Passing distance and CPA from a chart plot

What the rule requires

Plot the contact's successive relative positions and draw a straight line through them: that line is the relative-motion line, and the point where it passes closest to own vessel at the centre of the plot is the closest point of approach — the predicted passing distance. The time to run from the present position to that point is the TCPA. Bowditch Ch. 13 §1304

Two quantities, one construction. Nothing about own vessel's course or speed enters either answer; the relative plot already contains own motion, because own vessel is the fixed centre and everything on the plot moves with respect to it.

The trigger that decides the largest number of exam questions is the constant bearing. A contact whose compass bearing holds while its range decreases — the long-established shorthand is CBDR, constant bearing, decreasing range — will pass at a CPA of zero, which is a collision, and the situation calls for early and positive action. Note which bearing the test is stated on: compass, not relative. The device is sound but it is silent on that distinction, so hold the two words together as compass bearing steady.

You observe a contact bearing 145°T at 4.0 miles, and eight minutes later it bears 145°T at 2.6 miles. What is the predicted passing distance?

Zero. The compass bearing has not changed while the range has closed, which is CBDR, and the relative-motion line therefore runs through the centre of the plot. The plotted CPA is a collision, and it requires early and positive action rather than continued observation.

Getting the observations onto the plot

Every observation is a range and a bearing, and range and bearing to a target give a direct fix. Bowditch Ch. 13 §1301 That is what makes the contact plottable on a chart at all: each observation is a charted position, and successive positions are two charted points.

Once they are down, the construction is the ordinary chart-work you already own. Direction of the relative-motion line comes from walking parallel rulers from the line to the nearest compass rose and reading where the ruler crosses the outer, true ring. Length comes from dividers laid against the latitude scale at your working latitude, where one minute of latitude is one nautical mile. Bowditch Ch. 4 §406

Choosing the scale before you plot anything

The principal range scales in piloting are 0.5, 1.5, 3 and 6 nautical miles, and short ranges give the best discrimination for collision avoidance while long ranges give the best landfall detection. Bowditch Ch. 13 §1302 A CPA problem is a collision-avoidance problem, so a contact inside three miles is worked on the 3 or 1.5 mile scale. The 6 mile scale is the landfall tool.

What ARPA does and does not settle

ARPA computes target tracks automatically and puts predicted CPA and TCPA on the display. It does not replace bridge watchkeeping; it supplements the visual look-out. Bowditch Ch. 13 §1303 An exam option that has you relying on the readout in place of a look-out is wrong on that sentence alone.

Telling it apart

The criterion is what the plotted positions are referenced to. Get that wrong and the line you draw answers a question nobody asked.

Relative-motion line — the positions are referenced to own vessel, held stationary at the centre of the plot; the line is drawn through successive relative positions of the contact, and it yields CPA and TCPA. Bowditch Ch. 13 §1304 Most often misfiled as the contact's own course over the ground, which it is not.
True course line — the positions are referenced to the chart; the line joins two charted points, its direction is read at the outer ring of the compass rose, and it is labelled C above and S below. Bowditch Ch. 4 §406 Most often misfiled as something to label with the relative speed you measured off the plot.

CPA against TCPA

Same construction, different measurement, and candidates answer one when the question asked for the other.

CPA — a distance, in miles. Measured square across, from the plot centre to the relative-motion line.
TCPA — a time. Measured along the relative-motion line, from the present position to the foot of that perpendicular.

Working a question

Own vessel is making 7 knots on 015°T in the approaches, visibility three miles and closing. You acquire a single contact and take three ranges and bearings at three-minute intervals:

1000 — 041°T, 2.1 miles
1003 — 035°T, 1.65 miles
1006 — 025°T, 1.2 miles

Set the scale first. The contact is inside 2.1 miles and closing, so this is worked on the 3 mile scale, where discrimination is best for collision avoidance. Bowditch Ch. 13 §1302 Putting it on 6 miles halves the plotting accuracy of every position for no gain.
Confirm the bearings are compass bearings. They are given as true here. If you had lifted them off a head-up display as relative bearings, they must be converted before anything is plotted, because the steady-bearing test that decides risk is stated on the compass bearing. Bowditch Ch. 13 §1304
Plot the three positions from the centre, each as a range and bearing from own vessel, which is a direct fix of the contact. Bowditch Ch. 13 §1301 Label them with their times; an unlabelled plot cannot be worked, because you will not know which end is the past.
Draw the line through them and extend it well past the centre. Here the three plots lie on one straight line, and the direction from 1000 through 1006, taken to the compass rose with the parallel rulers, is 240°T. The third observation is cheap insurance: the relative-motion line is the line through successive relative positions, so a plot that will not lie on the same straight line as the other two leaves you with no line to measure from and you re-plot from the latest positions.
Read the relative movement. Dividers against the latitude scale give 0.5 mile between consecutive plots, where one minute of latitude is one mile. Bowditch Ch. 4 §406 Half a mile in three minutes is ten minutes of arc in a tenth of an hour, so the contact is closing along the relative-motion line at 10 knots.
Drop the perpendicular from the centre of the plot to the line, and measure it. It comes off the latitude scale at 0.7 mile. That is the CPA, the predicted passing distance, and it lies on the port bow: the contact draws forward from your starboard bow and crosses ahead.
Measure along the line from the 1006 plot to the foot of that perpendicular. It is 1.0 mile. At a relative speed of 10 knots that is six minutes, so TCPA is six minutes from the last observation — CPA 0.7 mile at 1012.

The decision falls out of step 6 and step 7 together, not from either alone. Seven tenths of a mile is a small passing distance and six minutes is a short fuse, and the plot has told you that in the time it took to draw one line.

On the same plot, what would you have answered if you had taken 1.2 miles as the passing distance?

You would have reported the range at the last observation rather than the CPA. The relative-motion line goes on closing past 1006 until it reaches the foot of the perpendicular from the centre; the passing distance is that perpendicular, 0.7 mile, and it lies ahead of your last plot in time.

Where candidates lose the point

Reads the smallest plotted range as the passing distance. It is the most attractive wrong answer on the whole topic, because that number is sitting in the question already and needs no construction. The CPA is the closest approach of the relative-motion line to own vessel at the plot centre, and unless the last observation happens to fall exactly at the foot of the perpendicular, it is a smaller number than any range you observed.

Measures the perpendicular from the last plotted position instead of from the centre. Own vessel is the centre of the plot. A perpendicular dropped from the contact's position measures nothing.

Extends the relative-motion line the wrong way. Direction of relative movement runs from the earlier plot to the later one, which is why every position gets a time against it. Extend the line backwards and you will produce a tidy CPA that was passed before you started plotting.

Answers "no risk" because the bearing on the display is opening. A head-up display shows relative bearing, and relative bearing changes when own vessel's head changes. The constancy test in the rule is on the compass bearing.

Treats CBDR as a close-quarters situation to keep watching. A steady compass bearing with decreasing range gives a CPA of zero, which is a collision, and the required response is early and positive action. Options offering continued observation, a call on VHF or a report to the master are all wrong for the same reason.

Accepts the ARPA readout as the look-out. ARPA displays predicted CPA and TCPA, and it supplements the visual look-out without replacing bridge watchkeeping.

Measures distance on the wrong scale. Distance comes off the latitude scale at the working latitude. A minute of longitude is not a mile except on the equator, and a candidate who spans the longitude scale will get a plausible-looking CPA that is wrong by the cosine of the latitude.

Check yourself

You plot a contact at 1200 bearing 210°T at 6.0 miles and at 1212 bearing 210°T at 4.4 miles. What do you report and what do you do?

CBDR: the compass bearing is constant while the range decreases, so the relative-motion line passes through the centre of the plot and the predicted CPA is zero. That is a collision, and it calls for early and positive action. Bowditch Ch. 13 §1304

A contact is acquired at 2.4 miles and you intend to plot it for CPA. Which of the principal range scales do you select?

The 3 mile scale. Short ranges give the best discrimination, which is what collision avoidance needs; the 6 mile scale is for landfall detection. Bowditch Ch. 13 §1302

Your relative plot gives three positions in a straight line running 265°T, and the perpendicular from the plot centre to that line measures 1.4 miles. Own vessel is making 9 knots. What is the passing distance?

1.4 miles. The perpendicular from the centre to the relative-motion line is the CPA, and own vessel's speed does not enter it — own motion is already accounted for by holding own vessel at the centre of the plot. Speed is needed for the TCPA, and then it is the relative speed measured off the plot, not the 9 knots.

Successive plots are 0.4 mile apart at four-minute intervals, and the foot of the perpendicular lies 1.2 miles along the line beyond your last plot. What is the TCPA?

Twelve minutes. Relative movement is 0.4 mile per four minutes, so 1.2 miles takes three intervals. Convert it that way rather than by chasing a relative speed in knots, and there is nothing to get wrong.

You measure the line joining two plotted positions of a contact with dividers, span the same dividers against the longitude scale in the chart border, and read 1.1 miles. Where is the error?

Distance is measured against the latitude scale at the working latitude, where one minute of latitude is one nautical mile. Bowditch Ch. 4 §406 A minute of longitude is shorter than a mile at every latitude away from the equator, so the reading is too large.

ARPA is tracking the contact and displaying CPA 0.9 mile, TCPA 7 minutes. Are you relieved of any watchkeeping duty?

No. ARPA computes and displays those predictions, and it supplements the visual look-out rather than replacing bridge watchkeeping. Bowditch Ch. 13 §1303

Your first and third plots fall neatly on a line, but the second sits well off it. What do you do before answering the CPA?

Nothing can be measured yet. The relative-motion line is the line through successive relative positions, and three positions that will not lie on one straight line do not define it. Take fresh observations and re-plot from the latest positions rather than drawing a line through two of the three and discarding the inconvenient one.