CPA and TCPA Explained: The Collision-Avoidance Math Behind Vessel Traffic

If you have stood a bridge watch with ARPA radar, you have read CPA and TCPA off the screen for every tracked target. They are the two numbers that answer the only question that matters in a developing meeting situation: are we going to hit, and if so, when? This guide explains what each one means, how the underlying vector math works, and how modern AIS-based dispatch and traffic systems compute them for every vessel pair at once.

The two definitions

CPA — Closest Point of Approach is the minimum distance that will exist between two vessels if both hold their current course and speed. It is measured in nautical miles. A CPA of 0.2 nm means that, on present headings, the two ships will at their nearest pass come within two-tenths of a mile of each other.

TCPA — Time to Closest Point of Approach is how long until that closest moment arrives, measured in minutes. A positive TCPA means the closest approach is still in the future. A negative TCPA means the vessels are already past their closest point and are now opening — the range is increasing — so there is nothing to act on.

Together they frame the risk. A small CPA with a large TCPA is a situation to watch. A small CPA with a small TCPA is a situation to act on now.

The vector picture

The math is cleaner than it looks. Place both vessels on a flat plane (over the short ranges that matter for collision avoidance — a few miles — the curvature of the earth is negligible, so a local flat projection is accurate to a fraction of a percent).

Each vessel has a position and a velocity. The velocity is built from speed over ground and course over ground:

velocity_east  = SOG × sin(COG)
velocity_north = SOG × cos(COG)

where COG is measured in degrees clockwise from true north. Now consider the relative position and relative velocity — vessel B as seen from vessel A:

relative_position = position_B − position_A
relative_velocity = velocity_B − velocity_A

From A's frame of reference, A sits still and B drifts along the relative-velocity vector. The problem reduces to: what is the closest that a point (B) moving in a straight line passes to the origin (A)? That is a standard minimization.

The formula

Let dp be the relative position vector and dv the relative velocity vector. The time of closest approach is:

TCPA = −(dp · dv) / (dv · dv)

where · is the dot product. If dv · dv is essentially zero, the two vessels have the same velocity, their separation never changes, and CPA is simply their current distance. Otherwise, plug TCPA back in to find where B will be at that moment:

position_at_cpa = dp + dv × TCPA
CPA = magnitude(position_at_cpa)

If TCPA comes out negative, the closest approach is in the past — discard it, the vessels are separating.

That is the entire calculation. It is what runs inside every ARPA unit, and it is what an AIS-based traffic system runs across every pair of moving vessels in view.

Why AIS makes this scalable

A radar tracks the targets its antenna can see and that the operator has acquired. AIS broadcasts position, SOG, and COG directly from each vessel's transponder, so a system receiving the AIS feed for an area already has the exact inputs the CPA/TCPA formula needs — no target acquisition required.

The practical consequence: a dispatch system covering a whole pilotage district can compute CPA and TCPA for every pair of moving vessels in the zone, continuously, and raise an alert whenever a pair crosses a risk threshold. A common threshold for an alert is CPA under 0.5 nm with a positive TCPA inside 30 minutes, with a tighter critical band — say CPA under 0.25 nm — escalating the warning.

Thresholds and false alarms

The art is in the thresholds. Set CPA too generous and a busy harbor generates constant alerts that get ignored; set it too tight and you lose the early warning that makes the alert useful. Real systems tune three knobs:

• CPA distance — how close counts as risk (often 0.5 nm warning, 0.25 nm critical)
• TCPA window — how soon counts as relevant (often 15–30 minutes; beyond that, courses will likely change anyway)
• Minimum speed — anchored and drifting vessels are filtered out, because two stationary ships do not have a meeting situation

A dispatch-level alert is an awareness tool, not a substitute for the vessels' own watchkeeping or for a Vessel Traffic Service. It tells the shore-side coordinator that two ships in the district are converging — useful context when assigning pilots and pilot boats.

Where this shows up in dispatch software

In a pilots-association dispatch board, CPA/TCPA runs in the background across the coverage zone. When two vessels are set to pass within the threshold, the board surfaces a collision-risk banner naming both vessels, the predicted CPA, and the minutes until it occurs. It is the same geometry a mate reads off ARPA, applied association-wide so the dispatcher sees a converging pair forming before either ship calls it in.

We describe the full dispatch picture — area AIS, ETA to the boarding point, the transit log, and these collision alerts — in Pilot Dispatch Software: What Modern Pilots Associations Actually Need, and you can see the live board on the Binnacle Passage page.