NATARAJACoastal chartwork — bearings, fixes & distance off
Learning outcomes & permitted supports
- Plot a fix by cross bearings and read the cocked hat as a MEASUREMENT of error — including what a common compass error does to it
- Work a running fix: transfer the first position line along the course and distance run, and say what the fix inherits from the log
- Judge a fix by its angle of cut — error ÷ sin(cut) — and refuse a fine one
- Take a distance off without a fix: rising/dipping range 2.095(√H + √h) and distance by vertical sextant angle d = h/tan θ
- Use the two-bearing problem, and show that doubling the angle on the bow makes the distance off equal the run
Supports in this lab: Chart, compass, sextant and a calculator. Position lines are straight on a local plane in nautical miles — the same approximation the chart makes, stated: at 51°N the residue is about four metres, well inside plotting accuracy. Golden 961/961.
Glossary: position line · cocked hat · running fix · rising & dipping · vertical sextant angle · dead reckoning
Three bearings, one ship — and the cocked hat
A bearing of a charted object is a POSITION LINE: you are somewhere on the reciprocal drawn from that object. Two lines cross at a point. Three almost never do — they leave a cocked hat, and the hat is not a failure of geometry but a measurement of your error. Slide a deliberate mistake into the mast's bearing and watch the triangle open up.
No error: the three lines pass through one point, and the fix is the point itself. The plotted fix here is the least-squares point: it splits the error between the lines instead of pretending one of them is perfect.
One object, two times: the running fix
When only one light is in sight you take its bearing, run your course and distance, and take another — transferring the FIRST position line along the run (equivalently: moving the object). The two lines cross at where you are NOW. The fix inherits every error in the run, so the log and the steering matter as much as the compass; and if the two lines meet at a fine angle the fix is weak no matter how good the bearings were.
Angle of cut now 21.7°. Under 30° the position lines slide along each other — a small bearing error becomes a large position error. Wait for the bearing to change, or find a second object.
Distance off — without any fix at all
Two lines you can take alone. Rising and dipping: the instant a light lifts over the horizon is a distance, because you are seeing the sum of two horizons — the light's and your own eye's. Vertical sextant angle: a charted height subtends an angle, and d = h / tan θ. Both give a circle of position around the object, and either one crossed with a bearing is a full fix.
| Your horizon | 7.84 nm — 2.095√14.0 |
|---|---|
| Nataraja Head lighthouse rises/dips at | 24.33 nm — 2.095(√62 + √14.0) |
| Distance off by vertical angle | 1.60 nm — 62 m ÷ tan 1.20° |
| Angle it would subtend at 3 nm | 0.639° — the inverse, for planning a wheel-over |
| Doubling the angle on the bow (30° → 60°) | distance off = the run = 4.00 nm; abeam 3.46 nm |
The doubling trick is not a rule to memorise, it is a triangle: when the second relative bearing is twice the first, the triangle is isosceles and the distance off equals the distance run. The golden suite proves it to machine precision at every angle.
| Fix by cross bearings | 12.885°N 74.845°E |
|---|---|
| Nataraja Head lighthouse | 310.2° · 6.51 nm · residual 0.000 nm |
| The radio mast | 280.6° · 6.54 nm · residual 0.000 nm |
| Church spire, Kadal | 246.2° · 5.95 nm · residual 0.000 nm |
| Cocked hat | none — the three lines are concurrent |