CPA and TCPA: what do they predict, and what do they leave out?

Begin with relative motion

In the simplest model, both vessels maintain constant ground-velocity vectors. Subtract your vessel's velocity from the target's velocity. This gives relative velocity: how the target moves when viewed from your moving reference point.

Use one coordinate frame and consistent units. If position is in nautical miles and velocity is in knots, time is in hours. A heading is an orientation; it is not necessarily the direction of motion over the ground. Current and other effects can make those different.

Let r be the target's relative position and v its relative velocity. Future relative position is r + v × time. The constant-velocity TCPA on the unbounded time line is −(r·v)/(v·v), provided relative speed is nonzero. CPA is the length of the relative-position vector at that time. This is a geometric calculation, not an assumption that real vessels cannot maneuver.

A worked example

The following is an invented point-motion example, not a real encounter or a maneuver recommendation.

Place the target 4 nautical miles east and 2 nautical miles north of your vessel. Let its relative velocity be 8 knots westward, with no north–south component.

After 30 minutes, it has moved 4 nautical miles west relative to your vessel. It is then directly north, still 2 nautical miles away. In this unchanged-motion model:

• TCPA = 0.5 hours = 30 minutes.
• CPA = 2 nautical miles.

The calculation does not say that two nautical miles is always a safe passing distance. The USCG notes that the navigation rules do not define one universal safe speed or passing-distance number. USCG navigation-rules FAQs

The horizon changes the question

If the display covers only the next five minutes, the predicted relative position at its endpoint is approximately 3.33 nautical miles east and 2 north. Its separation is approximately 3.89 nautical miles, and it is still closing.

Thus “minimum separation in the next five minutes” is about 3.89 nautical miles, while the longer constant-motion CPA remains 2 nautical miles at 30 minutes. Both values can be mathematically correct; they answer different questions. Label the prediction horizon rather than silently substituting one for the other.

A negative TCPA places the closest point in the past under the assumed motion. If asking only about the future, the minimum over the future interval may be the current separation. With zero relative speed, this formula has no unique TCPA; dividing by zero or displaying an arbitrary countdown is misleading.

What the simple calculation leaves out

• Maneuvers: a speed or course change alters the future path and requires a revised prediction.
• Response limits: a commanded change is not necessarily achieved instantly.
• Geometry: separation between reference points is not clearance between two hulls, a tow or a shoreline.
• Uncertainty: noisy, delayed or inconsistent observations can materially change the prediction.
• Context: visibility, traffic, waterway limits, draft and available maneuvering space matter.

The International COLREG framework requires appropriate observation and assessment, and warns against assumptions from scanty information. CPA/TCPA should therefore be interpreted alongside the relevant information and rules, not as an automatic stand-alone instruction. The linked USCG text marks differences between International and U.S. Inland rules; those variants must not be silently combined. Sources: IMO overview, USCG rules, especially Rules 5–8

Solve the bounded question explicitly

The unbounded TCPA formula locates the minimum over the entire mathematical time line. For a future interval from zero to a stated horizon H, use that time only if it lies within the interval. If it is negative, evaluate the current separation; if it exceeds H, evaluate separation at H. This is a minimization of the squared distance of r + vt on a closed interval, assuming constant nonzero relative velocity. It is not an operational maneuver rule.

In the existing example, H = 5/60 hour and the unbounded result is 0.5 hour. The bounded minimizing time is therefore five minutes. Naming it “minimum separation within five minutes” preserves its meaning. Calling it simply CPA without stating the horizon can hide the still-approaching situation. A report should store both the unbounded geometric result and the bounded result when both answer useful questions, together with the calculation timestamp.

Small relative speed changes the numerical problem

When relative velocity is exactly zero, separation remains constant in this model and every time has the same distance. There is no unique TCPA. When relative velocity is merely small, the denominator v·v is small and a modest velocity error can cause a large change in the predicted time. A distant TCPA with many decimal places is therefore not automatically a precise forecast.

For an original illustration, let relative position be 1 nautical mile east and relative velocity be 0.1 knot west. The mathematical closest approach lies ten hours ahead. If the assumed westward speed is instead 0.2 knot, it lies five hours ahead. The geometry is simple but the relevant motion assumption has to persist for an exceptionally long interval. Neither result supports a ten-hour operational prediction without considering future maneuvers, sensor quality and environment.

Use perturbations to reveal sensitivity

Return to the initial target at r = (4, 2) nautical miles and v = (−8, 0) knots. Now add an invented relative northward component of +0.4 knot, leaving everything else unchanged. TCPA becomes 31.2/64.16 hour, about 29.18 minutes; CPA becomes approximately 2.197 nautical miles. With a −0.4 knot northward component, TCPA is 32.8/64.16 hour, about 30.67 minutes, and CPA is approximately 1.798 nautical miles.

These perturbations are selected teaching cases, not confidence limits and not the probability distribution of a radar track. They show that a small transverse velocity component can materially change the predicted miss distance over the forecast interval. To convert input uncertainty into a justified probability statement would require an appropriate observation-error model, correlations and a model of subsequent motion. Merely computing several alternatives does not supply those ingredients.

Time alignment is part of relative geometry

Two target positions cannot safely be subtracted as though simultaneous if one is substantially older. A consistent calculation either uses synchronized observations or explicitly propagates them to a common time under stated assumptions. Heading, course over ground and speed through water must also retain their definitions. Combining one ship’s ground velocity with another ship’s water-relative velocity silently changes the relative-motion model.

The MCA’s guidance on electronic navigational aids discusses equipment limitations and cross-checking. The original analytical examples here are not quoted equipment accuracy specifications. A useful teaching record distinguishes sensor time, display time, model time and prediction time so that apparently conflicting results can be compared on the same basis.

Preserve the distinction between points and hulls

The calculation follows defined reference points. A vessel’s bow, stern, beam, tow or other occupied extent can lie away from that reference. During a turn, orientation changes the occupied geometry even if the plotted reference-point path looks clear. Turning a point CPA into physical clearance therefore requires geometry and time-consistent orientation, not simply an unexplained constant subtraction.

The USCG navigation-rules FAQ explains that the rules do not supply a universal safe passing-distance number. The analytical extension is to report what distance was actually calculated and leave its operational acceptability to the full situation and applicable procedures. This article supplies no alarm threshold, avoidance command or claim that a favorable CPA makes an encounter safe.

How to read a simulation plot responsibly

Look for the reference points, coordinate frame, units, timestamp, prediction model and time horizon. Check whether the plotted line is a prediction made earlier or the trajectory actually realized later. Ask whether maneuvering dynamics and vessel boundaries are included, and how missing or uncertain observations are represented.

A collision-free interval establishes only what occurred in that modeled interval. It does not prove an unfinished encounter will remain clear, that all possible target actions were considered, or that a navigation system is certified. A useful plot makes these limits visible rather than hiding them behind one reassuring number.

Sources and related reading

• IMO plotting-aid performance text, hosted by USCG
• USCG: Navigation-rules FAQs
• IMO: Preventing collisions at sea
• USCG: International and U.S. Inland navigation rules

Related library topics: heading versus course over ground; maneuvering response; prediction horizons. This is an educational explanation, not advice for handling a live encounter.