Visual piloting: three-bearing fixes and danger bearings

What the rule requires

The fix, and the triangle it produces

A line of position is a line along which the vessel is known to lie at a given moment from a single observation, and a fix is the intersection of two or more LOPs taken simultaneously Bowditch Ch. 7 §701. Simultaneity is the whole of the difference between a fix and a running fix, and it is the word exam writers hang the question on.

Three simultaneous LOPs to well-separated charted objects rarely cross at a point. The small triangle they form is the cocked hat. A small cocked hat indicates a good fix and is taken at its centre; a large one signals bearing or plotting error. Where the triangle lies near a hazard, the corner nearest the danger is assumed for safety Bowditch Ch. 7 §705.

That last clause is the one candidates skip. The centre is the default, not the rule.

Two bearings will cross and give you a position, but they will not tell you that the position is wrong. With a constant bearing error the two-bearing fix lies on the circumference of a circle passing through both charted objects and the observer, and a third bearing is required to determine the direction of the constant error Bowditch Pub. 9 Ch. 10 §1023. The third bearing is not redundancy; it is the error detector.

Angle of cut

Fix error from an error held constant across both observations is minimised when the bearings intersect at 90°. The error grows as the cosecant of the angle of cut, and the growth becomes rapid once the angle falls below about 30°; at 30° the fix error is about twice what it is at 90° . The strongest visual fixes cross their LOPs near 90 degrees, use objects positively identified on the chart, and are observed in quick succession with the beam-most object taken last Bowditch Ch. 7 §707.

Two easily identified objects that lie nearly in line make a weak fix. A tank and a stack close together on the same shore are worse than a tank ahead and a beacon abeam, whatever the plotter's confidence in the identification.

When several fixes on three objects produce triangles of about the same size, suspect a constant error in observing or plotting rather than sloppy work; if applying one constant correction to all bearings produces a pinpoint, apply that correction to subsequent fixes and then hunt the source in the gyrocompass, the repeater, or the bearing transmission system Bowditch Pub. 9 Ch. 10 §1024. The same technique run alongside the pier is a gyro check: three LOPs that plot as a cocked hat indicate gyro error, the total correction that eliminates it is the error, and more than 1.0° T is a gyro problem to investigate before piloting Bowditch Pub. 9 Ch. 10 §1007.

How often, and what to do when you cannot

The preferred piloting fix is taken from visual bearings to charted fixed NAVAIDS, plotted on the primary plot. In restricted waters the interval between fixes should usually not exceed three minutes, and a well-trained plotting team should need no more than 30 seconds to measure, record and plot three bearings to three separate NAVAIDS. If poor visibility obscures the visual NAVAIDS, shift to radar piloting on the primary plot; if neither visual nor radar piloting is available, consider standing off until visibility improves. Never continue a passage through restricted waters if the vessel's position is uncertain Bowditch Pub. 9 Ch. 10 §1013.

The three minutes is a ceiling on the interval, not a target for how long plotting may take. Most of the interval belongs to interpreting the plot and making recommendations to the conning officer.

Danger bearings

A danger bearing keeps the vessel clear of an off-lying hazard using a single charted object. Draw the line from the object tangent to the edge of the hazard — or tangent to a danger circle scribed at a safe distance around it — and label it NMT (not more than) or NLT (not less than) that bearing. So long as the object's observed bearing stays on the safe side of the danger bearing, the vessel remains in safe water Bowditch Ch. 7 §706.

No position is plotted. That is the point of the technique: it answers "am I clear?" without answering "where am I?"

The leading object must satisfy three conditions — visible to the eye, indicated on the chart, and its bearing from the danger area in the same general direction as the course of the boat as it proceeds past the area. Label the line DB followed by the direction, take frequent visual bearings, and draw a series of short lines on the danger side of the bearing so the hazardous side is unmistakable. In the Boat Crew Seamanship Manual example the danger bearing is DB 311° M: bearings greater than 311° M put the boat in safe water, and an observed 300° M means the boat is standing into danger. Every crewmember must know which side the danger lies on — whether the danger is all degrees less than the danger bearing or all degrees greater Boat Crew Seamanship Manual (M16114.5C) Ch. 14 — Piloting ¶8.

The sense reverses with the geometry. In the Bowditch worked case, a track passing close to a shoal has a line drawn from NAVAID H tangent to the shoal at 074.0° T, and so long as H bears less than 074° T the vessel will not ground; the dangerous side is hatched and the line labelled NMT 074.0° T. The tangent is not sacred — a navigator wanting margin may draw the line at 065° T instead. Lay down a danger bearing from any suitable NAVAID near any hazard, and ensure the planned track does not cross one. A danger range is the same idea expressed as a standoff range from an object Bowditch Pub. 9 Ch. 10 §1007 ¶1.

Your danger bearing reads NLT 128° M and you observe the light at 124° M. Safe or standing into danger?

Standing into danger. NLT means the safe water lies at bearings not less than 128° M, so 124° M is on the hazardous side. Read the label, not the number — the hatching on the chart and the NMT/NLT prefix carry the sense, and it reverses depending on which side of the track the hazard sits .

Distance off with the sextant

For the three-point problem, three charted objects are selected for horizontal sextant angles, one object common to both angles. Each pair of objects defines one circle of position passing through the two objects and the observer, so two nearly simultaneous angles establish two circles intersecting at two points, and the observer knows which of the two is his. The solution is worked by laying the arms of a three-arm protractor over the three objects on the chart; the centre of the protractor disk then sits over the observer's position Bowditch Pub. 9 Ch. 11 §1101.

A vertical angle answers a different question — how far off. Table 16 gives distance from an object of known height above sea level: measure the angle between the top of the object and the visible sea horizon, correct for index error and dip only, and enter the table with the corrected angle and the difference between the object's height and the height of eye. Where the visible horizon is not available, measure to the bottom of the object and use dip short of the horizon from Table 15, which may take several approximations before the same distance comes up twice Bowditch Pub. 9 Ch. 11 §1109.

Telling it apart

The separating question is what the observations produce: a position, a degraded position, or only a limit.

Fix — two or more LOPs observed simultaneously; a position Bowditch Ch. 7 §701.
Three-bearing fix — three simultaneous LOPs; a position plus an error indication in the size of the cocked hat Bowditch Ch. 7 §705.
Running fix — LOPs that cannot be taken simultaneously, the first advanced along the DR track to the time of the second; a position, and less accurate than one from simultaneous LOPs Bowditch Ch. 7 §702. Most often misfiled: two bearings to different objects taken minutes apart get plotted and circled as a fix.
Danger bearing — one object, one line, no position at all; labelled NMT or NLT and monitored by frequent bearings Bowditch Ch. 7 §706.
Danger range — the same limit expressed as a standoff range from an object rather than a bearing to it Bowditch Pub. 9 Ch. 10 §1007 ¶1.

Working a question

You have one usable object: a light on the beach. At 1000 you observe it by compass at 240°, corrected to 237° M, with no other well-defined object available for a bearing. Since plotting that LOP the boat has run at 28 knots on compass course 030°. At 1030 the light bears 325° C.

Time interval. 1030 − 1000 = 30 minutes.
Distance run. D = S x T/60 = 28 × 30/60 = 840/60 = 14 nautical miles.
Step off 14 NM with dividers from the latitude or nautical mile scale and lay it along the course line in the direction travelled.
Advance the 1000 LOP, moving it parallel to itself, forward along the course line by 14 NM. Label the new line 1000–1030 to show it is an advanced LOP.
Correct the second compass bearing, 325° C, to 322° M.
Plot the 322° M bearing. Where it crosses the advanced LOP is the running fix Boat Crew Seamanship Manual (M16114.5C) Ch. 14 — Piloting ¶8.

Step 4 is where the accuracy lives, because advancing the LOP assumes the course and speed were held exactly between the two observations Bowditch Ch. 7 §708. Step 2 is where the safety margin lives. An unknown head current means the distance actually run is less than 14 NM, so a fix advanced the full 14 NM plots the vessel farther from the beach than she really is. Assuming a following current pushes the running fix farther from the NAVAID than the vessel's actual position; assuming a head current plots it closer. To keep a margin of safety when running a fix off a shore NAVAID, always assume the current slows the vessel's speed over ground, which plots the fix closer to the shore than she actually is Bowditch Pub. 9 Ch. 10 §1019.

Where the second bearing is taken to a different object, maximise the speed estimate instead if that object is on the same side and farther forward, or on the opposite side and farther aft, than the first object was when observed. Both cases assume the danger lies on the same side as the object observed first.

Two shortcuts exist when only one object is available and you want range rather than a plotted intersection. Doubling the angle uses two relative bearings, the distance run between them equalling the distance to the object at the second bearing Bowditch Ch. 7 §703. Table 18 generalises it: enter with the difference between course and first bearing along the top and course and second bearing at the left, then multiply the distance run between bearings by the first tabulated factor for the distance at the second bearing, and by the second factor for the distance when the object is broad on the beam. On course 050° at 15 knots, a light bearing 024° at 1130 and 359° at 1140 gives factors 1.04 and 0.81 on a 2.5-mile run — 2.6 miles at the second bearing and 2.0 miles abeam. The method is accurate only if the helmsman steered a steady course and the navigator used speed over ground Bowditch Pub. 9 Ch. 10 §1021.

Where candidates lose the point

Plotting the centre of the cocked hat when the triangle straddles a charted shoal. The centre is right for an ordinary fix; with a hazard close aboard the corner nearest the danger is assumed Bowditch Ch. 7 §705.
Treating a large cocked hat as a fix with generous error bars. A large triangle signals bearing or plotting error, and triangles of similar size fix after fix point at a constant error to be found and eliminated Bowditch Pub. 9 Ch. 10 §1024.
Believing the vessel's own heading or position changes the value of a danger bearing. The number comes off the chart, from the line drawn between the leading object and the tangent to the hazard; the only thing that changes underway is the observed bearing you compare against it Bowditch Ch. 7 §706.
Memorising one example's direction and answering that greater than the danger bearing is always safe. In the Boat Crew Seamanship example bearings greater than DB 311° M are safe water, while in the Bowditch case safety lies at less than 074.0° T. The NMT or NLT prefix and the hatching on the danger side settle it Boat Crew Seamanship Manual (M16114.5C) Ch. 14 — Piloting ¶8.
Picking a running fix as equal in reliability to a two-object fix because the plotting was careful. It rests on course and speed having been held exactly Bowditch Ch. 7 §708.
Accepting a narrow angle of cut because both objects were unmistakable. At 30° the fix error is about double that at 90°, and it worsens quickly below that Bowditch Pub. 9 Ch. 10 §1023.
Pressing on through a channel while sorting out a plot that will not close. Slow or stop and fix the vessel; misidentification of aids to navigation and failure to slow down when in doubt of the ship's location are both on the list of errors behind most groundings Bowditch Pub. 9 Ch. 10 §1022.

Check yourself

You take three simultaneous bearings entering a harbour. The resulting triangle is about 100 yards on a side and its western corner falls on the 3-fathom edge of a shoal you draw too much water for. Where do you plot the fix?

At the corner nearest the danger — the western corner on the shoal edge. The centre of the cocked hat is the normal choice, but where the triangle lies near a hazard the corner nearest the danger is assumed for safety Bowditch Ch. 7 §705.

You have four charted objects in sight: a stack bearing 010°, a tank bearing 025°, a beacon bearing 095° and a church spire bearing 170°. Which three do you shoot?

The stack or the tank, plus the beacon and the spire. The stack and tank differ by only 15°, so using both wastes an observation on a pair that cuts at a poor angle; fix error is least when LOPs intersect near 90° and rises rapidly below about 30° Bowditch Pub. 9 Ch. 10 §1023. Take them in quick succession with the beam-most object last Bowditch Ch. 7 §707.

Fog closes in and the visual NAVAIDS are gone. Four minutes have passed since your last fix. What are you required to do?

Inform the captain, shift the primary plot to radar piloting, and get a fix down as soon as possible. The interval in restricted waters should usually not exceed three minutes. If neither visual nor radar piloting is available, stand off until visibility improves; do not continue with the position uncertain Bowditch Pub. 9 Ch. 10 §1013.

At 0900 a lighthouse bears 195° M. You steer 070° M at 12 knots and at 0920 the same light bears 260° M. How do you obtain the 0920 position, and what will you label on the chart?

Run a running fix. The distance run is D = S x T/60 = 12 × 20/60 = 4.0 NM. Advance the 0900 LOP parallel to itself 4.0 NM along the course line and label it 0900–0920, then plot the 260° M bearing; the intersection is the running fix Boat Crew Seamanship Manual (M16114.5C) Ch. 14 — Piloting ¶8.

Same problem, with a suspected current of unknown strength and the light on the shore you are closing. Which current assumption keeps you safe?

Assume the current slows your speed over ground and advance the LOP a shorter distance. That plots the running fix closer to the shore than the vessel's actual position. Assuming a following current would place the fix farther from the NAVAID than the vessel really is, which is the dangerous error Bowditch Pub. 9 Ch. 10 §1019.

Your last six three-bearing fixes have all produced a triangle of roughly the same size and shape. What does that indicate, and what do you do about it?

A constant error in observing or plotting the bearings. Apply the same correction, in the same direction, to all three bearings; if that yields a pinpoint fix, carry the correction forward to subsequent fixes and then track down the source — gyrocompass, repeater or bearing transmission system — and check the adjusted fixes against a satellite or radar position or the charted sounding Bowditch Pub. 9 Ch. 10 §1024.

You want to pass a wreck off your track without maintaining a continuous plot. Which object do you choose for the danger bearing, and what do you draw?

An object visible to the eye, shown on the chart, and bearing from the danger area in the same general direction as your course past it. Draw a line from that object tangent to the near edge of the wreck's danger area — or tangent to a danger circle scribed at a safe standoff — hatch the dangerous side, and label it DB with NMT or NLT as the geometry requires , then check the planned track does not cross it Bowditch Pub. 9 Ch. 10 §1007 ¶1.