CPA and TCPA Calculator (Closest Point of Approach Between Two Vessels)
How close another vessel will pass and how many minutes until then, from its bearing, range, course and speed and your own — the numbers a radar or AIS shows.
Results
What this tool does
Collision avoidance works on relative motion: subtract your velocity from the other vessel's, and its track relative to you is a straight line. The closest point of approach (CPA) is the shortest distance from you to that line, and the time to CPA (TCPA) is how long until it is reached; a negative TCPA means the vessels are already moving apart. A small CPA with a positive TCPA means a close encounter if nobody changes course, whatever the bearing looks like. Radar and AIS compute this continuously, but knowing the sum helps you judge the display and the rules of the road.
Formula
relative position P, relative velocity V = V_target − V_own · TCPA = −(P·V) ÷ |V|² · CPA = |P + V × TCPA|
Variables
| Symbol | Meaning | Unit |
|---|---|---|
br | Bearing of the other vessel from you (degrees true) | ° |
rg | Distance to the other vessel | NM |
oc | Your course (degrees true) | ° |
os | Your speed | kn |
tc | Course of the other vessel (degrees true) | ° |
ts | Speed of the other vessel | kn |
lm | Closest distance you consider safe | NM |
CP | Closest point of approach (CPA) | NM |
TC | Time to CPA (negative = already passed) | min |
RS | Closing speed (relative speed) | kn |
RK | Is the CPA below your limit? | — |
Worked example
- Bearing of the other vessel from you (degrees true)45 °
- Distance to the other vessel6 NM
- Your course (degrees true)0 °
- Your speed12 kn
- Course of the other vessel (degrees true)270 °
- Speed of the other vessel10 kn
- Closest distance you consider safe1 NM
- Closest point of approach (CPA)0.54 NM
- Time to CPA (negative = already passed)23.0 min
- Closing speed (relative speed)15.6 kn
- Is the CPA below your limit?Yes — risk of a close encounter
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- For work that must comply with a standard or be signed off, check the result against the applicable code and have it reviewed by a qualified engineer.