair of charted objects or features can provide a range.

A FIX is an accurately located position determined without reference to any assumption or estimation of prior position.

A RUNNING FIX (R FIX) may also be derived from two LOPs, one of which has been advanced (or retarded) from a different time—thus introducing an element of estimation.

An ESTIMATED POSITION (EP) is the best position obtainable short of a fix (or running fix). It is the most probable position, determined from incomplete or questionable data relating to course run, speed, current set and drift, etc.

Lines of position are the basic element of position determination. An observer lies somewhere along the length of a line of position. If two LOPs intersect, the observer must be at the intersection, which is the only place where he or she can be on both lines at once. A fix is usually determined by crossing two or more LOPs.

When LOPs are drawn on charts, they should be no longer than necessary and be clearly and consistently labeled. Labels should identify them completely, but superfluous data can be confusing. The label must specify the time the LOP was observed or measured and its basic dimension, such as direction toward or distance from the reference object.

A bearing is a line of position that has both time and direction. Time is always shown above the line of the bearing and direction below it; use plain, block letter styles. Remember to write time as a four-digit figure in the 24-hour system. Record the direction as a three-digit group with leading zeros as necessary. Follow the figures with a space and the letter “M” if the direction is magnetic; the absence of a letter indicates a true direction (i.e., the noting of “T” for true is unnecessary and redundant). Note that the degree symbol (°) is omitted since all three-digit numbers in labels are obviously directions; see Figure 18-04.

Figure 18-04 Prompt and correct labeling of lines of position is important; unlabeled or mislabeled lines cause confusion. Time is shown above the line in the 24-hour clock system. Direction is shown beneath the line as a three-digit number, with “M” added if it is a magnetic direction.

A circle of position has dimensions of time and distance and may be plotted as a full circle or an arc. Time is written above the line, and distance (and units of distance) below (which may be “inside” or “outside” the circle or arc); see Figure 18-05.

Figure 18-05 Circles of position are labeled in the same way as lines of position, with time written horizontally above the line and distance below (unit of distance may be shown). Time will be inside or outside the circle as determined by the curvature of the arc.

A range is an LOP whose direction is selfevident from the two points that establish it. In this case, the label need show only time, which is written above the range line; refer to Figure 18-03. The line need not be plotted completely through both objects of the range—draw it only long enough to make clear which two objects define it. This will minimize the need to erase lines drawn over important chart symbols.

An LOP that has been advanced from an older line is labeled with both times above it—the original time first and the time to which it has been advanced, separated by a dash; see Figure 18-06.

Figure 18-06 An advanced LOP is labeled with the original time and the time for which it was replotted.The direction remains the same and is repeated.

Label your LOPs immediately to avoid mistakes. Unlabeled lines on a chart can be a dangerous source of confusion.

A fix is an accurately located position. On many occasions in small-craft piloting, position will be determined by passing close by an identifiable object, often an aid to navigation. When such a fix is established, the skipper should note the time horizontally on the chart; see Figure 18-07.

Figure 18-07 An excellent, yet simple, determination of position occurs when a boat passes close to an aid to navigation or other identifiable point. Note the time on the chart written horizontally, no circle needed.

A fix that is obtained from lines of position will be the intersection of two or more such lines; see Figure 18-08. Note that the angle of intersection of two LOPs affects the accuracy of the position determination. When two lines cross at right angles (90°), an error of a couple of degrees in one or both LOPs will have the least effect. Where LOPs cross at small angles, however, an error in one or both observations or measurements will have far more serious implications; see Figure 18-09.

Figure 18-08 A fix is an accurately determined position for a vessel based on concurrently observed lines of position or other data. See Figure 18-13 for labeling.

Figure 18-09 Note the difference in position of the intersection that a change of 2° in one LOP makes when the lines cross at 90° (right).

Two lines of position should intersect as nearly as possible at right angles, and the angle should never be less than 60 degrees, if possible. A fix resulting from LOPs intersecting at angles smaller than 60 degrees may not be a fix at all and should be regarded with doubt.

Even lines that intersect at large angles will cover what are known as AREAS OF UNCERTAINTY. While we label our lines with specific directions, we know that there may be uncertainty in each line of 2 or 3 degrees. If the uncertainty is potentially dangerous, it should be represented graphically by adding two dashed lines on either side of the LOP that define the limits of the inaccuracy. LOPs from various sources will have different levels of inaccuracy—only an experienced navigator can judge these levels.

If both questionable LOPs are drawn with their dashed lines at the maximum and minimum possible values, the dashed lines will enclose an area of uncertainty at their intersection. The area of uncertainty is not the quadrilateral enclosed by the lines but an elliptical area; it is circular only if the two LOPs cross at exactly 90 degrees; see Figure 18-10.

Figure 18-10 When both LOPs have a possible inaccuracy of several degrees, there is an area of uncertainty around the intersection point (A). This area grows as the intersection angle narrows (B). Be very cautious with intersection angles of 30°or less.

Whenever possible, a third line of position should be drawn to reduce the uncertainty. Of course, a fourth and fifth LOP might also be available, but this is seldom the case and almost always unnecessary.

If the observer is on all three lines of position at the same time, then they should all intersect. The more common result is that they nearly intersect, and instead of defining a point, they form a small triangle. (Theoretically the three bearings should be taken simultaneously, but in practice they can be taken in sequence, as the distance traveled by the boat in this short interval is too small to show up on the chart plot.)

The best result is obtained when all three LOPs form 60°) angles. If this is the case, their inherent inaccuracy will have the least effect; see Figure 18-11.

Figure 18-11 An optimum set of three LOPs cross at 60° (left), but this is seldom possible. Alternatively, you might choose objects whose LOPs would cross at 90° as possible) ensuring a good fix, and adding a third LOP at about 45° as insurance, if there is time and a suitable object available for the third observation.

In practice, two bearings should be taken promptly, one after the other, spaced as closely as possible to 90 degrees, and, if time permits, a third chosen to split them as evenly as possible.

The triangle that almost always results when three LOPs are plotted is known as the TRIANGLE OF POSITION. A large triangle suggests a serious inaccuracy in at least one of the LOPs, and a check should be made on the observations for errors of reading, deviation, calculation, or plotting.

A small triangle (how small is a relative judgment depending on the proximity of danger) can be eyeballed to find its center, and this center used as the fix; see Figure 18-12.

Figure 18-12 Plotting more than two LOPs increases the accuracy of the resulting fix. If they were perfectly accurate (A), all three would intersect at the same point. However, in actual navigation they will almost always form a small triangle (B).

Labeling the Fix
A fix is plotted on the chart as a small circle with a dot in its center; the dot, however, should be omitted from the symbol when it is placed at the intersection of two LOPs. The word “fix” is not needed, but the position must have a label, written horizontally, that records the time of the fix; see Figure 18-13 and refer to Chapter 16, Figure 16-23. A running fix is shown with the same symbol, but the label should contain “R FIX” in addition to the time. When the fix is obtained by passing close by an aid to navigation, the aid’s symbol may take the place of the dot and circle. The usual distance off (50 to 100 yards or meters) is insignificant at most chart scales.

Figure 18-13 A fix is labeled with the time, written horizontally, but the word “fix” is not used. For a running fix, however, the label “R FIX” is added along with the time of the second observation, again written horizontally.

While a fix is typically made with two or more LOPs, don’t underestimate the value of a single LOP. Of course, a single LOP cannot tell the skipper where the boat is at the moment, but it can tell him (within its limits of accuracy) where the boat is not. This information, though lacking detail (and meaningful only in a negative sense), can nevertheless be reassuring where hazards may be close by.

Further, a single LOP can often be combined with a DR position to obtain a useful EP (estimated position). This EP is the position along the LOP that is closest to the DR position for the same time that the LOP was observed. To obtain such an EP, draw a line from the DR position, perpendicular to the LOP, until the lines intersect. An EP is marked with a square. A square is used exclusively for EPs, so the label does not have to say “EP.” Time may also be omitted because it is the same as the time that appears on the LOP or at the DR position; see Figure 18-14.

Figure 18-14 An estimated position can be obtained from a single LOP. It is the point on the LOP that is closest to the DR position for that time (left). If a beam bearing is used, the EP will be on the track line (right).

It is also possible to obtain an “EP with current.” This requires an estimate of the set and drift of the current, either from predictions or from accurate measurements of the effects of the current on the boat’s course and speed made good.

From the DR position for the specified time, an EP with current would be obtained as follows. A line is drawn representing the effect of the current during the time period since the DR track was started from the last fix. The line is drawn in the direction of the set of the current and to a length equal to the total drift. (The length of the line would be drift multiplied by the elapsed time since the last fix.) This total drift line is labeled, and from its end another line is drawn to intersect the LOP at a right angle (just as in a regular EP plot). The intersection obtained is your EP with current; see Figure 18-15.

Running fixes, and the use of a single LOP with depth measurements, will be discussed later in this chapter.

Figure 18-15 If information about the current is available, an improved “estimated position with current” can be plotted. The offsetting effect of the current (which includes wind effects) since the start of the DR track is calculated and plotted. The nearest point on the LOP from this current-influenced point is the EP.

In typical small-craft navigation, the primary source of lines of position will be visual observation. Observations will include bearings, ranges, and even horizontal and vertical angle measurements. Correct identification is essential. Although other methods and other equipment, such as GPS (covered in Chapters 16 and 17), radar bearings and distance measurements (covered later in this chapter), and depth measurements (also covered in this chapter), may also provide LOPs, visual observations are the indispensable foundation of piloting.

The aids to navigation depicted on charts comprise a system consisting of fixed and floating objects with varying degrees of reliability. A prudent boater, therefore, will not rely on any single aid to navigation, especially a floating aid, to fix his or her position.

If a navigator has a choice of objects to sight for bearings, he or she should select the nearer ones provided they would make good intersection angles on the plot. The reason is that a measured angular inaccuracy, if one exists, will be the same whether the object is near or far. However, the linear inaccuracy will be amplified as the LOP is extended over a greater distance. For example, an angular error of one degree extending to one mile will result in an error of about 100 feet (30.5m). At two miles, the error will double. In a case where a light is being observed at night over a distance of a dozen miles, a one-degree error is potentially quite serious.

Perhaps the simplest and most accurate visual observation is a bearing taken dead ahead and read from the vessel’s steering compass; see Figure 18-16. Swinging the boat off course to point it at a charted object rarely has much impact on the DR track, since the boat is held on this heading only long enough for the boat to be lined up and for the compass to settle down and be read—perhaps only 20 or 30 seconds. Of course, this is somewhat more difficult on a sailboat because the dead-ahead alignment from the helm is more likely to be obscured by the mast.

The skipper must be sure of his general location and of surrounding traffic so that the offcourse swing can be made safely. This is a quick, accurate, and simple way to get a bearing, and has the additional advantage that it can be accomplished by the helmsman alone.

Figure 18-16 A simple, accurate way to take a bearing is to “aim” your boat directly at the sighted object and read the compass. Brief off-course swings will not affect the DR plot materially, but be sure that there are no hazards alongside your course. Correct compass bearings to true or magnetic directions before plotting them on your chart.

If the charted object on which a bearing is to be taken is on the beam, a swing off course to line it up with the bow may not be feasible. Perhaps a limited channel width or the presence of other vessels restricts such a deviation from the DR track. In this case, choose some part of the boat that will give you an accurate right angle from the centerline. This might be a bulkhead, a seat back, etc. While the helmsman maintains course on the DR track, the bearing taker waits until the object comes into view along this line. Such sights will determine when the object is dead abeam—either 090 or 270.

If waiting until the object comes abeam is not feasible or desirable, then the heading of the craft can sometimes be temporarily altered slightly to bring the sighted object on the beam more quickly. Should such a temporary change of heading be made, be sure that the 90 degrees is added to or subtracted from the altered compass reading at the moment the sighted object is abeam; do not use the base course.

It is extremely important to note that any bearing taken from the steering compass is a compass bearing and, as such, is subject to both deviation and variation. If you are to plot them as true bearings, you will have to correct for both (refer to Chapter 13).

In the case of the 90-degree adjustment for sightings abeam, remember that 90 degrees must be added or subtracted after, not before, the compass reading has been corrected for deviation (and for variation), because, while variation will be the same on any heading, deviation will differ from one heading to another. The deviation is found on the deviation table opposite the boat’s heading (because deviation varies with heading), not opposite the direction for the bearing.

A bearing in almost any direction can be taken by using one of several instruments: a handbearing compass (refer to Figure 16-07), a pelorus, or a binocular or range finder with an internal compass (Figure 18-17). Care must be taken to use the value of deviation of the boat’s heading at the time of observation, not the deviation value for the direction of the bearing.

The direction measured for a bearing is from the boat toward the object. When the plot is being done, the position of the boat is not yet known, but it is still possible to draw the bearing line of position so that it will have the correct direction and will lead to or through the point on the chart that locates the object.

This is done by drawing a line outward from the position of the sighted object toward the observer’s position using a reciprocal of the corrected bearing. It is important to remember to correct the bearing for deviation before converting it to a reciprocal by adding or subtracting 180 degrees. (You must also correct for variation if directions are being plotted as true values.)

Figure 18-17 To take a reading with a hand-bearing compass, sight along the lubber line and view the compass card numbers in the small prism.

You can plot bearings, courses, and other directions as true or magnetic values, but you should be consistent in your work. There are advantages and disadvantages to both procedures. If you plot magnetic values, you must always include variation in your calculations unless you measure directions using the inner circle of a compass rose. Do not plot compass values for bearings or courses.

Getting a bearing either dead ahead or abeam may not always be safe or convenient, especially under sail. Bearings can also be taken without changing the boat’s heading. Such bearings may involve the boat’s steering compass, a handbearing compass, or a pelorus.

On many boats, it may be possible to take a bearing in most directions by sighting over the steering compass. Sometimes the accuracy of such a bearing is increased by the use of sighting vanes placed over the compass, but usually the sighting can be taken across the card itself. Once again, be sure to correct for deviation and variation, and use the deviation value appropriate to the boat’s heading, not the direction of the bearing.

Very often, visual bearings are made with a handbearing compass; refer to Chapter 16, Figure 16-07. While such a compass is also vulnerable to deviation, the usual practice is to choose a location on board that is sufficiently distant from magnetic influences that a deviation correction is not needed. Some experimentation might be required to find a spot that provides a secure platform and is free from magnet influence. (Remember, the metal frame of your eyeglasses, a grommet in your cap, or the batteries in the flashlight you may be using to illuminate the card can affect the compass.) Readings can be plotted directly if you are working with magnetic directions, or they can be corrected by applying variation if you are plotting true directions.

A pelorus can also be used to obtain bearings, usually as relative bearings; refer to Chapter 16. A pelorus can be used at any on-deck location from which the object is clearly in view and where the pelorus card can be accurately aligned with the boat’s fore-and-aft axis. In practice this might mean predetermining two or three convenient positions on deck.

A pelorus card can be adjusted (after fore-and-aft alignment) to the boat’s compass heading, and sights made directly as compass bearings, but this is less desirable than using the pelorus to read relative bearings (see the following entry). In all cases, compass bearings must be converted to true or magnetic values before plotting.

To take a relative bearing with a pelorus, the pelorus scale is set with 0° dead ahead. The helmsman is alerted and requested to steer a steady course, reading the compass continuously. Before the pelorus observation is made, the reader calls “stand by,” and as it is made, the reader calls “mark.” At his call, the helmsman notes the reading on the steering compass and calls it out to the recorder. If the steering compass reading has not been accurate, the pelorus reading must be discarded. If it is very difficult to steer a steady course, the helmsman can instead call “mark” each time he is confident of an accurate reading. The pelorus reader makes his observation at each “mark.”

A relative bearing must be corrected to a true or magnetic bearing for plotting; see Figure 18-18. First the compass reading is corrected for deviation and (if plotting in degrees true) variation, then the pelorus (or sighted) reading is added. (If the result exceeds 360, then 360 is subtracted.)

Once again, avoid a common error. Always be sure to use the deviation value appropriate to the boat’s heading—not the bearing direction.

Let’s look at the examples shown in Figure 18-18. In the first case, an observer is on a boat that is heading 040 (true) and takes a relative bearing on a buoy at 055°. To determine the true bearing of this buoy, simply add the two numbers, 40 +55 = 95; the bearing is 095°.

Here’s a more complicated example: A boat is heading 303° by compass when a relative bearing of 317° is measured. The variation for this location is 6°W and the deviation for this particular compass at 303° E. Convert the boat’s heading to true this way: 303 – 6 + 2 = 299°. The sum of the relative bearing and the true heading is 317 + 299 = 616. Subtract 360 and the true bearing is 256°.

Figure 18-18 Relative bearings must be converted to true or magnetic bearings before they are plotted. The relative bearing value is added to the boat’s heading at the moment of observation; if the sum exceeds 360°, subtract 360°.

Lines of position from ranges are of exceptional value in position determination. They are free from all of the magnetic effects that might cause errors in bearings taken with reference to a compass (whether sighted on the bow, across the boat, or with a pelorus).

LOPs from ranges are also much more easily obtained than bearings. No matter how small the boat or how rough the weather, if you can see both objects, you can line them up with absolute accuracy; refer to Figure 18-03. The accuracy of your LOP depends only on the accuracy of the range locations on the chart. You now have half of a very accurate fix.

Ranges fall into two groups. First, there are those that consist of two aids to navigation constructed specifically to serve as a range and are charted with standard symbols. The direction of such a range can be measured from the chart but is better determined from information in the Light Lists. However, any two charted, prominent objects can be used for ranges—ordinary aids to navigation, spires, towers, radio towers, stacks, the centers of bridges, and even clearly demarcated edges of natural features; see Figure 18-19. Be careful using natural features, because shoreline edges may change with changes in water level, and exact points may be hard to determine. If possible, avoid using buoys, as their charted locations are not as exact as those of fixed aids to navigation.

Figure 18-19 Navigators are not limited to ranges that have been established specifically as aids to navigation—many natural and man-made features make excellent incidental ranges. For example, these four stacks at Northport, Long Island, NY, provide a range that is almost exactly magnetic south.

Taking an LOP from a range requires no more than the observation of the time when your boat comes into alignment with the two objects. The LOP is plotted by lining up the symbols with a straightedge and drawing a light, solid line over the portion of the chart where your DR track or other LOP is likely to cross. The actual direction will not ordinarily be noted, but the LOP should be labeled with the time as soon as it is drawn. The range LOP can be crossed with another range or an LOP from any other source. If none is available at the time of alignment, the range line of position can be advanced at a later time and used as part of a running fix. In this case, the direction should be noted so that the advanced line can be related to the original and easily labeled.

Figure 18-20 Caution must be used when following a range because there is always a limit to the extent of safe water. The range line on the chart will be solid for the length for which it is to be used and dashed where it is not to be used. Often buoys or minor lights mark the beginning and end of a range-marked channel.

The U.S. Coast Guard and other authorities often establish ranges to indicate the center of a hazardous narrows, dredged channel, or simply an important waterway, especially if there are strong currents to contend with. Such a range line is printed on the chart, and it is often used for direct steering rather than as an LOP. Two cautions are needed. First, a vessel traveling in the opposite direction on the range will be on a collision course. Second, a range can be followed for too great a distance into dangerous waters. Buoys will often mark the limits of such a range, but study your chart carefully to determine when to turn away from a range to avoid danger; see Figure 18-20.

A way to obtain a fix without having to correct for compass error is to measure two horizontal angles having a common side. The two angles are taken using three objects identifiable on the chart. A sextant, normally used to measure vertical angles in celestial navigation, can be held sideways and read to a very high degree of accuracy; see Figure 18-21.

Figure 18-21 A sextant, such as the one seen above, is usually used for measuring vertical angles in celestial navigation, but it can also be used in piloting to measure horizontal or vertical angles to calculate distance off.

Horizontal angles can be plotted with a threearm protractor; see Figure 18-22. The two measured angles are set on the outer arms in relation to the center arm. The protractor is moved on the chart until all three arms intersect the locations of their respective sighted objects, the centerline of each arm representing a line of position. The center of the protractor (the apex of both angles) is the position of the observer. You plot it by placing a pencil point though a hole.

Figure 18-22 A three-arm protractor is a special plotting instrument for finding a position using two horizontal angles—in this case the angles between a buoy and the well-defined edges of two islands. The angles are carefully set on the protractor, then it is moved about on the chart until the arms lie over their respective objects.

The same plot can be accomplished with tracing paper. After drawing the angles on the tracing paper, they are moved around in the same way until all three are intersecting the locations of their respective objects; see Figure 18-23.

Figure 18-23 You can use horizontal angles to determine position by drawing them on tracing paper. Move the paper around on the chart until the lines intersect the objects that were sighted upon. The apex of the lines is your boat’s position when the observations were taken.

Be careful when selecting the three objects to be observed; if they and the boat all lie on the circumference of a circle, no fix can be obtained. When you study Figure 18-24, you will see three sighted objects—X, Y, and Z—and an observer at both A and at B. Both observations will give the same angle values, and will do the same almost anywhere between X and Z.

Figure 18-24 The horizontal angle between Y and Z will be the same when observed from point A, point B, or anywhere else between X and Z on the circle that intersects X, Y, and Z—and the same is true of the angle between Y and X. If you’re on or near the circle that sweeps the objects sighted, you’re in a “revolver” situation. The resultant ambiguity in your fix can be cleared up with another LOP; see the text.

This situation, known as a “REVOLVER,” can occur if the center object sighted is farther away from the observer than the other two. By selecting a center object that is either closer than the other two or equidistant with them, you can avoid an indeterminate situation.

A revolver should be avoided, but if one does develop it can be made usable by the addition of a single LOP, such as a single bearing on one of the three objects or on any other point.

The geographic coordinates of a position can be calculated from the known latitude and longitude of the three objects and the measured horizontal angles between them, but the complexity of the mathematics makes this procedure impractical for ordinary cruising.

A fix can also be determined from LOPs derived from two horizontal angles that don’t share a common point. Each angle can be used to establish a circle of position.

Consider one angle at a time. There is a circle that contains both of the objects sighted upon (in measuring the horizontal angle) as well as the observer. The radius of this circle can be calculated using the distance between the sighted objects.

Here’s how to proceed. First, measure the distance between the two sighted objects on the chart. Find the sine of the measured horizontal angle using your electronic or plastic circular calculator or a trigonometric table from Bowditch. Next, multiply the sine by two and use that product to divide the distance between the objects. This is the radius of the circle on which your boat and all the sighted objects lie. (It’s easy enough to find with a calculator.)

The next step is graphical. Set a pair of dividers (or a drafting compass) to that radius (on the chart scale), and scribe two intersecting arcs, one centered on one object and one on the other. That intersection is the center of the circle along which your boat and the two sighted objects lie, so put the compass point there and scribe as much of the circular LOP as you think you need.

Do the same procedure on your second set of objects. You now have two circular LOPs, and you are at their intersection. (Actually, if you were to draw very long circular LOPs, there would be two intersections, but common sense or a third LOP will tell you which one is your position.)

While this procedure sounds complex, it is actually quite useful, especially aboard a boat in circumstances where the compass bearing is very hard to read because of rough weather. A horizontal angle greater than 90° creates a situation in which the intersection of the LOPs will be on the far side of the sighted objects and may create an invalid fix.

All of the points (in a plane) that are a particular distance from an object lie in a circle—the distance is the radius of a circle. Such distances are often found by measuring the vertical angle from the bottom to the top of a known height, then using a simple trigonometric formula. For this reason, the heights of many man-made and natural features are carefully measured and recorded on your chart.

Such features include towers, lighthouses (usually measured to the light, not the top of the tower so as to be useful at night), radio towers, and bridges. Before you use such a recorded height, determine whether it has been measured from the base of the object or from some standard datum plane such as mean high water. Frequently, a correction will be required depending on the tidal level at the time of the observation (refer to Chapter 17).

Using a sextant or some other device for measuring sighted angles, get an accurate measure of the angle from bottom (water level or base) to top. Call this angle “A” and the known height “h.” This distance “d” can be calculated as: d = h / tan A.

Again, use the appropriate table in Bowditch or a calculator to find the tangent.

There are also specialized optical range finders, like the one shown earlier in Figure 18-17, that are adjusted to provide a reading from a scale.

Safety can often be ensured without a complete fix. As noted previously, a single LOP has value—it can tell you where you are not. In many situations, a line of position can be chosen that will keep a boat in safe waters without defining its position.

A bearing line can be chosen that will divide a safe area from an unsafe area. If you stay on the correct side of such a line, your vessel will be safe; crossing it will invite danger. Take the examples shown in Figure 18-25: Shoals are indicated on the chart but not marked by any aid to navigation. A lighthouse on shore can be identified on the chart just beyond each shoal. The danger bearing is indicated, therefore, by a line that extends from the lighthouse toward your vessel, tangent to the shoal. This line is drawn on the chart and its direction is measured. The line is labeled with the direction preceded by the letters “NMT” for “not more than,” or “NLT” for “not less than.” Time is not included in the label, as this is not an observed line. Add hachures on the danger side or use a red pencil to emphasize the line’s importance.

As the vessel approaches the area, a series of observations is made on the lighthouse (or whatever object was chosen). It should be beyond the danger area and on the same side of the boat. If the shoal lies to port, then any bearing on the lighthouse (or chosen object) greater than the danger bearing means that the boat has not yet crossed from unsafe to safe waters. Extra care should be taken until the bearing decreases below the NMT value.

If the hazard were to starboard, the opposite would apply in relation to the NLT value.

Danger bearings are not always possible. They require a prominent object (which could be natural) that can be identified on the chart. It should be beyond the danger area and on the same side of the boat.

Figure 18-25 A danger bearing can be used to avoid an unmarked hazardous area. In A above, any bearing on object X that is more than 337° indicates danger. In sketch B above, the opposite is true—a bearing less than 008° indicates danger.

As we have seen, a horizontal angle measured between two objects (identifiable on the chart) defines a circle of position, or circular LOP. In Figure 18-26, left, an observer at either X or Y would measure the same angle between point A and point B. He would also measure the same angle at any point along that segment of the circular arc. If the observer were on the other side of the center of the circle, at X' or Y', or at other points along that segment, a constant (but different) angle would be observed. Note that the angle will be greater than 90 degrees in the first case (observer and objects on the same side of the circle), and less than 90 degrees in the second case; see Figure 18-26, right.

Such a circle becomes very useful as a vessel approaches a hazard. It can establish a boundary between positions of safety and danger. When such a circular LOP is established from a horizontal angle, it is known as a HORIZONTAL DANGER ANGLE. A single horizontal danger angle is used to avoid a danger area such as an unmarked shoal.

Figure 18-26 The points at which there is a constant angle between the lines of sight to two objects form a circle as shown above. The angle between the lines to A and B is the same at X and Y, and all other points on the circle.

When you’re passing a shoal between you and the shore, the problem is how to stay far enough offshore to avoid the shoal. Find two prominent, identifiable objects that lie on the other side of the shoal from the area of safety. There’s no need to draw the circle. Measure the angle on the chart from the most seaward point of the shoal to the chosen objects. This is the horizontal danger angle, shown in Figure 18-27. As you approach the shoal, frequent measurements of the horizontal angle between the two objects will reveal whether or not you are standing outside the danger area. Angles less than the danger angle indicate that you are on a circular arc of a radius that is greater than the radius of the arc on which the danger lies. In other words, you’re farther offshore than the tip of the shoal—and you are safe.

Figure 18-27 The horizontal angle between A and B is frequently measured as the boat proceeds along the track line toward the hazardous area. Any angle less than 55° means that the boat is in safe waters.

On the other hand, if the measured horizontal angle becomes greater than the danger angle, you are closer inshore than the tip of the shoal and may be in danger.

It is preferable to measure the angles with a sextant, as you will have to take frequent and accurate measurements, but they can also be determined by taking either relative or compass bearings on the objects.

The same technique can be used to pass inshore of a danger area—just make sure that your angles remain greater than the angle of the threatening extent of the shoal; see Figure 18-28.

Figure 18-28 A horizontal danger angle can be used to pass between an offshore hazardous area and the shore itself. Here we see that the measured horizontal angle must be greater than the measured horizontal danger angle of 40°.

The technique described above can be used to establish two horizontal danger angles where the problem is to pass safely between two offshore hazards. The principle is the same, as you will note from Figure 18-29. The safe angle lies between the upper and lower danger limits.

Figure 18-29 Two horizontal danger angles can be used to pass safely between two unmarked hazardous areas. The frequently measured angle must be between a minimum and maximum value—in the situation above, not less than 48°.

Where only a single object is available, it may be possible to use a vertical angle to establish a boundary between safe and hazardous waters. Using the vertical angle technique, a circle is drawn that has the identifiable object of known height at its center and encloses all of the hazards. The radius is measured from the chart and used in the formula d = h / tan A, where “d” is distance, “h” is height, and tan A is the tangent of the angle. The angle itself can be found with a calculator or looked up in Bowditch, which also provides a table to substitute for the whole calculation (Table 15, 1995 or 2002 editions). The danger circle is labeled with the VERTICAL DANGER ANGLE.

On approach, a series of vertical angle measurements is taken, and as these values approach the danger angle, course is altered to maintain or decrease the measured values. If the angle were maintained, the course would describe a circular arc around the sighted object (outside the area of hazard). A decreasing angle would mean that the boat was getting farther away from the sighted object and also from the hazardous area.

Vertical angles may also be used in pairs in a manner similar to that described for horizontal angles. Two vertical angles can be used to define a safe passage between two hazards, one indicating the inshore limit and the other indicating the offshore limit. Range finders can also be used in this situation; refer to Figure 18-17 and see below.

Coincident (split image) reticle and LIDAR (light detection and ranging) range finders are useful on boats. Coincident range finders present two images of a sighted object that are fused into one by adjusting the range control knob. Distance is then read from the knob’s calibration marks. A reticle range finder superimposes a graduated vertical scale on a distant object of known height. LIDARs send a pulse of laser light toward a distant object and measure the time it takes for the reflected laser energy to return to the device. Measurement ranges can be as far as 1,500 yards (1,370 meters) to reflective targets and 400 yards (365 meters) to nonreflective objects such as flags.

Knowing where you are at all times is fundamental to boating safety. A fix is the best statement of position and should be obtained as often as possible. Of lesser significance, but not without value, are running fixes and estimates of position.

A basic fix is obtained by crossing two lines of position, and a third LOP is desirable. Such a fix assumes that the observations or measurements for these LOPs are made simultaneously, but the usual situation on a small boat is that there is only one bearing-taker. Observations are taken sequentially rather than simultaneously, but if they are taken quickly, the distance traveled between them is negligible and too small to be plotted on a chart. Adherence to the procedures described here will minimize the error from sequential observations.

As the boat moves along its course, a bearing angle will change. Those on the beam will change more rapidly than those on the bow or stern; bearings of closer objects will change more rapidly than those of distant objects. To obtain the most accurate fix from two LOPs, deal first with the object whose bearing will change slowly—the one that is farthest forward or aft. Then take a sight on the second, faster-changing object and finally, do a check on the first. You now have three bearings on two objects. Plot them by using the average of the two bearings on the slowly changing object and the single bearing on the faster-changing object. If circumstances permit only two sightings, take the slowly changing one first.

If you are able to sight three objects, take them in order of rate of change—the slower-changing ones first—and consider using the averaging technique if you are able. The time of the fix should be the approximate middle of the sequence. At typical cruising speeds and chart scales, a difference of a minute or two will not be significant. At 12 knots, for example, a boat will travel 0.2 miles in one minute; on a 1:80,000 scale chart, that’s a distance of less than 3.16 of an inch.

As described in Chapter 16, there are generally accepted standards of precision for the description of position. If geographic coordinates are used, latitude and longitude (in that sequence) are stated to the nearest tenth of a minute on charts with scales of 1:50,000 or smaller, and to the nearest second on charts of larger scale.

If the position is stated with respect to some aid to navigation or landmark, direction from that point is given to the nearest degree (true) and distance to the nearest tenth of a mile.

In some cases it is possible to get a good sighting on one object, but no second object is available. In this situation a standard two-LOP fix is impossible, but a running fix (R FIX) can be used instead. A running fix is usually not as reliable because it incorporates an element of dead reckoning rather than simple observation.

The technique is to observe and plot the single LOP that is available. Some time later, after the vessel has traveled a known distance, that plotted LOP will be advanced (or retired) to a new position on the chart. Think of the LOP sweeping along across the water, without changing its angle, as your boat moves forward. To complete the running fix at a later time, a second LOP is plotted from a sighting on the same object or a new object and the “old” LOP is advanced to a new position—the position it would have reached in the period between the time it was sighted and the time the second LOP was sighted according to the speed and direction of the boat. You will note the dependence on an accurate measure of the speed and direction of the vessel in the intervening time.

Here is how the technique is actually accomplished. The first LOP is observed and plotted, its time and direction carefully noted. A DR track is maintained. At a later time, a second sighting on the same object or some other object is made and plotted, time and direction recorded.

Figure 18-30 A line of position is advanced by moving forward any point on it an amount equal to the boat’s motion during the time interval, and redrawing the LOP through the advanced point. The intersection of this line and the new LOP for the given time is a running fix.

The “old” LOP must be “advanced” to a new position according to the DR track, so a point is selected somewhere on the older LOP and moved forward in the direction of the boat’s travel for a distance representing the distance covered since the first sighting was observed. A new line is drawn through that advanced point parallel to the old LOP. This new, advanced LOP is labeled as soon as it is drawn with the original time followed by a dash, the new time, and the original direction. The point where the advanced LOP crosses the LOP of the second observation is the position of the boat at the time of the second observation—the running fix; see Figure 18-30.

Sometimes, the LOP is an arc of a circle, such as when a vertical angle is used to establish distance away from an observed object. In that case, the point chosen to advance the old LOP to the new position is the center of the circular arc. The advanced LOP is scribed with the same radius as the old, but on an advanced center point; see Figure 18-31.

Figure 18-31 When a circular LOP has been used (as with a distance-off measurement), it can be advanced by moving its center by the same distance and in the same direction as the movement of the boat according to the DR track. The line C-C' is equal and parallel to the track line segment between the two DR positions.

If the track of the boat in the interval between sightings is a straight line of constant speed, the DR is simple. However, the boat may vary both direction and speed between observations. This complicates the DR track, of course, but it does not change the principle of advancing an LOP. Typically, the DR plot would be maintained through these variations of speed and direction. When it is time to advance the point on the LOP, it is advanced along a line parallel to and equal in length to a line drawn between the original DR position and the DR position at the time of the running fix; see Figure 18-32.

Figure 18-32 When advancing an LOP, you must take into account all changes of course and speed during the time interval. The net effect is determined by drawing a light dashed line (X-Y) between the two DR positions that define the time interval. You then advance the LOP for the length and direction of this line on a dashed line parallel to it.

Current may also play a role in the advancement of the LOP. If current is predictable, the advanced point is moved according to the DR track, and then offset by the appropriate direction and distance according to the estimated drift and set of the current—both calculations using the elapsed time between observations; see Figure 18-33.

In all cases, a fix that is obtained from two simultaneous observations is preferable to a running fix, because the accuracy of the original LOP can only be decreased by errors and uncertainties in the DR data for the boat’s speed and direction during the interval between the first and second observations. When current is also a factor, this data becomes less and less certain.

Figure 18-33 If the current is known or can be estimated, the advancement of an LOP can be made more accurately if that effect is included as shown above.

Obviously, increasing the length of the time between observations decreases accuracy. Yet enough time must be allowed to pass in order to pick up a second object, or to achieve a substantial angular change for the second observation of the same object.

In piloting, this interval is rarely more than 30 minutes, while in celestial navigation on the high seas, the interval might be as much as several hours, since inaccuracy is less dangerous in those circumstances.

It is a matter of judgment whether a DR track should be interrupted and restarted on the basis of a running fix.

The running fix can be one form of observation on a single object (a running fix can also use two objects at different times), but there are other techniques for dealing with one available object. Successive observations of one object can use a method of bow-and-beam bearings, doubling the angle on the bow, two bearings and run between, or two relative bearings. Each of these different techniques is described in detail in this section.

Bow-and-Beam Bearings This technique uses two sightings that are easy to make with good accuracy even if no special tool is available—a 45-degree sighting off the bow and a 90-degree beam sighting off the same side. Both can be made with crude tools, even a 45-degree triangular drafting square, or no tools at all—just objects on deck that define a 45-degree and a 90-degree angle with the centerline. For this reason, bow-and-beam bearings can often be made by the helmsman singlehandedly.

As the object comes into view, the observer waits for it to reach the 45-degree alignment. As usual, the time is recorded. Course and speed are maintained. The same object is observed when it reaches the 90-degree alignment and the time is then recorded; see Figure 18-34.

Figure 18-34 With only simple tools, and without an additional person to take bearings, a helmsman can determine his position. The time is noted when the sighted object bears 45 degrees on either bow. The boat continues on with steady course and speed. The time is noted when the same object is broad on the beam. The distance traveled between A and B is calculated, and is equal to the distance that the boat was off the object, B to L at the time when it was abeam.

Picture a 45-degree drafting triangle—the long edge corresponds to the LOP of the first (45-degree) observation. The other two edges are the same length. One of these edges is the LOP of the second observation and the other is the DR track of the boat between observations.

You have recorded the time of the first observation and of the second, so you can calculate how far the boat traveled in the interval. That traveled distance is equal to the distance from the object at the time of the second observation. You now have a distance and an LOP, so you can plot a fix.

Doubling the Angle on the Bow The technique of doubling the angle on the bow makes use of a similar principle as the bow-and-beam method, but it requires that relative angles be accurately measured, which means that two people are usually required.

As the boat approaches an observed object, a relative bearing is taken. Let’s assume that the bearing is 20 degrees to port. The boat continues along a DR track until the observed object bears 40 degrees to port. The time interval and the distance traveled are calculated. Again, as with the bow-and-beam method, the distance traveled between observations is equal to the distance from the object at the time of the second observation; see Figure 18-35.

Figure 18-35 If two observations are made so that the second relative bearing is twice that of the first, as measured from the bow to either starboard or port, the distance to the object at the time of the second bearing is the same as the distance made good between the sightings—L to B equals A to B.

There’s an advantage with this method in that the boat’s position can be fixed before the object is abeam. A hazard that may be present can more easily be avoided because the boat’s course can be projected forward from the early known position. Typically, a headland can be sighted and given a wide berth well in advance.

Two Relative Bearings A more generalized solution from two relative bearings can be used. Table 18 of Bowditch uses two items of information—the angle between the course and the first bearing (the first relative bearing) and the difference between the course and the second bearing (the second relative bearing). Columns of the table are in terms of the first item, and lines of the table are in terms of the second item above; the interval between tabular entries is 2 degrees in both cases; see Figure 18-36.

Figure 18-36 Distance off at the time of a second bearing, L-B, and distance off when abeam, L-C, can be found by using multiplying factors to the distance run, A-B, between the taking of the two bearings. These factors for various pairs of angles are given in Bowditch, Table 18. An extract of this table is shown at right.

For any combination of the two relative bearings within the limits of Table 18, two factors will be found. The first number is a factor by which the distance run between the bearings is multiplied to obtain the distance away from the sighted object at the time of the second bearing. The second factor of the same entry in this table is the multiplier to be used to determine the distance off when the object is abeam, assuming, of course, that the same course and speed are maintained.

Two-Bearings-and-Run-Between Yet another method is available when only a single object can be observed. In this procedure, a bearing is taken on the object as before. The boat proceeds along its course. A second bearing is taken after the angle has changed by at least 30 degrees. The second bearing may be taken either before or after the object has passed abeam. A distance run between observations is calculated from the boat’s DR track.

Both LOPs are plotted along with the course of the boat. A pair or dividers is opened to a span representing the distance run between observations. Then the dividers are placed in such a way as to have a point touching each LOP and have the line between them parallel to the boat’s course. In other words, the boat’s course is moved “sideways” until the distance traveled exactly spans the decreasing distance between the two LOPs. The divider points now indicate the boat’s position at the time of each observation; see Figure 18-37.

Note that the distance run must be the distance over the bottom—suitable corrections having been made already for current. Accuracy depends on the usual factors, but it also depends on the accuracy of the course steered in the time interval between the observations. The net effect is to make this procedure somewhat less desirable than those described earlier.

Figure 18-37 If two bearings are taken on a single object as a boat passes by, and these are plotted on the chart, only one point can be found on each bearing line where the boat’s course and distance traveled will make a good fit. The second point is the boat’s position at the time of the second observation. (The dashed lines on the chart extract above are shown only to illustrate how an improperly placed track line would be either too short or too long; they would not be used in actual piloting.)

Several decades ago, radio direction finding (RDF) was a commonly used piloting technique for offshore cruising. The U.S. Coast Guard and comparable agencies in other countries operated hundreds of marine radiobeacon stations, and offshore cruising boats were equipped with RDF receivers. Large ships were also fitted with radio direction finders, often automatic (ADF) models.

Bearings were taken of radiobeacons located onshore and on offshore aids to navigation; these were plotted on charts in a manner similar to visual bearings. Aeronautical radiobeacons were shown on charts if they were located so that they could aid in marine navigation. Bearings could also be taken on AM broadcast stations, and charts often had symbols marking the location of transmitting antennas. The fixes so obtained were less accurate than visual fixes, but had the advantage that they could be obtained at night and in conditions of reduced visibility, such as fog; they could also be obtained at distances far in excess of visual range. Bearings could also be taken on the radio signals of other vessels; this capability often aided rescue craft in “homing” in on a scene of distress.

With the switch to VHF and single sideband (SSB) for communications, and the use of satellite systems for navigation, radio direction finding as described above fell into disuse. It would be very rare indeed to find an RDF set on a modern recreational or commercial small craft. The U.S. Coast Guard has decommissioned many radiobeacons and converted the remainder into stations transmitting differential corrections for GPS. The call signs and frequency of many AM broadcast stations and the locations of their transmitting antennas are still printed on charts, as are some aeronautical radiobeacons. However, these are not now used for direction finding for lack of onboard equipment—and the availability of better navigation systems.

Radio direction finding has survived into the modern age in a specialized form. Automatic RDFs are available that indicate the direction of a received VHF signal to an accuracy of about 5°. Such sets are commonly found on Coast Guard utility boats and cutters and on some assistance-towing craft; see Figure 18-38. These are normally used in a “homing” mode to expedite the locating of a vessel requiring help; vessel operators are often asked to give a “long count from one to ten and back” while a bearing is taken. Coast Guard and other enforcement units can use a VHF ADF to locate a source of interfering signals, such as a stuck microphone button. Bearings may also be taken on the 121.5 MHz signals of an EPIRB (if the ADF model being used covers this frequency); refer to Chapter 20.

Figure 18-38 Automatic VHF radio direction finders (ADFs) will often be seen on assistance towboats that use them for locating craft in distress or needing assistance generally.

VHF direction-finding antennas may also be seen on some sport fishermen that are homing on radio transmissions of vessels reporting successful fishing activities. It must be noted, though, that such actions are illegal. The use of intercepted radio messages not addressed to you for your own benefit is prohibited by federal law (47 USC 605 and 18 USC 2511) and FCC regulation (47 CFR 80.88).

The Coast Guard does not operate any radiobeacons on the VHF band, but the continuous transmissions of the NOAA weather stations provide a good signal for determining an LOP; the range would be the same as in regular reception of weather information, with an accuracy of 5 to 10 degrees. You must know the location of the transmitting antenna and have that plotted on the chart you are using; this information might be found in marine forecasts, on NOAA weather radio websites, or from local weather forecast offices. This will normally provide only one LOP, but in limited areas it may be possible to receive two weather stations and, therefore, to obtain a rough fix. Coast Guard stations and Marine Operators normally have multiple antenna sites, and it is not possible to use their signals for taking bearings for lack of knowledge of which antenna was currently being used.

Figure 18-39 Radar ranges are more precise than radar bearings. When getting a position from radar alone, the best fix will be from the use of distances to two (or more) objects that can be identified on the radar display and on the chart.

Equipment for VHF direction finding may be either a self-contained unit or an add-on for a conventional communications transceiver. In the latter case, no modification is required of the basic radio, and it can continue to be used for its normal functions. But in both cases, a direction-finding antenna as in Figure 18-38 is required to sense the direction from which the VHF signals are coming; an automatic switch protects the RDF unit when the set transmits. The display can be either the relative bearing of the source of the signals, accurate to about 10 degrees, or the crosstrack error for heading directly to the other vessel.

Figure 18-40 Radar is an excellent piloting tool, as it can provide both direction and distance information on a single object. A fix from a radar observation is thus plotted as a line and an arc, with the fix labeled with the time and the block lettering “RADAR.” If the direction information is from a visual bearing, the fix is more accurate.

A boat equipped with a properly installed VHF/DSC radio can use digital hailing to call other vessels and shore stations. If the hailing call includes the vessel’s position, the receiving radio will display the MMSI and position of the caller and plot the position and identification of the calling station on a chartplotter. (See Chapter 20.)

The features and capabilities of radar are introduced and discussed in Chapter 16. Radar serves two purposes on board ships and boats. Although often thought of as a collision-avoidance device, radar can be an essential tool in navigation. Radar has a unique advantage in navigation in that a single instrument can measure both direction and distance to moving and still targets—and can do so under any condition of visibility. Excellent detailed information on radar can be found in NGA Publication No. 1310, Radar Navigation and Maneuvering Board Manual (now available only in digital format on the Internet at the NGA Maritime Safety Information Division website, http://msi.nga.mil/NGAPortal/MSI.portal).

Radar measurements of direction are not as precise as those made visually, but radar can almost always make a measurement despite fog, light rain, or darkness. Heavy rain can decrease and may even prevent radar from detecting targets.

On the other hand, as mentioned in Chapter 16, measurements of distance are quite precise and accurate—much more so than vertical sextant measurements or optical range-finder readings.

Direction and distance fixes determined with radar are used in much the same manner as LOPs from visual sources. There are three different procedures for obtaining a position by radar; they are, in descending order of accuracy: radar measurement of distance to two objects, as in Figure 18-39; bearing on one object with a corresponding radar distance to that object, as in Figure 18-40; and two radar bearings. When the opportunity exists, an excellent combination is a visual direction LOP and a radar distance to the same object. Any fix involving radar is labeled with time and “RADAR.” Radar distances can also be used in the manner described for vertical angles in order to avoid unmarked hazardous areas.

Radar observations of isolated objects such as buoys, offshore lighthouses, etc. are usually unambiguous. When the radar target is onshore, however, some identification problems may arise; there must be positive identification of the object on both the radar display and the chart.

The capabilities of LCD radar displays make radar piloting more convenient and accurate than ever. See Chapter 16 for a discussion of the use of electronic bearing lines (EBLs) for taking bearings and variable range markers (VRMs) for obtaining ranges. These features are now standard on radar units. There is a lot to be said for a device that can show you accurate bearings and ranges to shorelines, prominent features, and aids to navigation and detect other vessels that are out there with you in the night or fog. Some mariners consider radar even more useful than GPS—but modern multi-function displays that overlay radar imagery on a digital chart (see Figure 16-37) or display the two side by side on a split screen (see Figure 16-55) make it ever more convenient to benefit from both.

Figure 18-41 Many electronic chartplotters can also display radar, engine information, and fishfinder data. See also Chapter 16.

The depth of water at a vessel’s location is valuable piloting information. A single depth reading cannot tell a boater where he is, but it can tell him where he is not. If an accurate depth reading is 26 feet, you may be at any number of places that have a 26-foot depth, but you are not at a place that has significantly greater or less depth. When sufficient depth data have been obtained and have been corrected for the height of tide, they can be used for positive position determination in combination with an LOP from another source, or on their own.

As discussed in Chapter 16, most depth sounders display the depth from the bottom of the water to the transducer, which is normally located a few feet below the water line. This distance must be added to get the depth of the water from the surface, the information needed for piloting. Some models, however, can have the display reading offset so the reading indicates the water depth below the keel, or preferably, depth from the surface, better information for use in piloting. Remember that depth figures on charts are for conditions at mean lower low water.

Given a bottom with relatively uniform slope, it may be possible to get position information with depth and a single LOP. Typically this would be a beam bearing and a single depth reading. The LOP is plotted, then a point along the line is found that matches the depth indication on the chart, corrected for tidal stage; see Figure 18-42. Such a position should be regarded only as an estimated position and marked with a square symbol. If there are two or more locations along the LOP having this depth, you will not be able to estimate your position.

Figure 18-42 A single LOP can sometimes be combined with depth information to provide an estimated position (EP). The LOP shown above is a beam bearing. A depth reading of 31 feet indicates that the estimated position is somewhat inshore of the DR track.

In some instances a rough estimate of the boat’s position can be obtained by matching a series of depth readings—one that corresponds to the density of depth figures on the chart in question. Mark your depth readings on tracing paper and move the paper around on the chart, keeping the line of soundings parallel to a line representing the direction of your course steered while the soundings were made. Look for the best match between your series, corrected for tidal stage, and a series on the chart. Don’t expect to get an exact concurrence; see Figure 18-43.

This technique is not suited to shorelines foul with offshore rocks and areas of irregular or uniform depth. Confirmation of position cannot be positive, but certain denial can be useful.

Figure 18-43 A series of soundings can be plotted on a piece of tracing paper, each depth noted with time and spaced according to the boat’s movement over the bottom. The tracing paper is then moved about on the chart until as good a match as possible is made between the measured depths and the depths shown on the chart. Remember to correct for the state of the tide.

When cruising over a bottom of uniform slope, piloting can be simplified by choosing a depth curve that follows a safe path. The boat may be steered manually to track the depth contour or automatically guided along the contour by an autopilot. Don’t use this method when there is any chance of meeting hazards by covering more ground than anticipated or by distorting the curve at sudden, sharp changes in direction.

Changes in depth resulting from tidal action cause a horizontal movement of water. A vessel thus travels in a body of water that is itself moving with respect to the bottom. This motion of the water—current—must be considered in determining a vessel’s position. The effect of currents on the piloting of boats is covered in Chapter 17.

Beyond the basic techniques described in the preceding pages, as well as in Chapter 17 in the discussion of currents and piloting, there are many other specialized techniques that are useful in specific situations. These techniques should not be regarded as shortcuts. Their use requires a thorough understanding of principles, and they should not be approached until the basics have been learned.

Although the bow-and-beam technique has already been covered earlier in this chapter, you should know that there are other pairs of bearings that are not as easily obtained, but are easily used to plot a fix.

The following sets of bearings have a relationship to each other such that the run between the first bearing and the second will nearly equal the distance away from the sighted object when it is passed abeam:

Note that these are pairs of relative bearings to port as well as to starboard. In the preceding table, “20°-30° can be either relative bearing 020° and 030°, or 340° and 330°, “31°-56° can be either RB 031° and 056°, or RB 329° and 304°, etc.

If the observations are made when the relative bearings are 30° on either bow, simple calculations will give two useful items of information. The distance run between the observations is equal to the distance to the object at the time of the second bearing (remember doubling the angle on the bow). Also, this same distance multiplied by 7.8 (or more precisely, 0.866) is the distance that the boat will be off from the sighted object when it is broad on the beam, provided that the vessel’s course has not changed; see Figure 18-44.

Figure 18-44 The “seven-eighths rule” is a special application of doubling the angle on the bow. If 30° and 60° are chosen as the first and second relative bearings, we know two things—the distance away at the second bearing is equal to the distance made good since the first bearing, and the distance off when abeam of the object will be seven-eighths of the distance between the first and second bearings.

A comparable situation to the seven-eighths rule earlier is one in which the relative bearings are 22.5° to port or starboard. In this case, as before, the distance away from the sighted object is equal to the distance run between bearings, but the multiplier to determine distance off when the object is abeam is 0.7.

When determining the relative motion of another boat, it is necessary either to convert relative bearings to compass (or true) bearings, or to maintain a steady course. In practice, it is easier to maintain a steady course and observe the change in relative bearings. The actual relative bearing is far less important than the direction and rate of change.

What we want to know in any crossing situation is: Will we cross ahead or astern, or will we collide? Because we are interested in relative and not absolute motions, it is convenient to think of our own boat as being stationary.

The first situation, the other boat passing ahead, is shown in versions A and B in Figure 18-45. The successive positions of the two boats show a steadily decreasing bearing. Version A shows the boats in absolute motion and version B shows how it looks from our own boat if we imagine it to be stationary.

Figure 18-45 Shown here are two views of the same crossing situation. An observer directly overhead would see version A, while version B suggests the view from our own craft—the one that appears to be stationary. Crossing situations are much easier to predict if we imagine our own boat to be stationary and the other boat to be moving relative to our own.

The line connecting the two boats in version B is the line of relative motion, the course and distance traveled by the other boat in relation to our own. Provided both boats maintain course and speed, it is a straight line. In this situation, when the line of relative motion is extended, it passes the bow of our “stationary” boat; therefore, the other boat following that line will cross ahead. The distance x is the minimum distance that will separate the two boats in this case. It occurs at the closest point of approach (CPA).

In the second case, shown in Figure 18-46, versions C and D cover the situation when the relative bearing moves aft. The line of relative movement passes aft, and so will the other boat.

If the relative bearing does not change, the line of relative motion will pass through our boat’s position. You might say that the CPA is zero, but you would be more likely to say “He’s going to hit us!”

Figure 18-46 The other boat was going to cross ahead of our own in Figure 18-45. We could make this prediction because its relative bearing moved toward our bow. Here in versions C and D, the other craft’s relative bearing moves aft. We know it will pass astern. In both cases, the separation at the closest point of approach is the distance x.

It is important to note that these illustrations are not actual chart plots of the boats. The position of the other boat cannot be plotted unless distance and bearing information were available, and this is unlikely unless you are using radar.

The significance here is that bearing information only, which is available to any skipper, will indicate how the two boats will converge and whether they will collide. The separation at CPA cannot actually be known unless a plot is made, but this is not significant.

• Observe the relative bearing of the other boat only when you are on your specified compass course. You must be on the same course each time you take a relative bearing.

• Watch the other boat for changes in course or speed that would obviously change the relative motion.

• If the relative bearing moves ahead, the other boat will pass ahead. If the relative bearing moves aft, the boat will pass astern. If the relative bearing is steady, there is a very real danger of collision; see versions E and F in Figure 18-47.

Figure 18-47 In this crossing situation, versions E and F, the relative bearing remains constant—unless one craft or the other alters course or speed, they will collide. The line of relative motion passes through our boat and there is “no separation” at the point of closest approach.

Relative bearings can also be taken to help in compensating for the effects of offsetting currents or wind. In versions G and H of Figure 18-48, the skipper has put his boat on a course that he believes will put a buoy close aboard and avoid a shoal. At the time of the first bearing (“0 minutes”) the buoy bears 340° relative. Five minutes later, the relative bearing is 350°. It is clear that the boat is being set down more than expected and that the line of relative motion of the buoy will pass ahead of the boat (i.e., the boat will pass on the shoal side). If the bearing on the buoy remained steady, the boat would pass close to it, and if the bearing shifted gradually away from the bow, the skipper would know that he would clear the shoal safely. In Figure 18-48, version G shows the problem in terms of the boat moving relative to the buoy, while in version H, the boat is considered to be stationary and the relative motion shown is that of the buoy. Note that the lines of relative motion in each case are parallel to each other, equal in length and opposite in direction.

Figure 18-48 A plot of relative motion is also useful when the other object is stationary. In this situation, we can use the plot to determine whether we will leave the buoy to port or to starboard if we make no change in course or speed. In version G, our motion is relative to the buoy. In version H, its “motion” is relative to us.

A very useful technique in making a landfall confidently is to use a deliberate course offset to ensure that you miss your target on a known side. When you lay your course, you set it decidedly to one side of your objective. The advantage of this procedure lies in the fact that when you do not arrive at your destination, you know which way to turn. If you had aimed directly at your destination but failed to arrive there due to unpredicted current or steering errors, you would not be sure whether it lay on your port or starboard bow. With a deliberate offset, you can make a confident alteration of course, usually parallel to the shoreline.

A SAILOR’S EYE

One of the easiest ways to acquire what is often referred to as a “sailor’s eye” is to learn the simple but important principle of relative motion. For example, you find that your boat is converging on another boat that has the right of way. If you hold your course and speed, will you clear? What if your sailboat is on port tack and must yield right of way to a starboard tack boat? Do you hold and clear ahead, or pass under its stern? You have laid a course for a buoy, making allowances for current: Have you made the right allowance?

Relative motion is simply the motion of two boats in relation to each other. A boat approaching you on an opposite parallel course at 5 knots while you are making 4 knots has a relative motion of 9 knots. When meeting, if the skipper of the other boat turns to run alongside your boat to chat, your relative motion will be 1 knot, and you will see his boat going slightly faster than yours.

The relative bearing of an object can be estimated within 5 or 10 degrees with practice. When the direction of your own boat changes, the relative bearing also changes, though its true bearing remains the same. For example, if you are on a course of 050° relative, the true bearing of the object will be 140° (50 + 90). If you change your course by coming left to a course of 020°, the relative bearing of the object will now be 120°. But the true bearing remains the same at 140° (20 + 120).

Let us say, for example, that after several hours of trolling, drifting, and circling around, your position is uncertain. Just as you decide that your fishing luck has run out and it’s time to head back to Block Island, the fog closes in. Your best position estimate is that you are east of Bell Buoy 1, which is marked as point B in Figure 18-49. You believe that you are somewhere near point A on the chart.

Figure 18-49 If you fail to make your predicted landfall, which way should you turn? To avoid this dilemma, make a large, deliberate “error” to one side or the other; then the choice is clear.

The solution to your problem is a deliberate offset north of the harbor entrance, which is marked by lighted beacon “3.” For one thing, you think this offset will take you close to the off-lying bell buoy, and if you see or hear it you will know where you are. But even if there were no buoy, there is still merit in laying the course decidedly off to one side of the ultimate objective—Block Island Harbor.

Suppose you lay your course A-B to the bell buoy and miss it by ⅛ mile to the north. When you pick up the 3-fathom (18-foot) curve at G, you know you are north of the harbor—because you would have picked it up much sooner if you were south of the harbor entrance.

Had you missed the bell buoy by ⅛ mile to the south and picked up the 3-fathom curve west of D, you would still be able to follow the curve southeastward to lighted beacon “3,” deriving added assurance of your general position north of the harbor when you find depths holding generally at 3 fathoms on your course southeastward. If your calculations were completely wrong and you ran southeastward from any point below the harbor, depths would increase.

Let us consider, still in reference to Figure 18-49, what would have happened if you had laid a course directly for lighted beacon “3” and had missed your objective, E, by ¼ mile to the south. Picking up the 3-fathom curve at F, you could not be sure whether you were north or south of the entrance.

Your first conclusion might be that, since you have run a few minutes overtime before reaching the 3-fathom curve, you are north of the harbor. But don’t forget that you didn’t accurately know your starting point. Suppose you had been several minutes eastward of where you thought you were when the fog set in. In this event, the additional few minutes’ running time before reaching the 3-fathom curve would tell you nothing.

On the assumption that you are indeed somewhat north of the harbor, you turn south, proceeding slowly and carefully. A few minutes pass, but still no beacon or harbor entrance. How far to continue? That beacon or the entrance breakwater may be just ahead, obscured by the fog. On the other hand, doubt creeps in. You couldn’t have been that far off in your reckoning—or could you? So you’re in the fog—literally and figuratively.

In the first example, where you laid a deliberate offset, the intended track was about ⅜ mile to the north of the objective. How much deliberate offset should be made is a matter of judgment. On a long run in toward a long beach that runs for miles in either direction with few identifying marks, even if visibility is good, you may prefer to make allowances of a mile or more. But don’t use this technique blindly. Study the chart carefully for possible dangers that may lie along the beach.

The same procedure may be used in other uncertain situations, such as crossing the Gulf Stream. Rather than attempting to make an exact allowance for the distance that the stream will set northward, make a somewhat overgenerous allowance. If you don’t pick up your target landfall after the allotted passage time, you can turn northward with confidence rather than facing a 50-50 chance of being wrong.

If a boat’s heading changes, the relative bearing of an object is changed by the same amount. From the situation at A to that at B, the true heading of the boat has changed from 050°. The relative bearing has changed by the same amount, from 090° to 120°. Note: the sum of the heading and relative bearing remains constant (disregarding any forward boat movement).

If you are anticipating fog, you can take most of the worry out of the process by planning and plotting your run in advance. This same “timetable” technique can make a night run a pleasure instead of a trial.

Don’t wait until you are underway to lay out your course, distances, running times, etc. Go over the entire trip in advance and acquaint yourself with each leg of the cruise, the aids to navigation that you will pass, the characteristics of lights, etc.

If you are piloting a powerboat and you can predict your speed with precision, set up a timetable. Set your time of departure as 0000 and note, in orderly fashion, the predicted time of arrival abeam of every light and buoy on both sides of your intended track. Alongside each entry, show the characteristics of the navigational aid and the compass course at the time. You can also enter into your timetable the approximate times that major lights can be expected to become visible.

Use a watch or luminous clock with a luminous dial; set it at 0000 (12 o’clock) and start it when you leave your point of departure. The elapsed time of each event along the track will provide a quick aid to identification for each light or sound signal along the way.

By using elapsed time rather than actual time, you build some flexibility into your plan. A late or early departure will not upset your entire timetable. Any discrepancies that creep in can be compensated for by a reset of the clock.

The lights of towns along shore can be used to estimate distance offshore. An experienced boater may be able to judge his distance by whether or not he can see the glow from the street and sign lights, or whether he sees the lighting directly. This might be a useful technique on a long, straight shoreline of consistent elevation. Of course, the height at which the lights might be seen directly will vary with the observer’s height of eye and should be tested by observing the shoreline at known distances.

You may also be able to follow a shoreline (without the help of a depth sounder) by noting the point at which the beach or shoreline disappears from view. This requires a straight shoreline of consistent elevation, as well as some experience, but it can be quite a useful check; see Figure 18-50.

Figure 18-50 Many coastal areas with regular slopes toward deeper water offer depth information that can determine distance off. A rough but useful estimation of distance can also be made by using your height of eye at the helm and observing the buildings ashore.

Let’s say you wish to parallel the New Jersey or Long Island beaches. From the deck of a typical small boat, at about 4.5 miles offshore, the beach will be under your horizon. If you can just see the beach, you are about 4 miles off. Fog and haze will make this observation unreliable, and the height of your eye on your particular boat will also vary this distance.

You can also use your view of buildings on shore to estimate distance. For example, if you can make out individual windows on houses, you are probably about 2 miles offshore. With this rule of thumb, you could confidently stay from 2 to 4 miles off a long beach with no other reference.

An interesting variation on the technique for estimating distance off a beach is one used in Alaska and British Columbia, where the problem is often to maintain a distance off a rocky cliff face. It consists of sounding a short blast on the whistle or horn and timing the return of the echo.

The interval in seconds (preferably measured on a stopwatch) is divided by two because the sound wave goes to the cliff face and back. Multiply that figure by 1,100, which is a rough value for the speed of sound in air in feet per second. (Multiply by 340 for the distance in meters.)

For example, you sound a blast and the interval is 5 seconds. Multiply 2.5 seconds (the one-way trip) by 1,100 and you have 2,650 feet, or a distance off of slightly less than 0.5 nautical mile.

In some cases, a pilot might use an echo from both shores to stay in the middle of a passage simply by determining which one returns first, then altering course away from it.

This is another instance in which a convenient mental calculation can be made to reach a safe approximation. While it should never act as a substitute for position determination, this kind of calculation can be useful in making a decision while cruising shorthanded.

The rule-of-sixty provides a simple way of determining a new course to clear an off-lying danger area by the desired amount. You need to know your present course, your present distance off, and the distance by which you would like to clear the danger area.

Let’s assume that you are in the position shown on the chart. You have come out of Portsmouth Harbor and are running a southerly course down the coast to Cape Ann. Somewhere off New -buryport you pick up the light on Straitsmouth Island, dead ahead; see Figure 18-51, at the bottom center, marked C. Your course made good along the line A-C has been 164° true. A bell buoy has been placed 1.5 miles eastward of the light, marking a number of rock ledges. You want to make a course change to clear the shoals. By how many degrees should you change course?

You could get out the chart and plot your position—which, if you are able to do so, is what you should do. However, you can also apply the rule-of-sixty and determine your new course in your head.

You know that the light you are seeing on the horizon is visible for 6 miles. (We’ll assume that it is a clear night; if it isn’t, you will have to establish distance by some other means.) The procedure is to divide 60 (the rule) by 6 (the distance) to get 10. Since you want to clear the light by 1.5 miles, multiply 10 by 1.5 and get 15—this will be the number of degrees by which you will have to change course. Thus your new course is 164° true. Long before you need to be concerned about the rock ledges, you will pick up the bell buoy north of Flat Ground.

The rule-of-sixty is never a substitute for proper plotting, but in rough weather or other difficult circumstances, it can be a reassuring guide.

Figure 18-51 The “rule-of-sixty” can provide a shortcut for calculating the change in course required to clear an obstacle. It can be applied without making a chart plot.

A pilot is often required to measure a total distance along a course that consists of several short, irregular legs. Of course, each leg can be measured and added, but there’s a much easier graphic method.

In Figure 18-52, the boat is at A. You need to know the total run to F along the track B, C, D, E, F. The figure shows guidelines, which extend each leg outward, but after you become familiar with the procedure you can use your eye to make the same extensions.

Place your divider points on A and B. Swing the A point to A1 (which lies along an extension of the next leg). Plant the divider point at A1 and stretch the other leg to C. Again, swing the A1 point to A2 (the extension of the next leg.) Plant the point at A2 and stretch the dividers to D. Repeat in this way until the dividers are planted at A4 and stretched to F.

Now you can lay that divider span along the graphic scale or the latitude scale at either side of the chart. You can make the measurement quickly and more accurately this way than by measuring and adding each leg.

Figure 18-52 A pair of dividers can make quick work of adding the lengths of several straight segments of a curved course. Span the first leg, A to B, then swing the point of the dividers from A to A1 in line with B and C and plant it. Next, move the point of the dividers that was at B to C; this adds the lengths A-B and B-C. Continue this procedure until you reach the final point, F in this example. Use the total span of the dividers on the chart’s graphic scale to get the total distance.

The principles of piloting a small sailboat are no different from piloting a powerboat. However, much of successful piloting is not principle but practice, and the practice aboard a sailboat may differ in several respects.

Most important is the fact that sailboat speeds are much more difficult to predict. While a powerboat pilot might establish a timetable for a particular set of legs of his trip (refer to “Using a Predicted Timetable”), a sailboat skipper would be much less likely to be able to predict the boat’s speed several hours in advance along a particular leg of the trip.

In addition to dealing with the built-in complexity of a particular track, a sailboat navigator also makes extra calculations and plots to account for the fact that his boat will tack upwind and downwind too.

For the skipper of a powerboat, this leg may be a simple straight line with an easy-to-predict speed and direction. But for a sailor it becomes a series of doglegs, each with its own requirements (or estimates) and other variables to do with steering the fastest possible course following a shifting wind, rather than a straight line; see Figure 18-53.

Figure 18-53 The direct course (the one that a powerboat would take) is known as the rhumb line. The actual course that a sailboat might follow, both upwind and down, is considerably longer and more difficult to reckon. The values shown above are typical of a welltrimmed, deep-keel craft.

This makes it far more important for a sailboat navigator to keep track of his DR track at every tack and jibe, and to make an estimate of position just before the course is changed. It may be possible to sail equal times of each tack to simplify the calculation, but this is useful only when the destination is directly upwind, or down.

Since sailboats generally travel upwind with a “leeway angle” (refer to Chapter 8), allowances must be made for the difference between the compass orientation to the centerline of the hull and the hull’s actual direction of travel through the water. It might vary by 2 or 3 degrees—much smaller than a typical steering error, but of potential significance as an additional factor over a long leg.

A rule of thumb for sailing a tacking leg when the wind is not directly from or to your destination is to sail the longest leg first, except when the shorter leg may provide an opportunity for a fix that would be missing on the longer leg.

Finally, the actual chartwork aboard a sailboat can be more difficult due to a number of factors. The navigator may be working at an extreme angle of heel, for example, and is much more likely to be soaking wet.

Position determination cannot be learned totally “from the book.” Study is essential, but you must put into practice the various procedures and techniques learned. It’s a good practice and good entertainment to “overnavigate” during daylight and in good weather so that you will be experienced, capable, and confident when the sun sets or the weather closes in.

GPS and other electronic navigation systems have made unnecessary the routine use of many of the procedures described above, but remember, equipment can fail and power can be lost; you had better be prepared to navigate the “oldfashioned” way!

20°-30°	21°-32°	22°-34°
23°-36°	24°-39°	25°-41°
27°-46°	29°-51°	30°-54°
31°-56°	32°-59°	34°-64°
35°-67°	36°-69°	37°-71°
38°-74°	39°-77°	40°-79°
41°-81°	43°-86°	44°-88°