Radar Plotting: the OAW Triangle, CPA and TCPA

Why plot by hand?

In short: ARPA plots for you, but a senior officer keeps manual plotting alive — as a backup and as a way to understand the picture.

• At management level, advise your OOWs not to give up manual plotting, and make sure they practise it regularly.
• It is your backup if ARPA fails.
• It builds situational awareness: you see how your own course and speed changes affect each encounter.
• ARPA is simply an automated O-A-W plot — the same triangle, done by the computer.

Manual plot or ARPA?

• Manual plotting is slow and less accurate. With many targets or high speeds it cannot keep up.
• That is why manual plotting is not one of the three key sources in heavy traffic — those are radar TT (ARPA), AIS and TT–AIS association.

On the bridge

• Your first manual plot can take a very long time. With practice it takes minutes — practise until the triangle is automatic.
• Keep a plotting sheet ready on the bridge.

What the plot can and cannot trust

In short: the relative plot comes straight from the radar echoes. The target’s true motion also needs your own heading and speed — feed in wrong data and the true answers are wrong.

• CPA, TCPA, BCR and BCT come from the relative plot only → they are independent of own course and speed.
• The target’s true course, speed and aspect need own course and speed through the water → GIGO (garbage in, garbage out). Wrong heading or speed input → wrong target data, by hand and by ARPA.
• GIGO applies to radar, ECDIS and AIS alike. The one part of the system that can catch bad input is the watchkeeper.
• If any input fails, ARPA still gives you at least CPA and TCPA.

Why, on the triangle

• OA is measured from the radar echoes themselves.
• WO is built from the gyro (heading) and the log (speed).
• Gyro error → WO points the wrong way → W is in the wrong place → WA is wrong.
• Log error → WO has the wrong length → W is in the wrong place → WA is wrong too.
• OA does not change, so CPA and TCPA stay right.

Why sea-stabilised for collision avoidance

• A sea-stabilised display uses gyro heading and log speed through the water.
• The target’s true vector then shows its heading through the water, which gives you its aspect — the key to which COLREG rule applies.

Plotting maths

In short: three formulas, one proportion and a scale. Work one step at a time: time → NM → cm.

Speed, distance and time

• D = S × t ÷ 60 (D in NM, S in kn, t in min). The 60 is always there: knots are per hour, the plot works in minutes.
• Speed: S = D × 60 ÷ t · Time: t = D × 60 ÷ S.
• Read the interval first. If O→A covers 12 min, own run = 2 × the 6-min distance.

The rule of sixes

• 6 min = 0.1 h, so just move the decimal point: 12.5 kn → 1.25 NM in 6 min → 2.5 NM in 12 min (double it).
• It is the old UK method from before calculators. British practice is to plot at 6-min intervals; American practice sometimes uses 15 min, depending on distance and speed.
• Some UK oral examiners still ask you to plot without a calculator — so learn it. A calculator has been allowed in the Malta assessment, but check on the day.

Proportions — for TCPA and BCT

• At constant relative speed, distance and time are proportional.
• d₁ : t₁ = d₂ : x → x = d₂ × t₁ ÷ d₁.
• Example: the target ran 3.1 NM in 12 min; 6.2 NM are left to CPA → x = 6.2 × 12 ÷ 3.1 = 24 min.
• Another: 2 : 12 = 3 : x → x = 3 × 12 ÷ 2 = 18.
• Sanity check: double the distance → double the time.
• Simple proportions are easier than formula triangles. If rearranging is the problem, make x the subject and move the terms across. Round numbers make interpolation easy.
• The same proportion gives TCPA and BCT: time is proportional to distance along the relative line.

NM to cm on the sheet

• Other sheets: measure the radius first → 1 NM = radius ÷ range.

The O-A-W triangle

In short: three points, three vectors. OA is what the radar shows, WO is what you are doing, WA is what the target is really doing.

• O = first plotted position of the target · A = last position.
• OA = relative approach (arrow O→A).
• WO = own ship’s run in the interval (“way of own”).
• WA = the target’s run in the interval (“way of another”) = its true vector.

Set up the sheet

• Write own course and speed at the top of the sheet (the exam sheet gives them) and read the interval.
• Pick the scale from the biggest distance on the sheet. Cover or cut off the part of the scale you do not use, so you cannot read it by mistake.
• Draw a small reference triangle in a corner: O · A · W, with every arrow direction marked. Add O1 = alteration of course and O2 = reduction of speed (O3 if you like). It is your reference whatever the plot looks like.
• Do not draw extra lines — the plot only gets messy.

Plot it step by step

1. Plot the target at fixed intervals (usually 6 min). First position = O, last = A.
2. Relative speed = OA × 60 ÷ interval.
3. From O, lay own ship’s run in the interval along the reciprocal of own course → that point is W. WO = own vector.
4. WA = target’s true vector: target course = direction W→A · target speed = WA × 60 ÷ interval.
5. Extend O→A past own ship → relative motion line (RML). From it you get CPA, TCPA, BCR and BCT.

Why three observations

• One or two dots cannot show you whether the target is holding its track. With three you can see if the track stays consistent or changes.
• Plots not perfectly in line? Do not take the middle. Draw the line through the most dangerous one (the closest CPA). When in doubt, assume the worst case.

CPA and TCPA

In short: CPA is the shortest distance between own ship and the relative motion line; TCPA is how long the target takes to get there.

• CPA (closest point of approach) = perpendicular (90°) from own position to the RML. Mark that point X.
• The CPA is not the length of O–A.
• TCPA (time to CPA) = distance A→X × 60 ÷ relative speed. Add it to the time of A for the clock time.
• Or by proportion: (time O→A) : (distance O→A) = TCPA : (distance A→X).
• Example answers on a typical exam sheet: CPA 1.2–1.5 NM · TCPA ≈ 25 min · BCR 2.4 NM · BCT ≈ 23 min.

Worked example

• Own ship 130° / 28 kn, range scale 12 NM.
• Target at 1500: 090° / 12 NM → at 1510: 093° / 6 NM (interval 10 min).
• Own run in 10 min: 28 ÷ 6 = 4.67 NM.
• OA ≈ 6 NM in 10 min → relative speed 36 kn.
• CPA ≈ 0.6 NM · TCPA ≈ 10 min after 1510 → about 1520.
• Target’s true course and speed ≈ 216° / 25 kn.
• COLREG: the target is crossing from port, so it is the give-way vessel (COLREG Rule 15). Own ship stands on but keeps watching and is ready to act (COLREG Rule 17).

BCR and BCT

In short: where and when the target crosses ahead of you.

• B = the point where the relative motion line cuts own heading line.
• BCR (bow crossing range) = how far ahead of you the target crosses your bow (own ship to B).
• BCT (bow crossing time) = when it gets there — by proportion along the relative line, like TCPA.
• ARPA also gives bow crossing range and time, alongside range, bearing, course, speed, CPA and TCPA.
• The examiner may point at the plot and ask: “What is X? What is B? How do you calculate it?”

True vectors to relative vectors

In short: a standard paper exercise — you are shown true vectors and must draw what the relative vectors would look like.

• The task: own ship in the centre · relative motion, north-up, true vectors of 12 min shown → “If you changed the vectors to RELATIVE, draw the new vector for each target.”
• The rule: relative = target true − own true.
• The drawing: at the tip of the target’s true vector, draw own vector reversed (same length, opposite direction). Join the target echo to that point → relative vector.

Shortcuts

A practice sheet

Own ship 000°, about 10 kn (approximate values):

• Stationary echo ahead → comes straight at you.
• Target to the east, heading west → collision course.
• Target to the ESE, heading north at your speed → zero relative vector.
• Target to the SSW, heading north, slower → short, pointing astern.
• Target to the NW, heading SE → passes west of you at about 4.5 NM.
• Target to the west, heading east → passes astern at about 1.8 NM.

Constant bearing, constant range

Exam-style scenario: own ship 000° / 13 kn, relative vectors.

• Target B keeps a constant relative bearing of 045° and a constant range of 4 NM over 6 min → it is on the same course and speed as you. It is not stopped.
• Target A ahead keeps a steady bearing with decreasing range → collision course (CPA ≈ 0) → risk of collision (COLREG Rule 7) → early, substantial, seamanlike action (COLREG Rule 8).
• A target crossing on your starboard bow makes you the give-way vessel (COLREG Rule 15). Take early and substantial action (COLREG Rules 16 and 8) and do not cross ahead → alter to starboard and pass astern.
• To find the true motion of a target, build the O-A-W triangle: lay own vector W→O, then W→A is the target’s true course and speed.

Avoiding action on the plot

In short: decide the CPA you want, draw the new relative line, then read the new course or speed off the triangle.

• Typical question: “The Master wants to pass at least 2 (or 3) NM off — find the new course, or the speed reduction.”

Method

1. Draw a circle of the wanted CPA round own ship, like a radar range ring.
2. From A, draw the tangent to that circle and extend it → the new relative line.
3. Transfer it to the triangle: O1 = new course (alteration) · O2 = reduced speed on the same course.
4. Measure the new direction or length. Work out the speed by proportion.

Things to watch

• Choose the side first: starboard and port give different results. In restricted visibility check COLREG Rule 19(d).
• Course, speed, or half and half can give the same CPA. Whatever you choose, the alteration must be large enough to be apparent to the other ship, visually or by radar (COLREG Rule 8).
• Stuck on orientation? Rotate the sheet so your course points up.
• In the exam you may not need a full alteration plot, but you may be asked what happens if you reduce speed or alter course, and where the new relative line goes. Know the triangle.
• Do not learn fixed recipes. Understand what is happening, then choose what is easier to compute.

Why only your vector changes the picture

• Relative motion needs two bodies, and you can only change your own vector.
• If the other ship stops, only your motion remains → its relative vector = your vector reversed. In true motion its vector disappears.

Aspect and overtaking

In short: aspect tells you whether you are overtaking, being overtaken or crossing — and so which rule applies (COLREG Rules 13, 15, 19).

• COLREG Rule 13: overtaking = coming up from more than 22.5° abaft the beam. In doubt, assume you are overtaking (COLREG Rule 13(c)).
• At night, seeing the stern light together with a sidelight (white + green or red) puts you at the edge of the sector → you are in doubt.
• Light arcs: sidelights 112.5° each, stern light 135° (112.5 + 112.5 + 135 = 360).
• By day, use all available means: the other ship’s heading and bearing, the chart, AIS and radar. Press TRUE and compare true and relative vectors.
• ARPA can combine relative vectors with true trails, or the reverse.

Aspect by calculation

• Aspect = (bearing ± 180°) − target true course → Red or Green angle from his bow.
• Example: target bears 030° (reciprocal 210°), target course 280° → 210° − 280° = −70° → Red 70.
• More than 112.5° Red or Green → you are abaft his beam, inside his overtaking sector.
• The examiner’s first question is often “How do you know it is overtaking?” — he gives you the bearing and the course. Work it out by numbers, not by eye.

No lights at all

• The vessel may be drifting: its true vector is small and points the way of the current.

Rule 19 on the plot — restricted visibility

In short: with no visual contact you are at your most vulnerable. Ask yourself: “Where can I go to create distance?”

• First question: is a close-quarters situation developing? In restricted visibility there is no stand-on vessel — both ships must act.
• COLREG Rule 19(d) (target detected by radar alone): avoid an alteration to port for a vessel forward of the beam (except one being overtaken), and avoid an alteration towards a vessel abeam or abaft the beam.
• Options: alter to starboard, reduce speed, stop or reverse the engines (COLREG Rule 8(e)), or a mix. In any situation: slow down.
• COLREG Rule 19(e): with a target forward of the beam and close quarters possible, reduce to the minimum speed to keep steerage; take all way off if needed.
• COLREG Rule 19(b): have the engines ready for immediate manoeuvre — tell the engine room as soon as visibility drops (ships on heavy fuel may need to change over to diesel).
• COLREG Rule 19(c): apply the Section I rules with due regard to the visibility. You cannot take visual compass bearings in fog → use systematic radar observation.
• ARPA data may be used for collision avoidance in restricted visibility, as long as you verify them by other means.

Two readings of “abaft the beam”

• One reading: under Rule 19, treat any target abaft the beam as the abaft-the-beam case — not the 22.5° arc of Rule 13.
• The other: use the Rule 13 definition of overtaking (more than 22.5° abaft the beam).
• The point is debated — read COLREG Rule 19 itself and be ready to justify your answer.

Exam-style case

• Dense fog, narrow channel. A container ship 2.5 NM ahead, CPA 0.8 NM, no reply on VHF.
• ✅ Reduce speed and prepare to take avoiding action. In collision situations, the safe answer always includes reduce speed.
• ❌ “Head-on” in the option is irrelevant (it only tells you the target is forward of the beam) · maintain course and speed · increase speed and overtake to starboard (you are not overtaking).

Exam technique and typical errors

In short: the logic matters more than perfect numbers — but the classic slips cost marks.

Typical errors

• Taking the O–A length as the CPA.
• Using the wrong scale on the ruler.
• Taking the middle of a scatter of plots instead of the most dangerous one.
• Not marking arrow directions.
• While learning, do not stress about exact numbers — get the logic first.

What not to say

• Never give “I’ll call him on VHF” as your collision-avoidance answer. It is not in COLREG, and MCA guidance warns about its dangers: misunderstandings, and pulling the OOW’s attention off the lookout and the plot (MGN 324 (M+F)). In practice many ships do call, but early action makes it unnecessary.
• Do not base collision avoidance on AIS. ARPA builds the triangle from your own sensors; AIS repeats the other ship’s GPS, with possible installation errors. AIS is still useful for fast data and for the ship’s name to call.

How to prepare

• Rules-of-the-road questions often decide an oral. Being able to prove whether you are overtaking is a classic.
• Read the COLREG text itself, not only flashcards or apps — you pick up the links between rules, and an app can carry its author’s mistakes.
• Practise plotting sheets until the triangle is automatic.