Three-Bearing Fix and the Cocked Hat

What the rule requires

Simultaneity is the whole basis of the method. Bowditch Ch. 7 §705 defines the three-bearing fix as three simultaneous lines of position to well-separated charted objects, and a line of position is only ever a line the vessel lies on at the moment of that one observation Bowditch Ch. 7 §701. Take the three bearings minutes apart on a moving vessel and you no longer have three lines valid at one time; you have three lines valid at three positions, and their intersection is not a fix of anything.

The three lines almost never meet at a point. The small triangle they enclose is the cocked hat, and its size is your quality check on the work: a small cocked hat indicates a good fix, taken at its centre, while a large one signals bearing or plotting error .

Reading the triangle

Two decisions come out of the triangle, and the exam tests the second one far more often than the first.

Where the cocked hat is small and nothing charted lies against it, the fix is the centre of the triangle. Where the triangle lies near a hazard, the corner nearest the danger is assumed, for safety. You do not average your way onto a shoal.

A large cocked hat is not a large position. It is a fault report. It tells you that one of the bearings is wrong, or one of the lines is plotted wrong, and neither the centre nor any corner of it is trustworthy until you find which. The usual culprit is an object that was not positively identified before the bearing was taken Bowditch Ch. 7 §707.

Choosing the objects and the order

The strongest visual fixes share three properties: their lines of position cross near 90 degrees, the objects are positively identified on the chart, and the bearings are observed in quick succession with the beam-most object taken last. Between fixes, danger bearings and clearing lines keep the vessel off charted hazards.

The order matters because of how fast a bearing changes. An object near the beam swings its bearing quickly as the vessel runs; an object near the bow barely moves. Take the slow ones first and the beam-most one last and all three bearings are, near enough, of the same instant. Reverse the order and the beam bearing is the stalest of the three by the time it reaches the chart.

The 90-degree preference is about how errors land. Two lines cutting at a fine angle produce a long thin sliver of uncertainty, and one degree of bearing error slides your position a long way along it. With three objects you cannot have every pair at right angles, so the working standard is well-separated objects with no pair anywhere near reciprocal.

Your 1015 three-bearing fix produces a cocked hat about 0.15 nm on a side, and a charted 3-fathom shoal lies just outside the western side of the triangle. Where do you take the position?

At the corner of the triangle nearest the shoal — the westernmost corner. The triangle is small, which would normally put the fix at its centre, but proximity to a hazard overrides that: the corner nearest the danger is assumed for safety . Taking the centre would place you 0.07 nm further from the shoal than you can prove you are.

Telling it apart

The criterion is whether the lines of position are simultaneous, and every method below sits on one side of that line or the other.

• Fix — the intersection of two or more LOPs taken simultaneously Bowditch Ch. 7 §701. No advance, no assumption about course and speed.
• Three-bearing fix — simultaneous, three objects, and the only one of these methods that gives you a built-in error check in the form of the cocked hat Bowditch Ch. 7 §705. Two crossing LOPs will always meet somewhere and tell you nothing about whether they are right.
• Running fix — used when LOPs cannot be taken simultaneously. The first LOP is advanced along the DR track to the time of the second observation, and the intersection is the running fix, less accurate than a fix from simultaneous LOPs Bowditch Ch. 7 §702. It is less reliable specifically because it assumes course and speed were held exactly between the two observations Bowditch Ch. 7 §708. Most often misfiled: three bearings taken to one object over ten minutes and called a three-bearing fix.
• Doubling the angle — two relative bearings to a single object, where the distance run between them equals the distance to the object at the second bearing Bowditch Ch. 7 §703. It yields a range, not a simultaneous fix, and the bearings must change substantially for it to work at all.

The practical consequence: a question that hands you three objects visible at once and three bearings at one time is a three-bearing fix question, and the presence of a course and speed in the problem does not make it a running fix. Course and speed only enter when a line has to be advanced.

Working a question

You are on course 090° at 8 knots along a charted coast. Three objects are in sight: a water tank bearing 095°, Gull Rock Light bearing 032°, and a church spire bearing 158°. The time is 0912.

1. Identify each object on the chart before touching the compass. All three must be charted objects you can positively name Bowditch Ch. 7 §707. An unidentified stack in a row of three stacks is the most common source of a large cocked hat later.
2. Check the spread. Adjacent bearings differ by 63° and 63°, and the outer pair by 126°, so the closest cut is 54°. No pair is anywhere near reciprocal, and the objects are well separated as Bowditch Ch. 7 §705 requires.
3. Fix the observation order from the relative bearings. On heading 090°, the tank at 095° is nearly dead ahead, Gull Rock is 58° on the port bow, and the spire is 68° on the starboard bow. The spire is beam-most, so it goes last; the tank goes first.
4. Take all three in quick succession at 0912 and note the single time. The whole point of the sequencing in step 3 is that one time stamp is honest.
5. Plot the three LOPs and label the intersection with the time. Three lines, one triangle.
6. Measure the triangle. Say its sides run about 0.2 nm. That is a small cocked hat, so the bearings and the plot hang together.
7. Look at what is charted around it. With clear water on all sides, the fix is the centre of the triangle. If a 2-fathom shoal lies 0.3 nm to the south, the fix is instead the southern corner.
8. Cover the run to the next fix. Lay a danger bearing on one of the same objects and keep the observed bearing on the safe side of it Bowditch Ch. 7 §706.

Had step 6 produced a triangle 1.5 nm on a side, nothing in steps 7 or 8 would be worth doing. A large cocked hat sends you back to step 1.

Same three objects, but you take the spire first at 0912, Gull Rock at 0914, and the tank at 0916. What have you plotted?

Not a three-bearing fix. The lines are not simultaneous, and at 8 knots the vessel has run about 0.53 nm between the first and last bearing. You have also taken the beam-most object — the spire, whose bearing changes fastest — first, so its line is the most out of date of the three. To get a position out of these observations the earlier lines must be advanced along the DR track to the time of the last, which makes it a running fix, less accurate than a fix from simultaneous LOPs Bowditch Ch. 7 §702.

Where candidates lose the point

Taking the centre of the triangle when a hazard lies against one side. The centre is the answer most candidates carry into the room, and it is right only in open water. Where the triangle lies near a hazard the corner nearest the danger is the assumed position Bowditch Ch. 7 §705. Read the chart extract before answering, not just the triangle.

Reading a large cocked hat as a bigger circle of uncertainty. The attractive distractor is some version of "the fix is less precise, so allow more margin." A large triangle signals bearing or plotting error, which means one of the inputs is wrong rather than merely coarse. The correct action is to re-identify the objects and re-shoot, and the wrong action is to accept the position with a caveat.

Taking the beam bearing first. Candidates work left to right along the horizon, or shoot whichever object they spotted first. The beam-most object is taken last because its bearing changes fastest Bowditch Ch. 7 §707.

Choosing the three objects that are easiest to see instead of the three that are best spread. Two lights close together on the same headland give a fine-angle cut, and the resulting near-parallel lines make an enormous triangle that looks like bearing error when it is really object selection. The standard is LOPs crossing near 90 degrees.

Calling a three-bearing fix on a single object. Three bearings on one lighthouse over twelve minutes is a running-fix problem, whatever the question calls it. The advance along the DR track is required, and the result carries the assumption that course and speed were held exactly Bowditch Ch. 7 §708.

Treating the fix as sufficient for the whole leg. A fix is good at its own time and nowhere else. Danger bearings and clearing lines supplement the fix to keep the vessel off charted hazards between fixes .

Check yourself

You plot simultaneous bearings on three charted objects and the resulting triangle measures roughly 1.2 nm on a side. What does this tell you, and what do you do?

It tells you there is bearing or plotting error in the work Bowditch Ch. 7 §705. A large cocked hat is a fault indication, not a position with wide limits. Re-identify each object on the chart, confirm each is the object you actually sighted Bowditch Ch. 7 §707, and take the round of bearings again. Do not work an EP out of the centre of that triangle.

Steering 145° at 10 knots, you can see a lighthouse 15° on the port bow, a radio tower nearly dead ahead, and a charted cupola 80° on the starboard bow. In what order do you shoot them?

Tower first, lighthouse second, cupola last. The cupola at 80° on the starboard bow is beam-most, and the beam-most object is taken last because its bearing changes fastest; all three go in quick succession .

Only one charted object is visible — a lighthouse on an otherwise featureless coast. Can you obtain a fix, and what does the method cost you?

You can obtain a running fix, not a fix. Take a bearing, run a timed distance on a steady course, take a second bearing, then advance the first LOP along the DR track by the distance run and lay it across the second bearing Bowditch Ch. 7 §708. It is less accurate than a fix from simultaneous LOPs because it assumes the course and speed were held exactly through the run Bowditch Ch. 7 §702.

Your 1430 cocked hat is small, and the nearest charted danger is a wreck 0.1 nm off the northeastern corner. Where is your position for the purpose of the next course you lay?

At the northeastern corner. Proximity to the wreck governs over the small-triangle case, and the corner nearest the danger is the assumed position . Lay the next leg from that corner, and back it up with a danger bearing on a charted object clear of the wreck Bowditch Ch. 7 §706.

Two of your three objects bear 048° and 231°. What is wrong with this fix, and why does the cocked hat get large?

Those two bearings are within 3° of reciprocal, so the two LOPs lie almost along the same line and give no useful cut. The strongest fixes cross their LOPs near 90 degrees , and the three objects must be well separated . A fraction of a degree of error on a near-reciprocal pair moves the intersection a long way, which inflates the triangle even when the bearings themselves are carefully taken.

You want to hold clear of a shoal patch during the run between two fixes, using one lighthouse. How do you set it up?

Draw a line from the lighthouse tangent to the edge of the shoal, or tangent to a danger circle scribed at a safe distance around it, and label it NMT or NLT that bearing. As long as the lighthouse's observed bearing stays on the safe side of that danger bearing, the vessel is in safe water .

A question gives you three bearings taken at 0800, 0806 and 0812, plus your course and speed. Is the cocked-hat rule the one that governs?

No. The bearings are not simultaneous, so there is no three-bearing fix and no cocked hat to interpret. This is a running fix: advance the earlier LOPs along the DR track to the time of the last observation before crossing them . The presence of course and speed in the data is the signal that an advance is required.